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~ ) P e r g a m o n

AppI. M ath. Let t . Vol. 10, No. 1, pp. 31-34, 1997

Copyright@1997 Elsevier Science Ltd

Printed in Great Britain. All rights reserved

0893-9659/97 $17.00 + 0.00

P I I : S 0 8 9 3 - 9 6 5 9 ( 9 6 ) 0 0 1 0 6 - 1

M o d e l l i n g t h e V o l a t i l i z a t i o n o f O r g a n i c

S o il C o n t a m i n a n t s : E x t e n s i o n o f t h e J u r y ,

S p e n c e r a n d F a r m e r B e h a v i o u r A s s e s s m e n t

M o d e l a n d S o l u t i o n

R . S . A N D E R S S E N A ND F . R . D E HOOG

CSIRO Division of Mathema tics a nd Statisti cs

GPO Box 1965, Canberra, ACT 2601, Australia

B . R . M A R K E Y

Conta minate d Sites Section

Environment Protection Authority NSW

P.O. Box 1153, Chatswood, NSW 2057, Australia

(Received and accepted March 1996)

Communicated by G. C. Wake

A b s t r a c t - - I n the analysis of the volatilization of organic soil contamination, the Behaviour As-

sessment Model (BAM) of Jury, Spencer, and Farmer [1] has proved to be a valuable exploratory

tool, because it has an analyt ic solution which is easily and quickly evaluated. However, because

the surface boundary condition in the BAM is homogeneous, its applicability is limited to situations

where the above ground concentration of the volatilant is zero above a boundary layer. The impor-

tan t situa tions of the accumulation of the volatilant below buildings, in vegetation or below material

stored on the ground are thereby excluded from consideration. This paper derives an analytic so-

lution for the nonhomogeneous surface boundary condition extension of the BAM which allows its

exploratory potential to be extended to the more realistic scenarios mentioned above. This analytic

solution contains the BAM solution as a special case.

Keyw or ds -- Conv ec ti on -d if fu si on , Explicit analytic solution, Nonhomogeneous surface boundary

conditions, Soil contamination, Volatilization.

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

In assessing the po tential health r isk associated with emissions of organic soil contamin ants , the

major ini t ial step is the modell ing and solution of the volat i l izat ion processes which generate the

emissions.

A num ber of models have been proposed and uti l ized by various authors for simulati ng the

volat i l izat ion of organic soil conta mina nts (see [2-5]) . They include the one-d imensiona l semi-

infini t e-depth model of Jury et al. [1,5] for the p roto type s itua tion where the de grad ati on rate

con sta nt of the cont ami nan t, its effective diffusion coefficient D E , and its effective solute con-

vection velocity V~ are assumed to be constant . I t is of part icular interest , s ince the authors

derive an explici t analytic solution which has found applicat ion in conta minat ed si tes exposure

assessment (cf. [6]).

Typeset by .AA4,S-~X

31

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3 2 R . S . A N D ER S SE N e t a l .

T h e p u r p o s e o f t h i s p a p e r i s t o e x t e n d t h e J u r y et al. [ 1, 5] m o d e l t o a m o r e r e p r e s e n t a t i v e o n e -

d i m e n s i o n a l s i t u a t i o n , a n d t o c o n s t r u c t , f o r t h i s e x t e n s io n , a n a n a l y t i c s o l u ti o n w h i c h c o n t a i n s

t h e J u r y et al. [ 1, 7] s o l u t i o n a s a s p e c i a l ca s e . T h e e x t e n s i o n p r e s e n t e d h e r e i s m o r e c o m p r e h e n -

s iv e t h a n t h a t c o n s i d e re d b y L i n a n d H i l d e m a n n [ 8] , i n t h a t i t a ll ow s f or th e s u r f a c e - b o u n d a r y

c o n d i t i o n t o b e n o n h o m o g e n e o u s .

T H E B E H A V I O U R A S S E S S M E N T M O D E L [ 1 ]

I n a d e t a i l e d a n d c o m p r e h e n s i v e s t u d y o f t h e v o l a t i li z a t io n o f a s o il c o n t a m i n a n t , J u r y et al. [1]

p r o p o s e d t h e f o ll o w in g p a r a b o l i c p a r t i a l d i f fe r e n ti a l e q u a t i o n , w h i c h u n d e r p i n s t h e i r B e h a v i o u r

A s s e s s m e n t M o d e l :

O C _ O 2C O C

a - - T ~ c = D~-~z2 - v ~ - ~ z ,

w i t h i n i t ia l c o n d i t i o n

C(z,O)={Co, O O , (1)

C o c o n s t a n t ,

+ V E C( O, t ) = - H E C ( O , t) , t _> O,

2 )

(3)

(4)

C

D E

H E

d

D A

G

R G

v E

t h e t o t a l c o n c e n t r a t i o n i n t h e s o il o f t h e v o l a t i li z a t i n g c o n t a m i n a n t ;

t h e e f f e c ti v e d if f u s i o n c o e f f ic i e n t o f t h e c o n t a m i n a n t ;

h / R G , t h e g a s s t e a d y - s t a t e b e h a v i o u r o f t h e c o n t a m i n a n t ;

D A / d ;

t h e d e p t h o f t h e s u r f a c e - b o u n d a r y l a y e r o f ai r;

t h e d i f f u s io n co e f f ic i e n t f o r t h e v a p o u r t h r o u g h a i r;

t h e p a r t i t i o n c o e f f i c ie n t f o r t h e g a s e o u s c o n c e n t r a t i o n r e l a t i v e t o t h e t o t a l ;

t h e e f fe c ti v e s o l u te v e l o c i t y o f t h e c o n t a m i n a n t ; a n d

t h e d e g r a d a t i o n r a t e c o n s t a n t o f th e c o n t a m i n a n t .

V o l a t i l i z a t i o n a t t h e s u r f a c e i n t h e J u r y et al. [1 ] mode l , i s a s s umed to t ake p lace v ia d i f fu s ion

t h r o u g h a s t a g n a n t b o u n d a r y l a y er i n t o a n o v e rl y i n g a t m o s p h e r e w h e r e t h e c o n c e n t r a t i o n i s

t a k e n t o b e z e r o. T h e a b o v e s u r f a c e - b o u n d a r y c o n d i t i o n , t h e r e f o r e , fo l lo w s f r o m F i c k ' s l a w

(c f. [9 , S e c t i o n 1 . 2] ), a f t e r a p p r o x i m a t i n g t h e s p a t i a l d e r i v a t i v e o f t h e c o n c e n t r a t i o n C a c r o s s

t h e b o u n d a r y l a y e r ( se e [ 1 ]) . T h e r e s u l t i n g s u r f a c e - b o u n d a r y c o n d i t i o n (3 ) r e p r e s e n t s a b a l a n c e

b e t w e e n t h e s u r f a c e v a p o u r f l u x ( t h e l e f t - h a n d s id e o f ( 3 )) a n d t h e e f f e ct o f t h e b o u n d a r y l a y e r

( t h e r i g h t - h a n d s i d e o f (3 )) . H o w e v er , th e h o m o g e n e i t y o f t h i s b o u n d a r y c o n d i t i o n e x c lu d e s f r o m

c o n s i d e r a t i o n th e i m p o r t a n t p r a c t i c a l s i t u a ti o n s , w h e r e t h e v o l a t i l a n t o n le a v i n g t h e g r o u n d ,

a c c u m u l a t e s b e l o w b u i l d i n g s , i n v e g e t a t i o n o r b e l o w m a t e r i a l s t o r e d o n t h e g r o u n d .

A s t h e s o l u t i o n t o t h e i r m o d e l , J u r y e t al. [ 1 ,7 ] g a v e , w i t h o u t p r o o f , t h e f o l lo w i n g e x p l ic i t

exp res s ion :

1

C ( z , t ; L ) = ~ C o e x p ( - t ) { C 1 ( z , t ) + C 2 ( z , t) + C 3 ( z , g)}, (5)

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Model l i ng t he Vo la t i l i za t ion 33

w h e r e

C l ( Z , t ) = e r fc 2 Dv/_D__ / e r fc k ~ ,

C 2< z ' t) : ( I + - ~ E ) e x p k , DE ] - 2 ~ ]

e r f c ( 2 - - D ~ E t ] ] '

Cj(z , t ) = 2 + ~ E e xp DE

× [ e r f c ( Z + ( 2 H E + V E ) t ) ( H E L ~ ( + L + ( 2 H E + V E ) t ) ]

2,/DEt

- - exp \ D E / er fc

z 2v/DE t .

H o w e v e r , n o f o r m a l d e r i v a t i o n o f t h i s r e s u l t w a s g i v e n b y t h e s e a u t h o r s .

E X T E N S I O N O F T H E J U R Y M O D E L [ 1 , 5 ]

T h o u g h t h e J u r y

et al.

[ 1, 5] m o d e l i s a q u i t e c o m p r e h e n s i v e r e p r e s e n t a t i o n o f t h e p r o c e s s e s

i n v o l v e d , i t h a s s o m e m i n o r , b u t n o n t r i v i a l , s h o r t c o m i n g s ; n a m e l y ,

(i) t h e i n i ti a l d i s t r i b u t i o n o f t h e c o n t a m i n a n t

C(z,

0 ) i n t h e s o i l i s a s s u m e d t o b e c o n s t a n t

d o w n t o a f i n i te d e p t h L a n d z e r o b e lo w ;

( ii ) t h e s u r f a c e - b o u n d a r y c o n d i t i o n (3 ) is h o m o g e n e o u s .

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

by

C(z, O) = f(z ),

0 < z < oo, (6)

~tnd

l o c i v z c ( o , = - H z C ( O , t) >_

(7)

- - D E ~ Z z = O

+

t

+

~c+ t ) ,

t O,

r e s p e c t i v e l y , w h e r e

C+(t)

d e n o t e s t h e g a s c o n c e n t r a t i o n a t t h e u p p e r s u r fa c e o f t h e s t a g n a n t

b o u n d a r y l a y e r t h r o u g h w h i c h t h e v a p o u r d i ff u se s a f t e r l ea v i n g t h e s oi l. I n t h e a b o v e d e r i v a t i o n

o f e q u a t i o n ( 7) , t h e J u r y et al. [1] f r a m e w o r k f o r t h e s u r f a c e - b o u n d a r y c o n d i t i o n h a s b e e n f o ll o w e d .

I n o r d e r t o s o l v e t h e a b o v e m o d e l , t h e f i r s t s t e p i s t o a p p l y t h e t r a n s f o r m a t i o n

( - - , / -D--EE' c(¢, t )=exp((p+c~2)t)exp(ct()C

x/-D~E( ,t , c~ - 2v/-D~, (8)

w h i c h y i e l d s

w i t h i n i t i a l c o n d i t i o n

~C C~2C

- - - c = c ( ¢ , t ) ,

0 < ( < ~ , t > 0 , (9)

at o ( 2

C ( ~ ,0 ) = e x p ( a ¢ ) f ( X / ~ E ~ ) ,

w i t h s u r f a c e - b o u n d a r y c o n d i t i o n

[ q c ]

+ h c ( O , t ) = e x p ( ( + c ~ 2 ) t ) C+(t)

- N ~ = 0 v ~D T~

a n d w i t h i n n e r - b o u n d a r y c o n d i t i o n

0 < ( < o o , ( 1 0 )

l i e 2 HE

h - 2 x / - ~ ' t _> O , i i )

c (oo, t) = 0, t > 0. (12)

T h i s c a n o n i c a l p a r a b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n h a s a n e x p l i c i t s o l u t i o n g iv e n b y e q u a -

t i o n ( 7) o f S e c t i o n 1 4 .2 i n [1 0]. T r a n s f o r m i n g t h i s s o l u t i o n b a c k t o t h e o r i g i n a l f r a m e w o r k d e f i n e d

b y ( 1 ) , ( 6 ) , ( 7 ) , a n d ( 4 ) , o n e o b t a i n s

o z )

C( z , t )= exp ~ e x p ( - (~ + ~2) t) {F l ( z , t )+F2( z , t ) -Fa( z , t )+F4 ( z , t ) -F s( z , t )} , (13)

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34 R.S. ANDERSSEN et

al.

where

2 ~--D--~Et exp ~ - / ~ ) j exp

~ V ~ E ] I ( z ) d z ,

F2(z , t ) -- 2 ~x/r~-D~t fo exp 4 / ~ 7 ]J exp \ x / - ~ , ] f ( z ' ) d z ' ,

v/-D--~E exp t h: + V ~ E / er fc [2 DVFD~E + h exp \x/-D--~E]

F4 (z, t ) = h ~o

exp

( - - z 2/4DE (t - - T)) hC+(T)

~ - - ~ exp ( (# + a2) ~) ~ d~-,

F5( z,t ) = h 2 exp h2 (t-- T) + erfc 2v /D E( t_ T) + h

× e xp + . 2 )

- -~ - -E d r .

The analytic solution (5) can now be derived as a special case of (13) (cf. [11]).

R E F E R E N C E S

1. W.A. Jury, W.F. Spencer and W.J. Farmer, J. Environ. Qual. 12, 558-564, (1983).

2. R. Mayer, J. Letey and W.J. Farmer, Proc. Soil Sci. Soc. of Amer. 38, 563-568, (1974).

3. W.A. Jury, W.F. Spencer and W.J. Farmer, J. Environ. Qual. 13, 573-579, (1984).

4. W.A. Jury, D.D. Focht and W.J. Farmer, J. Environ. Qual. 16, 422-428, (1987).

5. W.A. Jury, D. Russo, G. Streile and H. E1 Abd,

Water Resources Research

26, 13-20, (1990).

6. P.F. Sanders and A.H. Stern, Environ. Toxic. and Chem. 13, 1367-1373, (1994).

7. W.A. Jury, W.F. Spencer and W.J. Farmer, J. Environ. Qual. 16, 448, (1987).

8. J.-S. Lin and L.M. Hildemann,

J. Hazardous Materials

40, 271-295, (1995).

9. J. Crank, The Mathematics of Diffusion, Oxford Science Publications, Oxford, (1990).

10. H.S. Carslaw and J.C. Jaeger,

Conduction of Heat in Solids,

Clarendon Press, Oxford, (1986).

11. R.S. Anderssen, Modelling the below-ground volatilization and diffusion of contaminants: Validation and

extension of the Jury et al. (1983, 1987) Model, CSIRO Division of Mathematics and Statistics Report

DMS-E-95/35, Canberra, (1995).