near-field mass transfer in geologic disposal systems: a review

8/17/2019 Near-Field Mass Transfer in Geologic Disposal Systems: A Review

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f t M , ^ * , ^ - *

v A - - ;

-9se

LBL-23689

E 3

Lawrence Berke ley Laboratory

UNIVERSITY OF CALIFORNIA

E A R T H S C I E N C E S D I V IS I O N

Presented

at the

Materials Research Society

Symposium: Scientific Basis for Nuclear

Waste Disposal XI. Boston. MA.

November 30-December 3.

1987

Near-Field Mass Transfer in Geologic

Disposal Systems: A Review

T.H. Pigford

and

P.L. Chambre

November

1987

DISCLAIMER

This report

was

prepared

as an

account

of

work sponsored

by »n

agency

or the

United S

Govcrnmenl. Neither

the

United States Government

nor any

agency thereof,

nor any of

employees, makes

any

warranty, express

or

implied,

or

assumes

any

legal liability

or

res

bility

for the

uccuracy, completeness,

or

usefulness

of any

information, apparatus, produ

process disclosed,

or

represents that

its use

would

not

infringe privately owned rights. R

ence herein

to any

specific commercial product, process,

or

service

by

trud t name, trade

manufacturer,

or

otherwise docs

net

necessarily constitute

or

imply

its

endorsement, re

mendation,

or

favoring

by the

United States Government

or any

agency thereof.

The

and opinions

of

authors expressed herein

do not

necessarily state

or

reflect those

o

United States Government or any agency thereof.


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L B L — 2 3 6 8 9

DE86 007416

U C B - N E - 4 I 0 6

LB L-2 3 6 8 9

N E A R - F I E L D M A S S T R A N S F E R I N G EO L O G I C D I S P O S A L S Y S T E M S : A R E V I E W

T . H . P ig fo rd an d P . L . C h a m b r e

D ep ar t m e n t o f Nu c l ea r En g i n ee r i n g an d La wren ce B erk e l ey La b o ra t o ry , Un i v e r s i t y o f C a l i fo rn i a , B erk e l ey ,

CA 94720

A B S T R A C T

A p r i m ary p u rp o s e o f p e r f o rm an c e a s s es s m en t o f g eo l o g ic r ep o s i t o r i e s for r ad i o ac t i v e w as t e i s t o p red i c t

t h r ex t en t t o wh i ch r ad i o ac t i v e s p ec i e s a re r e leas ed fro m t h e wa s t e s o l i d s an d a re t r an s p o r t ed t h ro u g h

g eo l og i c m e d i a t o t h e en v i ro n m e n t . R e l i ab l e q u a n t i t a t i v e p red i c t i o n s m u s t b e m ad e

of

ra tes of re lease of

r ad i o n u c l i d es f ro m t h e was t e i n t o t h e ro ck , t r an s p o r t t h ro u g h t h e g eo l o g i c m ed i a , cu m u l a t i v e r e l eas e t o t h e

acces s i b le en v i ro n m en t , an d m a x i m u m co n cen t r a t i o n s in g ro u n d w a t e r an d s u r face wa t e r . Here we r ev i ew

t h eo re t i ca l ap p r o ach es t o m ak i n g t h e p red i c t i o n s of n ea r -f ie l d r e l eas e fro m b u r i ed was t e s o l i d s , wh i ch p ro v i d e

the source terms for far - f ie ld re lease . The ex tent to which approaches and i s sues depend on the rock media

an d o n r eg u l a t o ry c r i t e r i a i s d i s cu s s ed .

I N T R O D U C T I O N

Des i g n e r s o f g eo l o g i c d i s p o s a l s y s t em s fo r s o l i d was t e m u s t p red i c t t h e l o n g - t e rm t i m e-d ep en d en t r a t e o f

d i s s o l u t io n o f Loxic co n t am i n an t s i n g ro u n d wa t e r , t o p ro v i d e t h e s o u rce t e r m for p red i c t i n g t h e l a t e r t r a n s p o r t

o f t h es e co n t a m i n a n t s t o t h e en v i ro n m e n t . I f a co n t a i n e r i s p ro v i d ed t o p ro t ec t t h e was t e from g ro u n d wa t e r

for -in ex ten de d per io d , i t i s nece ssary t o precis : , the t im e when the co nta ine r a l lows co nta m ina nt s to b e

re l eas ed t o g ro u n d wa t e r , an d i t m ay b e n eces s a ry t o p red i c t t h e r a t e o f s u ch r e l eas es t h ro u g h p a r t l y f a il ed

co n t a i n e r s . M as s - t r an s fe r a n a l y s i s is b e i n g u s ed t o p red i c t r a t e s o f d i s s o l u t i o n an d r e l eas e o f r ad i o ac t i v e

cons t i tuents in fu ture repos i tor ies for h igh- level rad ioact ive was te , and i t

h a s

be en app l ied t o pre dic t the l ife

of a cop per co nta in er for h igh- level rad ioa ct iv e w as te .

Th e U . S . Nu c l ea r R eg u l a t o ry C o m m i s s i o n

1

has speci f ied numerical l imi t s for the ra te of re lease of

r ad i o ac t i v e co n s t i t u e n t s i n t o ro ck s u r r o u n d i n g p ack ag es for h ig h - l ev e l was t e , i n c l u d i n g u n rep ro ce s s ed s p en t

fuel and rep roce ss ing was te , and i t ha s speci fied the requi red t im e for su bs t an t ia l ly co mp lete co nta inm en t of

t h e r ad i o ac t i v e co n s t i t u en t s b y t h e was t e co n t a i n e r . O t h e r n a t i o n s h av e n o t i m p o s ed s u ch s p ec i f i c r eq u i r e

m en ts on the was te packa ge, bu t pre dic t ion of thes e sam e featu res wi ll be needed for the overal l p re d ic t ion

o f r ep o s i t o ry p e r fo rm an ce .

Here we s u m m a r i ze t h e ev o l u t i o n an d ap p l i ca t i o n o f n ea r -f i e ld m as s - t r an s fe r an a l y s es in h i g h - l ev e l -was t e

p r o g r a m s .

H I S T O R I C A L

I ' n t i l the in t roduct ion of tnass - t ransfer analys i s for predic t ing Jong- term performance of was te packages

in a repos i tory , it had been genera l ly assum ed tha t the ra te of re lease of co nt ; jn in an ts f rom a was te so lid

would

hf

cont ro l led by the ra te of chem ical react ion of th at w as te so l id wi th groun d w ater . Sam ples of

w.'irti*- sol ids were exp ose d to s imu late d gr ou nd wa ter in la bo ra to ry lea ch stud ies , wher e the ra te of release

in to the wa ter would be cont ro l led by chemical reac t ion , by degra dat ion of th t so l id , o r by f requency of

rep lacin g the leac han l . In 1978 N'e re tn ie ks

2

sugges ted that ra te of d i sso lu t ion in a geologic envi ronment

might be limi ted in s tead by so lubi l i t i es and by ra te of d i f fus ive-conve- t ive t ra ns po r t of d i sso lved species f rom

the was te surface .

Ap pl icat io n of mass t ransfer an alys i s in the U.S . pro gra ms was s t im ula ted by obs erv at io ns of the Na-

1


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t i ona l R esea rch C ounc i l ' s Was t e I so l a t i on Sys t em S t udy ,

3

f irst m ad e in 198 1, th a t the re was l i t t l e va l idi ty

i n a s sumpt i ons t ha t t he l ong- t e rm d i s so l u t i on ra t e s i n a geo l og i c repos i t o ry cou l d be p red i c t ed f rom l abo

ra t o r y da t a on le ach ra t e s o f was t e sampl e s . Lab ora t o ry l e ach d a t a a re emp i r i c a l l y cor re l a t ed , a t a g i ven

t em pe ra t ur e , wi t h t i me of exp osu re an d t he concen t ra t i on of d i s so l ved spec i e s i n t h e l i qu i d . T h e l a t t e r

dep end s on cor re l a t i ng pa ram e t e r s such a s t he ra t i o o f s am pl e sur face t o t he v o l um e of we l l mi xed l i qu i d ,

l iquid rep lace m ent f requency, e tc . T he ac t ua l proce sses of di sso lut ion a re reac t io n of th e wa ste sol id wi th

gro un d w ater a t th e wa ste surface a nd di ffus ion and c onv ect ion of th e dissolved spec ies f rom th e surfa ce into

t he g roun d w a t e r in t he sur rou nd i ng rock . In t h i s open sys t em of t he repo s i t o ry t h e re is no we l l-def ined

vol um e of we l l -mi xed l iqu i d a s i n t h e c l osed-sys t em l abo ra t o ry ex pe r i me nt s . He re t he con cen t r a t i on in t h e

sur face li qu id depe nd s on t he mass - t rans fe r p ro pe r t i e s o f t he su r ro un di n g m ed i um .

Reco gniz in g th a t th e waste-so l id m at r ix and m ost of i t s sol id co ns t i tu en ts a re of low solub i l i ty , we fi rst

soug ht an u ppe r l i mi t on t he ne t r a t e o f d i s so l u t i on by a s sum i ng t ha t t he so l u t i on a t t he w as t e sur face i s

s a t u r a t e d w i t h t h e i n d i v i d u a l w a s t e c o n s t i t u e n t s . T h e e x a c t a n a l y t ic a l s o l u t i o n s

4

for t he t i me -dependent

mass t r ans fe r r a t e f rom a was t e so li d in con t ac t wi t h sa t u r a t e d p orou s rock showed t h a t t h e l i mi t i ng m ass -

t rans fe r r a t e s for t he bur i ed w as t e a re so s low t ha t t h e ac t ua l d i s so l u t i on ra t e is expe c t ed t o be l i mi t ed

by di ffus ive-convect ive t ransport in the pores in the rock and not by chemica l reac t ion ra te a t the waste

sur face . B y com par i n g t he se p red i c t ed d i fTus ive -convec ti ve mass tr ans fe r r a t e s wi t h l abor a t or y d i s so l u t i on

da t a ob t a i n ed und e r condi t i ons whe re i n so l id - l i qu id reac t i on ra t e i s con t ro l l i ng , i t was conc l ud ed t h a t t h e

bas i c ra t e cons t an t s fo r t he so l i d - l i qu i d reac t i on we re rap i d enough , fo r t he was t e so l i ds t hen s t ud i ed , t ha t

t he d i s so lved con cen t ra t i on i n t he sur face l i qu id woul d be ve ry c l ose t o t he sa t u r a t i o n c on cen t ra t i o n . Th i s

conc l us i on was val id fo r boros i l i c a t e g l a s s was t e , which had been ex t ens i ve l y t e s t ed i n l abora t ory exp e r i m ent s .

I t has s ince been dem on st r a te d t o be va lid for th e di sso lut ion of ur an iu m diox ide in sp en t fue l .

M echan is t ic ana lys i s of ma ss- t ran sfer i s bas ed on wel l -es tab l i shed the ory of di ffus ive-convect ive t ran s

por t . I t s app l i c a t i on requ i re s exp e r i m ent a l me asu rem ent o f we l l-de fi ned p a ra m e t e r s such a s poros i t y , so l

ubi l i ty , di ffus ion coeff ic ient , and por e ve loc i ty. I t re lies on no ar bi t ra ry and ad jus t able pa ra m et er s f rom

empi r i c a l r a t e mea sure me nt s ; r e li ance on such pa ra m e t e r s woul d un de rm i ne t he re li ab i l it y o f t he t heory t o

pred i c t d i s so l u t i on ra t e s in t he l ong- t e rm fu t ure . Mass - t rans fe r t heory can be used t o p red i c t t he l ong- t e rm

s t ea dy -s t a t e d i s so l u t i on ra t e in a r epos i t o ry , a s we ll a s t he h i gh e r t r ans i en t d i s so l u t i on ra t e s t h a t c an ex i s t

for hu nd re ds and th ou sa nd s of ye ars a f te r the waste-sol id come s in co nta c t wi th w ater . T he f i rs t ana lys i s in

t he U .S . p ro gram a s sum ed a was t e so li d in d i rec t con t ac t wi t h po rou s rock . Th e more rea l i s t ic s i t ua t i on s of

backfi l l be tween the waste and rock, rock wi th di scre te f rac tures as wel l as pores , and the e ffec t s of waste

con s t i t uen t s o f h i gh so l ub i li t y we re amon g t he m any ex t ens i on s o f mass - t rans fe r ana l ys i s l ef t for fu t ure s t udy .

The f i rs t resul t s f rom this new m-iss- t ransfer theory were publ i shed in 1982,

4 , 5

and t he t heory was

h i gh l i gh t ed by t he Was t e I so l a t i on Sys t em S t udy

3

a s a re com me nde d app roa ch for p red i c t i ng t he l ong- t e rm

performance of waste sol ids in a reposi tory.

N e r e t n i e k s

2

used s i mpli f ied t e chni ques o f mass - t rans fe r ana l ys i s t o p red i c t t h e s t e a dy -s t a t e r a t e o f ox

i da t i on of t he coppe r con t a i ne rs and t o p red i c t t he ra t e o f was t e d i s so l u t i on fo r t he Swedi sh KB S s t ud i e s .

Th e va l i d it y of s eve ra l app rox i ma t i on s in h i s e s t i ma t e s c an now be eva l ua t ed f rom t he m ore accur a t e t i m e -

i lepe i ident ana lyt ica l solut ions developed by us .

Subsequent ly, near-f ie ld mass- t ransfer ca lcula t ions have been used for the three U.S. waste- i sola t ion

p r o j e c t s ,

b

-

7

'

n

- ° and some of t he new B e rke l ey ana l y t i c a l t o o l s

5 , 1 3

have been inco rpo ra te d in the Paci f ic No rth

w e s t L a b o r a t o r y ' s c o m p u t e r c o d e

1 0

for waste-p ackage perf orm anc e asses sm ent . Near-f ie ld m ass- t r ans fer

ana l yse s and ca l cu l a t i ons a re now i ncorpora t ed i n t he C anadi an

1 1

a n d S w i s s

1 2

r e p o s it o r y p r o g r a m s .

M A S S T R A N S F E R F R O M A W A S T E S O L I D I N T O P O R O U S R O C K

I I l l u s t r a t i v e r e s u l t s

Kigure J i l lus t ra tes som e resul t s ob ta in ed from mass- t r ansf er the ory . Here an equ iva len t spher ica l w aste

package con t a i n i ng spe n t fuel is su r rou nde d by , and i n d i rec t con t ac t w i t h , s a t u r a t e d poro us rock . Gr ou nd


4/20

wa ter flow i s s low enou gh th at near- f ie ld t r an sp or t is cont ro l led by mo lecu lar d i f fus ion , typ ic al of exp ecte d

co n d i t i o n s i n t h e U . S . r ep o s i t o ry p ro g ram s . Th e f r ac t io n a l r e l eas e r a t e , d e f in ed h e r e a s t h e m a s s r a t e of

d i sso lu t ion of a rad ionucl ide species d iv ided by the in i t i a l inventory of that species in the was te so l id , i s

p lo t te d ag ain s t the t im e s ince the be ginn ing of d i sso lu t ion . For long- l ived spec ies , such as U-234, Cs- 135 ,

an d M 2 9 , t h i s i s n u m e r i ca l l y eq u a l t o US N R C ' s fr ac t i o n a l r e l eas e r a t e b as ed o n t h e i n v en t o ry o f s u ch s p ec i e s

a t 1 0 0 0 y ea r s .

Th e l ower cu rv e is for t h e s o l u b i l i t y - l im i t ed d i s s o l u t io n o f s p ec i e s r e leas ed co n g ru en t l y w i t h u ra n i u m

fro m UO2 m a t r i x , ca l cu l a t ed f ro m t h e ex a c t an a l y t i ca l s o l u t i o n s

5

fo r t h e t i m e-d ep e n d e n t m a s s t r an s fe r ,

a s s u m i n g s a t u ra t i o n c o n ce n t r a t i o n o f t h e d is s o l v ed u ra n i u m i n g ro u n d w a t e r a t t h e wa s t e s u r face an d fo r

a u ran i u m s o l u b i l i t y o f 0 . 0 0 1 g / m

3

f or r e d u c i n g c o n d i t i o n s .

3

F rac t i o n a l r e l eas e r a t e s o f s u ch co n g ru en t l y

disso lv ing species wi th hal f l ives lower th an t ha t of ur an iu m wi l l de pa r t f rom an d fal l below th e u ra niu m

cu rv e a t t i m es o f t h e o rd e r o f t h e i r h a l f l i v es . Th es e f r ac t i o n a l r e l eas e r a t e s i n t o t h e ro ck a re we l l b e l o w

t h e U S N R C g u i d e l i n e o f 1 0 ~

5

/ y r . Un d er o x i d i z i n g co n d i t i o n s t h e u r an i u m s o l u b i l i ty can in c reas e m ark ed l y ,

b y f iv e o r si x o rd e r s of m ag n i t u d e , b u t n o t en o u g h for t h e p red i c t ed r a t e of n e t d i s s o l u t i o n t o ex ceed t h e

guidel ine l imi t .

Time, ye ars

Figure 1. F rac t iona l re lease ra t es in to rock as a funct ion of t ime (ear ly

con tain er fa i lure , 10 cm of void water , 1% of to ta l ces ium a nd iodine in

g a p .

D-

0 . 1 2 n r / y r ) .

T he up pe r curve s of F igu re 1 app ly to the re lease of the m ore so luble species , e .g . , ces iu m and iodine ,

that wore re leased f rom the U O T m at r i x d u r i n g r eac t o r o p e ra t i o n an d a re p res en t in U 0

2

g ra i n b o u n d ar i e s

an d p o res an d i n t h e fu e l - c l ad d i n g g ap s , Th e cu rv es a re ca l cu l a t ed f ro m t h e an a l y t i ca l s o l u t i o n s

1 3

for the

t ran s ien t d if fusive mas s t ransfer in to sur rou nd ing po rous rock , g iven a speci f ied in i t i a l am ou nt of so luble

3


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species dissolved in ground water that has filled voids in the waste package. Assuming that 1 percent of the

tota l inventory of these soluble species are in this readily soluble gap activ ity , the pred icted m ass rates of

release into the rock divided by the total initial inventory of each species are orders of magnitude greater

than the fractional releases from matrix dissolution. Because the 1000-year inventories of cesium and iodine,

expressed in curies, are each less than 0.1 percent of the total curie activity of the waste at 1000 years,

numerical limits for fractional release rate of these species can be calculated

14

from USNRC's numerical

criteria for low-inventory constituents. These calculated limits, applicable when the fractional release rate is

based on the inventory at the time of emplacement, are shown in Figure 1. The limits for Cs-135 and 1-129

are exceeded for an individual waste package only during the first few days of dissolution and may be easily

met for the average waste package if not all waste-package containers fail at the same time. The limit for

Cs-13 is exceeded if an appreciable number of waste containers fail not long after emplacement, but it is

easily met if all waste containers remain intact for 300 years or more.

The time-dependent mass release rates used to construct Figure 1 can also be used as time-dependent

source terms for far-field transport analysis.

Subsequent sections of this report discuss possible limitations of the analyses used to produce Figure 1

and also discuss the effects of additions to the waste-package-rock system that may be more realistic in some

situations, such as backfill, more rapid ground water flow, time-depend ent tem pera tures, rock fractures, etc.

Mast Tran sfer for Solub ili ty-Limited Spe cie i , Co ntr ol led b y Diffusion In Pore L iquid

Th e illustrative resu lts in Figure 1 are calculated from t he equatio ns for time-dep ende nt mass transfer

by molecular diffusion,

5

'

13

applicable for th e low groun d-wa ter flow rates typical for repositories in basalt,

MI IT. or salt. F.xact analy tical solutions for the time-d epend ent ra te of mass transfer from waste forms of

various shapes has been obtained.

5

The steady-state form of that equation for a low-solubility long-lived

species, assuming constant saturation concentration JV* in the liquid at the waste surface and assuming that

the waste solid is surrounded by porous rock, is:

wht>re

fi

is the fractional dissolution r ate of species i,

€

is the porosity of the rock,

D

is the diffusion coefficient

in pore water, and n, is the concentration of the species in the waste solid. 0 is a geometrical factor that

run be calculated from the waste-form dimensions and is 3/ R

2

for a spherical waste of radius R.

It is also important to examine the higher release rates that occur before steady state is reached and

that can occur with radioactive species, as predicted by the exact analytical solutions.

5

If the waste solid

ran i-pact with ground water rapidly enough to maintain near-saturation in liquid at the waste surface, the

early rate of mass transfer will be high, and it will decrease as the concentration profile penetrates farther

from the waste surface. Greater sorption retardation increases the time to steady state, and it increases the

transient dissolution rate because sorption steepens the concentration gradient.

Solubility-Limited Mass Transfer With Diffusion and Advectlon

We hav e

5

also developed the exact analytical solution for the time-dependent mass transfer when both

diffusive and convective transport are important. The mathematical analysis makes use of the known distri

bution of ground water velocities around an infinite cylinder and through the pores of the surrounding rock.

An asymptotic form of that analytical solution, applicable to steady-state mass transfer of a solubility-limited

-i.iblc species with steady flow around a long waste cylinder surrounded by porous rock, is:

8 ^ , - ( W ) - / ( .

+

« f f l

p c >

. ,

'flit* Pec let num ber Pe is given by V RjD, where U is the ground water pore velocity upstream of the waste

solid, i is the radius of the waste solid, and L is the length of the waste cylinder.

A


6/20

Equations for the steady-state dissolution rate that are similar in form, but not identical, to Equation

(2) have been obtained by others

2 , 7 , 8 , 1 5

using boundary-layer app roxim ations. However, only the exact

mathematical solution provides a criterion of validity of the asymptotic steady-state equation, Equation (2).

Ignoring the limit of validity and applying the steady-state asymptotic equation, or its equivalent

7

'

8, 1

, to

predict mass transfer rate at the very low velocities expected in U.S. repositories is not valid and will give

nonconservatively low estimates of the dissolution rate.

Correct and incorrect predictions of fractional dissolution rate over a wide range of Peclet numbers are

illustrated in Figure 2, calculated for solubility-limited mass transfer of UO2, from a spherical-equivalent

waste solid of radiu s 48.6 cm. Th e up per curve is estima ted at 100 years, with a retard atio n coefficient of 100.

The lower curve is for steady state. The dotted curve shows an incorrect extrapolation of the steady-state

P e

1

'

3

dependency from the high Pe region to the pore velocities of a few millimeters per year,

7

expected in

the tuff and basalt repositories, resulting in underestimates of the release rate by several orders of magnitude.

Th e porosity used was 0.1 , the diffusion coefficient 10 ~

5

cm

2

/sec (315 cm

2

/yr) and the UO2 solubility in an

oxidizing environment 5 x 10

5

g / c m

3

.

d

10

1 0 * -

o

2 .7

10

• •

1 0 '

Diffusion-Advection -.

100 yr

S t a t d y - s u t *

1

~ r " M H I | ^ I l l l l l l l I M U H I f l I imp • I I t i l

1 0 '

4

1 0 '

2

1 0 '

1

1 0° 10

1

1 0

1

G r o un d W a t e r P o r e V e l o c i t y , c m / y r

1 0

3

R

= 48.6 cm

t =

0.1

D

= 315 cm'/yr

N'

= 5 x 10

5

g /cm

3

10

Figure 2. Ex trapo lation s of fractional dissolution rates for UO2 to low flows

in an oxidizing environment.

For an assumed waste radius of 48.6 cm and an upper-limit value of the diffusion coefficient of 10~

5

cm

2

/sec (315 cm

2

/y r) , which neglects tortuosity corrections, the ground water pore velocity must be about 3

m/yr or greater for the steady-state equation (2) to be applicable. Although such velocities may be expected

in some repositories, the ground water velocities expected for prospective U.S. repositories in basalt, tuff, and

salt are orders of magnitude lower. However, if tortuosity reduces the diffusion coefficient a hundred-fold, the

steady-state equation (2) is applicable to pore velocities as low as 3 cm/yr, still larger than those predicted

for the U.S. repo sitorie s. For the lower velocities the equ ation s for diffusion-controlled release are applicable .

Equation-^ for the time-dependent fractional release rate for a radioactive species with a constant-

concentration b jundary condition are also given.

5

The early rate of mass transfer will be high, and it

5


7/20

will decrease as the concentration profile penetrates farther from the waste surface. With no decay, steady

state

is

reached

in as

Little aa

a

year

or

not until

a

few hundred years, depending on the magnitude of the

retardation coefficient

K.

Greater sorption retardation increases the time to steady state, and

it

increases

the transient dissolution r ate because sorption steepens the transient concentration gradient. Advective

transport shortens the time to reach steady state.

Increasing the radioactive decay constant X increases the rate of

m ss

transfer by steepening the concen

tration gradient near the waste surface. Increasing the modified Damkohler modulus

KXH/U

from zero, for

no decay,

to

10 cau ses

a

more than fourfold increase in the steady-state dissolution rate, and

it

decreases th e

time to reach steady state. For

a

mixture of stable and unstable isotopes of a given element, t he appro priate

half life

is

the effective half life

of

the isotopic m ixture

at

the time the decay correction

is to

be applied.

Decay corrections are more important when mass transfer

is

controlled

by

diffusion, i.e., when th e por e

velocity is so low that convection does not affect dissolution.

D e p e n d e n c e o n W a s t e -F o r m M a t e r ia l P r o p e r t ie s : So l i d-L iq u id R e a c t i o n R a t e

The analytical solutions discussed above for solubility-limited mass transfer have

a

clear meaning for

a

single-component waste solid that has

a

well-defined solubility app rop riat e t o the chemical environm ent

at

the waste surface. Solubility

is a

conservative upper-limit bou ndary conc entratio n, if effects of colloids can

be neglected. The result is a bounding estim ate of the release rate from the waste solid th at does not depend

on the parttcutar chemical form

of

the waste solid unless th at chemical form itself affects the solubility of

the single-component species. Effects on other constituents within the waste solid will be discussed later.

In many studies

it

has been assumed t ha t waste forms can be developed th at will perform b ett er in

a

repository because of low rates of chemical reaction of the waste solid with ground water. The proposition

is

reasonable bu t has been m ade witho ut quantification of the effect of chemical reaction ra te on the performance

of a waste solid in a repository. In the m ass-transfer analyses discussed above for solubility-limited species,

it

has been implicitly assumed that the solid-liquid reaction rate is rapid enough so that the dissolved material

is

at, or

very n ear, the solubility limit

at

the waste surface. Actu al chemical reaction ra tes do not enter

that analysis, and the bounding results from that analysis are not influenced

by

factors which can affect

solid-liquid reaction rate, such as interior cracks

in

the waste solid, devitrification

if a

glass, and other such

mechanisms that could increase the surface area for solid-liquid reactions.

To illustrate an approach to determine the effect of solid-liquid reaction rate on the rate of mass transfer

to ground water

in a

geological environ ment, an analytic al solutio n for the ra te of diffusive m ass transfer of

a dissolved waste into surrounding porous rock was developed,

17

using chemical reaction rate as

a

boundary

condition instead of assuming

a

solubility-limit boun dary concentration. Leach-rate d a t a

1 8

for

borosilicate

glass waste suggest that the solid-liquid reaction rate can be approximated by

a

zero-orde r forward reaction

an d

a

back reaction tha t

is

first order with respect

to

the con centration of dissolved silica. Th e continuity

boundary condition

at

the waste-liquid interface in a repository specifies that the current of dissolved species

al the waste outer surface equal the net rate of dissolution by solid-liquid reaction:

--*™=,0-̂ .

i > 0

where

.%'(/?. 0 is

the concentration of dissolved species in liquid

at

the spherical waste surface and

j

0

is

the

experimental forward reaction rate of the dissolved species per unit surface area. The forward reaction rate

is measured when the surface is

in

contact with

a

liquid th at contains none

of

th e dissolved sp ecies being

considered

in

E q.

(3)

Using the above boundary condition together with the diffusion equation and with other side conditions,

the tinif-dependent concentration

in

the liquid

in

the porous rock and the tim e-depende nt net dissolution

rate j{R,t) are obta ined

17

. At steady state the concentration and mass-transfer rate

at

the surface are:

6


8/20

> W ~ )

T

i

s

(5)

where , _

Eqs.

(4) and (5) show that when a is large

N(R,oo)

w

N'

an d

j(R,ao)

a

€DN*fR.

Under these conditions

the net dissolution rate is controlled by diffusion in the exterior field. When a < 1 the surface concentration

N(R

i

oo)

ts much less than the saturatio n concentration and th e net mass-transfer rate is controlled by the

solid-liquid reaction and is equal to j

a

For intermediate values of a one must use Eqs. (4) and (5) for steady

state and the analytical solution for transient values of

N(R,t)

and

j(R,t).

Th e analysis of the surface

concentration and dissolution rate with species decay in the waste form and in the exterior field has also

been treated.

To illustrate, we assume a waste glass cylinder of radius 0.15 m and length 2.4 m, resulting in an

equivalent sphere of radius 0.44 m. From the laboratory leach data

1 8

for PNL 76-68 borosilicate glass we

derive the following effective values for silica at 90°C: j

0

= 1.18 g S i0

2

/ m

2

day, and N* = 200 g S i 0

2

/ m

3

. For

an est im ate d diffusion coefficient of silicic acid in water a t 9 0°C of 2 x 10 ~

2

m

2

/d and neglecting tortuosity

corrections, and for a rock porosity of 0.01, we estimate a= 1240. Assuming no sorption of silica on the

rock (K = 1), the estima ted time to steady sta te is 320 years. From Eq. (4) N(R,oo) = 0.999 TV* for silica.

The steady-state mass transfer of silica will be controlled by exterior-field diffusion, and assuming saturation

concentration

N*

of silica at the waste surface is not only conservative but is also realistic.

Our more detailed illustration

17

shows the surface mass Mux of silica and the surface con centra tion as

a function of time. If satu ratio n at all times were assumed t he early mass-transfer rate would be infinite,

a result not physically reasonable. It is in these early times that the solid-liquid reaction rate is important

for the dissolution of a low-solubility solid matrix. The actual surface concentration is initially zero, and it

grows with time at a finite rate determined by the solid-liquid reaction rate. Within about seven minutes

after bare glass waste has been placed in contact with low-porosity water-saturated rock the concentration

of nonsorbing silica in the surface liquid will have reached 82 percent of saturation and the mass-transfer

rate will be within 5 percent of that predicted for constant saturation at the surface, assuming no silica

sorption in the rock. If silica sorbs with an assumed retardation coefGcient of 100, this time is increased to

700 minutes.

The time to reach near-saturation concentration will be longer if there is stagnant liquid between the

waste surface and the exterior porous medium . This tim e delay is easily added to the theory because the

intervening liquid is likely to be well mixed.

Van Luik

el

a/.

19

list values of

jo

for unirradiated and irradiated UO2 in deionized water and in brine

at various temp eratu res. They adopt a conservatively low value of j

0

= 0.005 g /m

2

day. Assuming that

uranium dissolution obeys a simple concen tration dependence as described in Eq. (3), adoptin g uranium

solubilities ranging from 2 x 10~

2

g / m

3

at pH 11 to 2 x 1 0~

8

g / m

3

at pH 6,

1 9

and for other parameters the

same as used in the above exam ple for silica, we obtain an es tim ated flu x ratio for th e spent-fuel matrix of 5

x 10

4

to 5 x 10

5

, greater by one to seven orders of magnitude than that estimated for silica from borosilicate

glass. If these data are correct, uranium dissolution in a geologic repository is likely to be controlled by

exterior-field mass transfer and not by chemical reaction rate. More data are needed on solid-liquid reaction

rate as a function of concentration of dissolved uranium and of temperature.

In a separate paper in these proceedings

30

we present a new analysis of the role of chemical reaction

when steady-state mass transfer from the waste solid can be controlled by both advection and diffusion.

For borosilicate glass waste in the temperature range of 57°C to r*50°C, the steady-state dissolution rate is

affected by chemical reaction rate only at Peclet num bers well above 100, far beyond any flow rate reasonably

expected in a repository.

7


9/20

Rele ase of Other Spec ies From * L ow -Solubl l lty Waste Matrix

I f the re is no preferent ia l leachin g of oth er chemica l e lem en ts co nta ine d in th e low-solub i l i ty w aste

m a t r ix , a l l c ons t i t ue n t s i n t he m a t r ix w i l l be r e l e a se d c ongr ue u t ly a s t he m a t r ix d i s so lve s . For a c ongr ue n t ly

r e l e a se d spe c i e s t ha t d i s so lve s w he n r e l e a se d f rom th e w a s t e so l id , t he i ns t a n t a n e o us f r a c t i ona l d i s so lu t ion

r a t e o f t ha t spe c i e s w i l l be i de n t i c a l w i th t he i ns t a n t a ne ous f r a c t i ona l d i s so lu t ion r a t e o f t he w ? »s t e m a t r ix

a nd w i ll be c on t r o l l e d by the m a t r ix so lub i l it y . Fo r r i gor, t he f r a c t i ona l d i s so lu t ion r a t e m us t be i n t e r p r e t e d

he r e a s t he i ns t a n t a n e ou s m a ss r a t e o f d i s so lu t ion o f a spe c i e s d iv ide d by th e i nve n to r y o f t h a t spe c i e s a t t h a t

sa m e ins t a n t . T h us , i f c ongr u e nc y o f r e l e a se a nd d i s so lu t ion c a n be e s t a b l i s he d , t he e qua t ion s fo r so lub i l i t y -

l im i t e d d i s so lu t ion o f a s i ng l e - c om pone n t w a s t e so l id c a n be a pp l i e d t o t he w a s t e m a t r ix , t he r e b y y i e ld ing

the f ra c t i ona l d i s so lu t ion r a t e s of t he c on s t i t u e n t s .

L e a c h ing e xpe r im e nt s on bor os i l i c a t e g l a s s c on ta in ing w a s t e a c t i n ide s ha ve show n tha t som e of t he

c on ta ine d a c t i n ide s d i s so lve l e ss r a p id ly t h a n w ould be p r e d i c t e d by c ong r ue nc y w i th t h e s i l i c a m a t r ix , a nd

sa tu r a t i o n c on c e n t r a t i o ns of ne p tu n iu m a nd p lu to n iu m in l e a c h so lu t ion s a r e f ound to e qua l t he so lub i l i t i e s

o f t h e s t a b l e c o m p o u n d s f o r m e d w h e n t h e s e a c t i n i d e s r e a c t w i t h t h e a q u e o u s l e a c h a n t .

2 1

T h i s s u g g e s t s t h a t

a c t i n ide s r e l e a se d w he n the s i li c a d i s so lve s fo r m pr e c ip i t a t e s a t t h e w a s t e su r f a c e . For a ny suc h c on s t i t u e n t

who se disso lut ion ra t e is l imi ted by i t s own solubi l i ty , the f rac t iona l diss olu t ion ra te can be pre dic ted by

a p p l y i n g t h e s i n g l e - c o m p o n e n t m a s s - t ra n s f e r e q u a t i o n s d e v e l o p e d b y u s , u s i n g t h e m e a s u r e d s o l u b i l it y o f t h a t

spe c i e s fo r i t s bou nd a r y c o nc e n t r a t i on . T h e r e su l t s w i ll be val id i f t h e i ns t a n t a n e o us f r a c t i ona l d i s so lu t ion

r a t e o f t h a t spe c i e s , ba se d on i t s ow n so lub i l i t y , is l es s t ha n th e i ns t a n t a n e o us f r a c t i ona l d i s so lu t ion r a t e o f

the w a s t e m a t r ix .

I f th e low-solubi l i ty c on st i tu en t wh ose re lease i s con t rol led by i t s own solu bi l i ty i s an i so top e of an

e l e m e n t w i th o th e r i so tope s p r e se n t , t he so lub i l i t y c on c e n t r a t i on i n t h e m a s s - t r a ns f e r e q ua t ion sh ou ld b e

r e p l a c e d by th e e l e m e n ta l so lub i li t y m u l t i p l i e d by the i so top i c fr a c t ion o f t he i so tope o f i n t e r e s t . T h i s c a n

be f o r m ula t e d e xa c t ly if t he r a d ionu c l ide o f i n t e r e s t i s a m inor i so top i c c ons t i t ue n t o f i t s e l e m e nta l spe c i e s i n

the w a s t e so l id . I t is a n a ppr ox im a t ion if t h e r a d ionuc l ide o f i n t e r e s t t s p r e se n t i n a n i so top i c c on c e n t r a t i on

s im i l a r i n m a g ni tu de t o t h a t o f a n o th e r i so to pe o f t h a t e l e m e n t bu t o f d i ff e re n t ha l f l if e.

I n a pp ly ing the m a s s - t r a ns f e r t he o r y t o t he d i s so lu t ion o f bor os i l i c a t e g l a s s w a s t e in a t yp i c a l r e po s i to r y

e n v i r o n m e n t , it is p r e d i c t e d

3 , 3 2

th a t t he a c t i n id e s i n t he w a s t e w i ll d i s so lve a t a r a t e l im i t e d by the i r o w n

so lub i l i t i e s . H ow e ve r , f or unr e pr oc e sse d spe n t fue l, t he l ow so lub i l i t y of u r a n iu m in r e du c ing c on d i t i on s

a nd the h igh c onc e n t r a t i o n o f u r a n iu m in t he w a s t e r e su l t i n a p r e d i c t e d f r a c t i ona l d i s so lu t ion r a t e o f t he

w a s t e m a t r ix i n a r e pos i to r y e nv i r o nm e nt t h a t i s m uc h low e r t ha n f o r bor os i l i c a t e g l a s s . If t he c on ta in e d

c on s t i t ue n t s a r e r el e a sed c on gr ue n t ly , t he y a r e p r e d i c t e d t o d is so lve a t a r a t e l im i t e d on ly by u r a n ium

solub i l i ty . T o make re l iable predic t ion s of the per form ance of sp en t fue l in a repo si tory , i t i s im po r ta nt to

e s t a b l i sh e xpe r im e n ta l l y t h e e x t e n t t o w hic h c on s t i t ue n t s in t he u r a n ium d iox ide m a t r ix d i s so lve c ong r ue n t ly .

T h e e xp e r im e nt s shou ld be pe r f o r m e d w i th a l e a c ha n t a t ne a r - sa tu r a t i on c o nc e n t r a t i o n w i th r e spe c t t o t he

w a s t e m a t r ix , be c a use t h i s i s t he su r f a c e -l i qu id c onc e n t r a t i on p r e d i c t e d f or t he r e pos i to r y e nv i r on m e nt .

T h e ins tan tan eo us f rac t iona l re lease ra t e of a spec ies l imi ted by i ts ow n solub i l i ty i s pre dic ted to be

pr op or t i o na l t o t he so lub i li t y o f t h a t spe c i e s d iv ide d by i t s c on c e n t r a t i o n i n t he w a s t e so l id . Spe c i e s p r e se n t

only in smal l or t race con cen t ra t i on s a re mo re like ly to be l imi ted in disso lut ion ra t e by the mat r ix solubi l i ty .

L ow - so lub i l i ty r a d ioa c t ive spe c i e s i n i t ia l l y p r e se n t i n su ff ic ie n t c onc e n t r a t i ons t o be d i s s o lu t io n- r a t e - l im i t e d

by the i r ow n so lub i l i t y c a n e ve n tua l ly de c a y to a c onc e n t r a t i on l ow e nou gh to be d i s so lu t ion- r a t e - l im i t e d by

the m a t r ix so lub i l i t y .

D i ff u si ve M a s s T r a n s fe r o f S o l u b l e S p e c i e s

In a spi 'n l ' fue i waste packa ge th e solub le ces ium and iodin e acc um ula ted in fue l -c ladding gap s , voids ,

a nd g r a in boun da r i e s o f spe n t fuel r ods a r e e xpe c t e d t o d i s so lve r a p id ly w h e n g r ou nd w a t e r p e ne t r a t e s t he

fuel c lad din g. Even tho ug h disso lut ion m ay be rapid , the ra te of re lease of the se solub le spec ies f rom th e

waste packag e wi ll be l imi ted by the ra t e of ma ss t ransfer of th e dissolved spec ies into th e su r ro un di ng

po rou s medi a . We have deve loped the ana l yt ic solut ion for th e f rac t iona l re lease ra te of a readi ly solu ble

r a d ioa c t ive spe c i e s

y

a s sum ing ins t a n t a ne ou s d i s so lu t ion of t h e so lub l e spe c i e s i n to a vo lum e of g r o und w a te r


10/20

that has penetrated in the waste package voids at t = 0, and assuming that the ground-water flow rate is

small enough tha t mass-transfer into surrou ndin g poruus rod ; is controlled by molecular diffusion.

The resu lts are illustrated in the up per curves in Figure 1, assuming th at one p ercent of the total cesium

and iodine is readily soluble gap activity and assuming void water equivalent to a 10-cm layer between

the fuel rods and the surrounding rock. With these assumed parameters, the release rates of the soluble

activity are more than 107-fold greater that: the congruent release rates of these same species from the

solubility-limited w aste mat rix. For early time s the fractional release rates of cesium-135 and cesium-137

are almost equal, but the cesium-137 release rate decreases rapidly because of radioactive decay, whereas

the release rate of 3 x 10

6

-year cesium-135 is characteristic of a long-lived or stable spe cies. Because of

the assumed negligible sorption of iodine in the surrounding rock, the early fractional release rate of 1.7 x

10

7

-year iodine-129 is lower than the rates of strongly sorbing cesium, but the less rapid early depletion of

iodine in the void water ?sults in a greater fractional release rate of iodine after abo ut 20 years .

OTHER ASSUMPTIONS IN THE MASS-TRANSFER THEORY

Effect of a Liquid-Fil led Ann ulus B et w ee n Wa ste and Roc k

The mass-transfer analyses discussed above assume that the waste solid is in contact with porous rock

containing ground water. The assum ption of a constant solubility-limit boun dary concentration for low-

solubility species has been discussed above. The theory assumes conservatively that the corrosion-resistant

container cf the waste package has Jailed and that its corrosion products offer no resistance to mass transfer.

If there is a liquid-filled annulus between the waste package and the rock, or between the waste solid and

the rock, the mass-transfer equ ations a re still applicable, provided th ere is no apprec iable convective flow

through the ann ulus. It is reasonable to assume that t he liquid between the

t

waste solid and rock is well

mixed, so the dimensions of the borehole should b e used for th e geometrical param eters in the mass-transfer

equations; The effects of convective transport in the annulus and convective flow through the waste solid

have not been analyzed.

E ffe c t o f Flo w D ir e c t i o n a n d Ge o m e tr y

Extensions t ha t include the effect of a porous backfill are described late r. Th e analyses described

above have assumed tha t ground water flows normal to the cylindrical axis of a waste package. Oth er

flow direction s are possible in a repository. We have considered flow parallel to th e axis, as well as other

waste-form geom etries, and have shown th at predicted dissolution ra tes are not significantly different.

Hydxodynamic Dispersion

It has been suggested by some

1 6

that because of hydrodynamic dispersion the effective diffusion coef-

ficient th at a ppe ars in the analytical solutions should be a value greater th an th at for molecular diffusion.

However, experimental data

2 3 , 2 4

for effective diffusion coefficients in packed colum ns show th at for th e ground

water velocities and dim ensions typical for individual waste packages the value for molecular diffusion should

be used. At locations far enough removed from th e waste surface for hyd rodyna mic dispersion to become

important, the concentration gradients are too small to aiiect the mass transfer from the waste surface.

However, if the same governing equations were to be applied to mass transfer over greater distances, the

uffect of hyd rodynamic dispersion should be considered. Th e effective diffusion coefficient will increase with

dis tance, requiring equ ation s for diffusive-advective tr an sport w ith a spac e-depen den t diffusion coefficient,

such as that developed by u s .

2 5

E f fe c t o f R a d i o a c t i v e D e c a y

The analysis discussed above were developed for radionuclides with no radioactive-decay precursors.

Some of the radionuclides important in repository performance assessment ure continuously generated by

decay of precursors, e.g., Ra-226 from U-234 and Th-230, and Th-229/fta-229 from Np-237 and U-233. It is

9


11/20

not necessarily conservative to apply the mass-transfer equations for a species with no precursor separately

to each member of the decay chain, assuming that each species is at its solubility limit in ground water near

the waste surface.

26

Decay of a precursor in the tra ns po rt field can cause solubility-limited concen tration

of its daughter to occur in the field further from the waste surface, resulting in greater rates of mass-

transfer into the rock. This is particularly important when mass-transfer theory is used to predict the rate

of tran spo rt throug h backfill, or through o ther intervening diffusing m edia, into surro und ing rock. Th e

greater penetration of a long-lived precursor through thft backfill is a means for a shorter-lived daughter to

appear at appreciable distances from the waste surface. A near-field mass-tr?.nsfer analysis applicable to the

simultaneous diffusive transp jrt of a radionuclide chain has been developed and ifl described later.

L oc al So rptio n EquiUbvium

Most of our analysis assumes local sorption equilibrium, w ith a constan t retard ation coefficient. Mass

transport without local chemical equilibrium between liquid and solid phases has been analyzed in a pre

vious report.

2 7

Mass transfer w ith local sorption eq uilibrium bu t with a concent ration-d epen den t so rption

distribution coefficient is discussed elsewhere in this report.

Surface Diffusion

Most of our ana lysis assumes no diffusion of sorbed species on or in th e solid phase. However, some

experiments with selected species and solids Have demonstrated appreciable surface diffusion of adsorbed

species. For accu rate pred ictions of mass transfer an experime ntally determ ined diffusion coefficient is

desirable, rather than using the upper-limit value usually adopted in our numerical illustrations. In making

such measu reme nts, it is im po rtant to check also the possibility of surface diffusion. Th e mass-transfer

equations can be reformulated if such measurements indicate that su7face diffusion need be considered.

I n t e r f e r e n c e f r o m O th e r W a s t e P a c k a g e s

Most of our analysis of mass transfer from individual w aste packages conservatively neglects the ov erlap

ping concentration fields resulting from species dissolved from nearby waste packages. For high-level waste

packages separated by several meters the concentration fields are not expected to overlap enough in the near

field to significantly reduce the mass transfer from an individual package.

Although nearby waste packages do not appreciably affect the dissolution and release rates, the over

lapping concentra tion fields do affect th e calculation of far-field tra ns po rt from array s of waste packages.

Our analyses of concentration fields from arrays of waste packages

2 7 , 2 8 , 2 9

show a near region in which the

concentrations vary greatly in the direction transverse to ground-water flow, an intermediate region in which

the array can be treated as an infinite plane source of dissolving species of the same areal source strength,

and a far-field region in which the array can be treated as a plane source of finite ex ten t. Th e array equa tions

have been developed for both porous and fractured media. These intermediate-field analyses will be useful

to repository projects in making detailed predictions of releases to the environment.

P o r o u s o r F r a c t u r e d Ro c k

In th e m ass-transfer analysis discussed above, and extensions of these solutions described in some of the

following sections, we assume that the reck surrounding the waste solid can be reasonably characterized as

a porous solid. The effect of rock fractures is discussed later.

Co n s t a n t Te m p e r a t u r e

The solutions described above assume that the waste solid and surrounding rock are at a constant

and spatially uniform tem peratu re. Extensions to account for time-dependent tempe ratures expected if

the waste solid is exposed to ground water when the repository is heated by radioactive decay have been

developed. Ou r m ass-transfer equations tha t pred ict the ra te of diffusive-controlled dissolution of a low-

solubility waste solid surrounded by porous rock can be applied in integral form to predict the effect of

10


12/20

a time-dependen t repository tem peratu re on the dissolution rat e. We assume, conservatively, tha t at any

time t the waste and rock are at a spatially uniform temperature equal to the temperature at the surface of

the borehole. Time-dependent rock temperatures at the borehole surface are specified from a calculation of

transie nt heat co nduc tion. From these data , from da ta on the change of solubility and diffusion coefficient

with tempera ture

31

, we have demonstrated the application of temperature-dependent mass-transfer analysis

of solubility-limited species from borosilicate glass waste in a basalt repository.

3 0 , 3 1

Calculation of the effect of repository heating on the rate of diffusive mass transfer of soluble species

into the rock is simpler

32

because the only param eter affected by tem pe ratu re is the diffusion coefficient,

if the re tard atio n coefficient ca n be assumed con stan t. At very early time s the higher diffusion coefficient

causes greater mass transfer rate from the waste-package water into the surrounding rock, depleting the

concentration in the waste-package water. This early depletion results in a later mass-transfer rate that is

luwer thun would occur if the temperature were always at the ambient temperature. Thus, for most time*

of interest. the release rate predicted by using a diffusion coefficient evaluated at ambient temperature is

conservatively higher than the release rates corrected for repository heating.

C o n s t a n t a n d U n i fo r m C h e m ic a l E n v ir o n m e n t

In predicting the dissolution rate of low-solubility species it is important to examine the possibility of

solubility changing with time , due to changes in the local chemical environm ent. For example, dissolution

of borosilicate glass waste causes local changes in pH, which affects solubility of several species. Also, the

oxidizing environment due to air during construction and waste emplacement can revert to a more reducing

environment after repository sealing and resaiurstion, resulting in large changes in solubility of uranium

and other species. If the chemical environment stabilizes within a few hundred years, while the protective

container is still effective, the solubility corresponding to the stabilized environment would be the appropriate

value to use in predicting release rates.

A longer-term mechanism for changing the chemical environment is the radiolysis of ground water

due to alpha particles from trans uran ics in the dissolving wa ste. If radiolytically-pro duced hydrogen can

escape, radiolytic peroxide can result in a locally oxidizing environment and greater solubilities of uranium,

particularly imp ortant in predicting the long-term release rate from spent fuel. N ere tni ek s

33

has analyzed

the progress of such a redox front for a spent-fuel repository in granite, determined by the rate of peroxide

generation, mass transfer of the peroxide from the waste surface, and reaction of peroxide with ferrous

compounds in the water and rock. The result is a moving and expanding redox front, a transition from the

oxidizing region of high uranium solubility near the waste surface to the low-solubility reducing region. To

predict the net rate of release of uranium across this front, the equations for mass-transfer of low-solubility

species can be applied, using the radius of the redox front as the geometrical parameter.

Alpha radioiysis can affect the release of other constituents from the uranium-dioxide matrix in spent

fuel. If the species dissolve congruen tly with th e uraniu m, th ey will dissolve at a rate controlled by the

higher solubility of uranium at the waste surface. If these species do not precipitate at the redox front, alpha

radiolysis will have resulted in a greatly increased release of contained radioactive species to the geosystem.

A partly failed iron c ontainer m ay prevent th e radiolytic ally generated oxida nt from moving into th e

rock, and corrosion products may decompose peroxide oxidants, but it is still possible that radiolysis can

create a locally oxidizing environment near the spent fuel surface that accelerates release of constituents

from the spent fuel matrix.

3 4

M SS TR NSFER WITH B CKFILL BETWEEN W STE SOLID ND ROCK

Steady-State Maw Transfer Through Backf i l l

We have obtained the solution to the steady-state mass transfer rate from a waste solid, through sur

rounding backfill, and into porous ro


13/20

mass transfer rate into rock because of the assumed zero flow in the backfill. For backfill of much greater

porosity than rock, as may be expected for backfill now contemplated in U.S. repositories, increasing the

backfill thickness increases the rate of m ss transfer into the rock, a consequence of the increasing area, of the

backfill-rock in terface a s the backfill thickness is increased. Unless the backfill can be m ade far less porous

than the rock, introducing backfill cannot be expected to result in large reductions in the steady-state rate

of release of stable and long-lived species into the rock, assuming that the chemical environment remains the

same.

When radioactive decay is included in the analysis of steady-state mass transfer, the retardation coef-

ficient also appears. The results demonstrate that a sorbing backfill does not necessarily reduce the steady-

state release rate into surrounding rock.

Time-Depende nt Mass Tran sfer of Solubi lity-L imited Ra dioa ct ive Spe cie s Through B ack fil l

The.

equation s for the time-dep enden t diffusive tran spo rt of radioactive species throu gh backfill have been

developed for solubility-limited species

2 2 , 3 5 , 3 6

and for the rapidly dissolving gap activity in spe nt f ue l.

13

Th e equation s are applicable for a radioactive species with no decay precursor and for diffusion-dominated

mass transfer through the backfill and the surrounding porous rock.

During the early times of transient dissolution a higher backfill porosity greatly increases the dissolution

rate,

because th e early resistance to mass transfer is entirely within the backfill. For typical backfill p roperties

and for a retardation coefficient of 1000 the mass-transfer rate into the rock closely approaches that into the

backfill after about 10

4

years, and steady state is reached after about 10

s

years.

Increasing sorption steepens the transient concentration gradient and increases trie fate of diffusive

mass transfer. Ste ady -state rates of mass transfer of stable species are not affected by sorptio n. For a given

backfill retardation, increasing sorption by rock increases the maximum rate of mass transfer into the rock,

showing that the rock properties and finite backfill must be included in the analysis, as has been done here.

For typical parameters, the maximum rate of mass transfer into the rock depends mainly on the backfill

retardation.

For neptunium-237, with a half life of 2.1 x 10

6

yeaw, corrections for decay are minor. For carbon-14,

with a half life of 5730 years, decay not only increases the rate of mass transfer into the backfill, because it

steepens the concentration profile, but it also increases the transient and steady-state rates of mass transfer

into the rock. For a radionuclide half life as short as 15.3 years , decay considerably increases the m ass

transfer rate into the backfill, but the half life is short enough that the species all decays while diffusing tn

the backfill. Th us , there is a range of half lives for which the r at e of mass transfer into the rock is gre ater

than the steady-state value for no decay. As illustrated here for carbon-14, some radionuclides important in

repository performance analysis will fall within this range.

Previous backfill analyses

3 7 , 3 8 , 3 9

have calculated the break throu gh time for a stable species at a

specified distance in a backfill of infinite thickness and have used that time to estimate the amount of

radioa ctive decay th at will occur during diffusion throug h finite backfill in a repository. Th is has led to

considerable overestimates of the possible retention of radionuclides in backfill. Mass-transfer an alysis t ha t

take into account the diffusive prop erties, of the rock should be used to pred ict release of radionuclides

through a backfill of finite thickness.

Time-Dependent Mast Transfer of Soluble Radioact ive Species Through Backfi l l

We have given analytical solutions for the time-depen dent diffusion of readily soluble gap activity

through backfill and into surround ing porous rock. Illustrative res ults '

10

for a 30-cm layer of backfill of

porosity 0.2 and I percent release of soluble species to the fuel-cladding gap show tha t nonsorbing iodine-129

arrives at the backfill-rock interface in less than a year, with a peak release rate about ten-fold less than the

equivalent release-rate limit calculated from the USNRC criterion.

1 , 1 4

The peak release rates of the sorbing

cesium isotopes into rock, with an assumed retardation coefficient of 1000, occur at about 100 years. The

normalized release rat e of cesium-137 is less than that of cesium-135 because of decay in the backfill. The

12


14/20

peak release rate of cesium-135 is about 10-fold less than its release-rate limit. The release limit is exceeded

for cesiurn-137 if an appreciable nu mber of waste containers fail no t long after em placemen t. For these

parameters, backfill several times thicker than in current repository designs would be needed for each waste

package to comply with the release-rate criterion if no container is assumed, but the present designs should

be adequate for cesium-137 if all waste containers last 300 years or more.

Effect of Non-Linear Sor ption on Ha s* Transfer Through B ack fil l

In the previous analyses we have assumed linear sorption in the backfill, as expressed by a retardation

constant independent of concentration. Some data show nonlinear sorption in bentonite, with a tendency

towards saturation of the sorption sites. For such backfill materials, the mass-transfer characteristics can be

far different than those analyzed above for linear sorptio n. Ppecies can be released sooner and more rapidly

into the rock if sorption satu ratio n o ccurs in the backfill. Analytical solutions for a mass transfer with a

simplified nonlinear sorption isotherm have been given.

2 2 , 4 1

E f f e c t o f F l o w I n B a c k f i l l

Th e above analyses have assumed no ground-flow in backfill. Flow in backfill is no t expected to be

imp ortan t in U.S. repositories because of the very low flow in the surrou ndin g rock. In the Swedish progra m,

where greater ground-water Sow is expected, the planned backfill is compacted bentonite, of low enough

permeability to result in the negligible backfill water flow that is assumed in their mass-transfer analysis

2

. From our analysis

4 2

of water flow through a spherica l shell of backfill surrounde d by rock, th e effect of

backfill water flow can be predicted.

M SS TR NSFER N FR CTURED ROC K

N e r e t n i e k s

2 , 4 3 , 4 4

has developed an equation for stead y-sta te mass transfer throug h backfill into ground

water in discrete fractures, using bound ary-layer app roxim ations. Th e initial purpose was to predict the

lifetime of a thick copper canister by predicting the rate of m ss transfer of sulfide oxidant from ground water

to the canister surface. T he b oundary-layer approx imation do es not apply for very small fracture-flow Peclet

numbers, characteristic of U.S. repositories.

We have developed the solutions for time-dependent transport through backfill into a fracture with

flow

45

and for transient diffusion from a waste solid with fractures in porous rock, with negligible water flow

in the fractures.

46

These new analyses allow greater realism in predictions for U.S. repositories, and they

also establish the region of validity of the earlier analyses of mass transfer into surrounding porous media.

MASS TRANSFE R OF RADIOACTIVE DECAY CHAINS

To predict the time-dependent mass transfer of

h

radioactive species that is a daughter in a radioactive

decay chain with long-lived precursors, it is necessary to include the simultaneous transport and decay of

the precursors. We have published elsewhere

47

the fundamental equations for advective transport of decay

chains in infinite media. Chambre has developed analytical solutions

36

to predict the release rates of decay

chains with arbitarily many members into porous rock from backfill of finite thickness. The predicted space-

depend ent concen trations at 1000 years after the beginning of dissolution, using Batem an-typ e boundary

concentrations, ate illustrated

4

*

1

for the decay chain

2 3

4

U

_ 2 3 0

T h

_ 2 2 6

R a

Appreciable concentrations of

2 2 6

R a near the backfill-rock interface are a consequence of the presence there

of its precursors. The greatest release rate of

K 6

R a occurs at about 2 x 10

5

years.

13


15/20

REL EAS E RATES F ROM W AS TE P ACKAGES IN A S AL T R EP OS ITORY

T h e M a s s- Tr a n s fe r A p p r o a c h

In 1985 we proposed

49

a realistic and direct approach towa rds predicting radionuclide release rates in a

salt repository based on mass-transfer analysis. Recognizing that within a few years after the emplacement of

waste packages in a geologic repository salt creep is likely to close the air gap between a waste container and

tliv borehole wall, we proposed that thereafter release rate of dissolved species from the waste solid is likely

'.o be governed by mass transfer into brine in grain boundaries in the surrounding salt and in intersecting

interbeds of other rock. Because of the low expec ted m igration velocities of brine in th e consolidated salt,

mass transfer dominated molecular diffusion was a likely possibility. If so , many of the m ass-transfer analyses

described above could be adapted to predicting release rates in a salt repository.

Subsequent analyses of creep closure and consolidation by Brandshaug

50

confirms that consolidation is

expected within a few years after emplacement.

B r i n e M ig r a t i o n V e l o c i t i e s

During the consolidation period most of the brine inclusions in salt crystals near the waste package

are predicted to have moved to grain boundaries under the influence of temperature gradients. After salt

consolidation brine in grain boundaries near the waste package can only migrate outward into the surrounding

sail, under th e influence of pressure gradients caused by transien t hea ting of the sa lt. Hot salt near the w aste

package expands against the waste package and surrounding salt, resulting in high compressive stresses near

the waste package. Grain-boundary brine expands more than does the salt and further increases the local

pressures and pressure gradients tha t cause brine to flow outward into the cooler sal t. Such outward flow of

brine relieves the pressure gradient on the fluid, which finally relaxes to near-lithostatic pressure.

To determine the extent to which advection by brine in grain boundaries is an important transport

mechanism for released radionuclides, we have estimated the time-dependent migration of brine after salt

cons.olidation. Based on current inter preta tions of experim ental da ta on the migra tion of grain-b ound ary

brine in natural salt deposits, it is assumed that the migration velocity is proportional to the local gradient

in fluid pressure. To find the u ne-dependent pressure profile, we have adopted the governing equations for

the theromoelastic stresses and deformation of salt due to thermal expansion of the salt and of its contained

grain-boundary brine developed by McTigue.

sl

Based on calculated time-dependent temperature profiles

in the salt after waste emplacement and utilizing thermoelastic and permeability parameters deduced by

McTigue for salt at the Waste Isolation Pilot Plant in New Mexico, our analytical solutions for the time-

dependunt profiles of pressure and brine velocity provide means to estimate the effect of brine migration

on radionuclide release to salt.

5 3

C onsolidation is assumed to have occurred in stantan eous ly. We assume ,

conservatively, tha t the w aste container has completely failed during consolidation or shortly therea fter.

W ith these para me ters , the brine migration velocity is nearly zero after ten years. Even the highest

velocity is of the order of millimeters per year and occurs only with a meter or so of the waste package.

The predicted velocities are so low that molecular diffusion may be the dominant transport process for

radionuclides in sa lt. Th is is likely to be tru e if the diffusion coefficient is near 1Q~

5

cu r/ se c, a typical value

for a liquid continuum, or even if it is reduced a hundred-fold by tortuosity.

More realistically, if the waste containe r is effective for several tens or h undreds of years, the tem per atur e

and pressure gradie nts will have relaxed and the migration velocities are expected to be so low that convective

transport seems even less likely.

These conclusions are tentative, however, because of the considerable uncertainties in the appropriate

properties of salt to be used in the calculation of pressure gradients and velocities.

Release Rates by Diffusion

As a result of the foregoing analysis of brine migration, we assume that after salt consolidation mass

14


16/20

transfer of dissolved species away from the waste solid is governed by molecular diffusion, so that the equations

for the time-dependent diffusive transfer of dissolved species from a waste solid into porous rock can be

applied. The waste container is conservatively to have failed during consolidation, or shortly thereafter, so

that brine at the waste surface begins to dissolve the spent-fuel waste and its contained radioactive species.

Assuming a was te solid 0.72 m in rad ius, a diffusion coefficient of 10~

7

cm

2

/sec, salt porosity of 0.001,

a uranium retardation coefficient of 20, and uranium solubility of 0-001 g/ni

3

, we estimate the fractional

release rate for uranium of the order of 10~

1 2

/yr and less.

9

Similarly, applying the equations for the time-dependent diffusive release of the readily-soluble gap

inventory of cesium and iodine, assuming that one percent of the initial iodine and cesium is in the readily

soluble form and assuming 0.45 m

3

of void water in the failed waste package, we obtain fractional release

curves for I-I29, Cs-135, and Cs-137 similar to those shown in Figure 1, but of lower magnitude because of

the expected lower diffusion coefficient.

To estimate the release rates from a waste package into more porous interbeds that may intersect the

borehole in a salt repository, we can apply our new analytical solution for the tim e-depe nden t tran spo rt from

a waste cylinder into a fracture with negligible water flow .

DIFFUSION-LIMITED RELEASE RATES FROM WASTE PACKAGES IN A TUFF REPOSITORY

The design for a waste repository in unsaturated tuff provides an air gap between each waste package

and the surrounding rock. Infiltration water can drip on the waste package. During the therm al period

the hot waste evaporates infiltration water at atmospheric pressure, resulting in no liquid pathways for

transpo rt of radionuclides from the waste package. Th e tuff pr oj ec t

5 3

estimates later liquid releases of

dissolved radionuclides from a waste package by multiplying saturation concentrations of waste constituents

by the volume flow rate of ground water contacting each waste package. The simple predictive theory seems

reliable and requires only da ta on flow rate and satu ration co nce ntration s. It also requires validation th at

the porous rock does not contact the waste package or that the annular gap does not fill with sediments.

Otherw ise, diffusion pathway s for dissolved species on th e waste surface into g round water in the partly

saturated pores of the surrounding rock can result in transient diffusive mass-transfer rates as much as

three orders of magnitude greater than the bulk-flow solubility-limited release rate at the low flow rates

predicted for the tufl repos itory, depen ding on the magn itude of the diffusion coefficient. The refore , key

issues of validating waste-package release rate predictions include the long-term integrity of the air gap and

the diffusion coefficient for dissolved species in tufF.

The tufT project also presents

53

an altern ative estim ate of the release rate from a waste package in contact

with moist tuff, based on a mass-transfer e qu at io n

7 , 8

developed from a boundary-layer approximation. As has

been pointed out earlier, it is invalid to apply such equations developed from boundary-layer approximations

to predict m ass transfer at th e low water infiltration Tates (ca. 0.003 to 1 m m /y r) e stimate d for the tuff

project. Also, it is nonconservative to apply th e equation for stead y-s tate m ass transfer, developed from

boundary layer theory, because the transient release rates will be much higher than the steady-state rate for

a very long period.

Our estimates for the possible release rates for a tuff repository, assuming conservatively that there

are liquid diffusive pathways into the surrounding rock, are based on applying the equations for diffusive-

controlled mass transfer discussed iiorein. The preliminary results for solubility-limited dissolution of spent

fuel, based on solubility and porosity used by the tufF projec t and assuming a diffusion coefficient of 1 0

5

c in

J

/s ec , predict diffusion-controlled release rates over 1000-fold greater than those estima ted else wh ere ,

53

but still within regulatory limits. Corrections for the time-dependent repository temperatures have not been

included in these preliminary estimates.

15


17/20

RELE SE R TES FROM W STE P CK GES N B S LT REPOSITORY

Most of the mass-transfer analyses and topics discussed herein have been illustrated with data expected

to be representativ e for a repository in bas alt. Prediction s from these analy tical solutions can be used as

an independent check of the numerical calculations made by the project. The more extensive mass-transfer

analyses reported herein, such as the eifect of repository heating and the transport of radionuclide decay

chains, as well as the effect of discrete fractures in the rock, can be applied to the performance assessment

of a basalt repository.

ACKNOWLEDGEMENT

Work supported in part by the U. S. Department of Energy via Contract DE-AC03-76SF00098.

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