This document discusses the transport of contaminants in saturated media when injected as a slug or pulse. Figure 2.12 shows the spreading of a tracer injected as a slug into a two-dimensional flow field at three time increments, based on a laboratory experiment. The plume spreads with time in the direction of flow. Figure 2.13 shows the spreading of a chloride plume from a large-scale field test, which is more complex than the idealized laboratory plume. De Josselin De Jong derived an analytical solution for modeling the transport of a contaminant injected as a slug into a uniform two-dimensional flow field.

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V
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2
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T h e in te gr a lin th e a b o v e e q u a tio n w a s s o lv e d by H a n tu s h (1956)T h is s o lu tio n is
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o f W ttD ,
B ]c a n b e fo tın d in H a n tu s h (1956) a n d a lim ite d s e r ie s is ta b u la te d in A p
p e n d ix C In w e ll h y d r a u lic s th is is k n o w n a s th e le a k y w e ll fu n c tio n ,
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/ lb]
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ta n k g o e s d ir e c tly n o r th A d r in k in g w a te r s u p p ly w e ll is lo c a te d a t a s p o t th a t c a n
b e lo c a te d b y g o in g 12 3 5 m d u e n o r th o f th e le a k in g ta n k a n d th e n 7 2 m d u e e a s t
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lo n gitu d in a l d isp e r s iv ity is 12 m a n d tłie la te r a l d is p e r s iv ity is 1 2 m a n d th e a q u ife r
th ic k n e s s is 1 00 m W n a t w o u ld th e b e n e z e n e c o n c e n tr a tio n b e a fte r 2 0 0 y e a r s ?
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e q u a tio n 2 3 1
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fo i n r flo w fie ld in a s h o r t p e r io d o f tim e ,
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FIG U R E 2 12 ın ie c łio n o fa s lu g o Fa tra c e r in to a łwimensional flo w fie ld s ho w n a f thre e
tim e in c re m e n ts Ex p e rim e n ta ıre s u ıts fro m J Be a r,
JOUrnol o fG e o p hy s ic o ıRe s e a rc h 6 6 ,
n o 8
(19 6 1):2 4 5 5 6 7 C o py rig ht by the A m e ric a n G e o p hys ic a lU n io n
w ith thn e T his r e s u lt is i끄u s tr a te d b y F ig u r e 2 12 a n d r e p r e s e n ts th e pattern o f c o n ta
m in a tio n a t th r e e in c r e m e n ts th a t r e s u lt Ir o m a o n e tim e s p ill F rgu r e 2 12 is b a s e d o n
th e r e s u lts o f a la bo r a to ry e x p e rim e n t c o n d u c te d b y B e a r (1961) F ig u re 2 13 s h o w s th e
s p r e a d o f a p lu m e o f c h lo rid e th a t w a s in je c te d in to a n a q u ife r a s a p a r t o f a la rge s c a le
fie ld te s t (M a c k a y e t a l 1986) T h e p lu m e th a t r e su lte d fr o m th e fie ld te s t is m o r e
c o m p le x th a n th e la b o r a to ry p lu m e d u e to th e h e te r o g e n e itie s e n c o u n te r e d in th e r e a l
w o r ld a n d th e fa c t th e p iu m e m a y n o t b e fo llo w in g th e d iffu s io n a lm o d e lo f d is p e r s io n
D e Jo s s e lin D e Jo n g (1958)d e r iv e d a s o lu tio n to th is p r o b le m o n th e b a s is o f a
s ta tis tic a ltr e a tm e n t o f la te r a la n d tr a n s v e r s e d is p e r s iv itie s B e a r (196 1)la te r verified
it e x p e r im e n ta lly If a tr a c e r w ith c o n c e n tr a tio n C o
is in je c te d in to a tw o d im e n s io n
a l flo w fie ld o v e r a n a r e a A a t a p o in t (x o , } o),
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a t
tim e t a fte r th e in je c tio n is
C (x , y ,
t) =
4 r rt (D L
D r Y/2
e x p
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p R O B Į EM o v e r tu r n s a n d s p ills a v o lu m e o f w a te r s u ffic ie n t to s a tu r a te a th in a q u ife r o v e r a n
a r e a o f 5 n ı
2
T he a q u ife r c o n ta in s gr o u n d w a te r flo w in g w ith a n a v e r a ge lin e a r
v e lo c ity o f 0 4 5 m /d a y A s su m e th a t th e v a lu e s o f D L a n d o f D T a r e 2 1 m
2
/d a y a n d
0 2 1 m
2
/d a y r e s p e c tħtlly
T he re is a n e a r by priv a te w e lla t a s e a s o n a lc o tta ge If th e c e n te r o fth e sp illis a t lo
c a tio n x o
- O a n d yo
- O,
th e n th e lo c a ń o n o f th e w e llis x - 72 m a n d y - 5 5 m Æ e o w n
e rs o f th e priv a te w e ll a r e a w a y fo r th e s e a s o n a n d w illn o t ret urn fo r a n o th e r 200 d a ys
1f th e r e is n o d e gr a d a tio n o r r e ta r d a tio n o f th e b e n z e n e a s it m o v e s th r o u gh th e a q u ife r ,
w ha t w illth e c o n c e n tra tio n o f b e m e n e b e in th e p riv a te w e llw h e n th e o w n e r s return
T h e c o r r e c t e q u a tio n to u s e is 2 32 fo r a s lu g in je c tio n o f c o n ta m in a tio n in to a
tw o d im e n s io n a l a q u ife r

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C ha pte r Tw o
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th e c e n te r o f th e p lu m e ,
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th e c e n te r o f m a s s o f a c o n s e r v
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EX A M P LE A tr u c k c a r r y in g d ilu te b r in e w ith 2 13 0 m g/L c h lo r id e fr o m th e c le a n u p o f a p o n dPR O B LEM c o n ta in in g w a s te fr o m a p r o d u c in g o il w e ll overturns a n d s p ills th e d ilu te b r in eo v e r a n a r e a o f 455 s q u a r e fe e t T h e u n d e r ly in g th in a q u ife r h a s a n a v e r a g e lin e a r
g r o u n d w a te r v e lo c ity o f 1 2 3 fu d a y W U e r e w o u ld th e c e n te r o f m a s s o f th e p lu m eb e in 133 d a y s ,
w h a t w o u ld th e m a x im u m c o n c e n tr a tio n b e a n d h o w fa r b e y o n da n d to th e s id e o f th e c e n te r o f m a s s w o u ld th e p lţ ım e s p r e a d ?
1 T he p lu m e w o u ld b e a d v e c te d b y th e flo w in g gr o u n d w a te r s o th a t th e c e n te r o fï n a s s w o u ld b e a t x -
v x
t
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2 T he m a x im u m c o n c e n tr a tio n a t th e c e n te r o fm a s s c a n b e fo u n d fr o m e q u a tio n 2 32 a
C . .
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fir s t w e n e e d to u s E X o . 2 82 to e s tim a te o ĹL
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T he r a tio o f lo n gitu d in a lto tr a n s v e r s e d is p e r s iv ity (c TL /c ľ T iin a n a q u ife r is a n im p o r
ta n t c o n tr o l o v e r th e s h a p e o f a c o n ta m in a n t p lu m e in tw o d im e n s io n a l m a s s tr a n s
P o r t T h e lo w e r th e r a tio ,
the br o a d e r th e s h a p e o f th e r e s u ltin g p lu m e w ill b e F ig u r e
2 14 sh o w s v a r io u s tw o d im e n s io n a ls h a p e s o f a c o n ta m in a n t p lu m e ,
w h e r e th e o n ıyfa c to r v a r ie d w a s th e r a tio o f lo n gitu d in a l to tr a n s v e r s e d is p e r s iv ity T M s íu u s tr a te s
th e fa c t th a t it is im p o r ta n t to h a v e s o m e k n o w le d g e o f th e tr a n s v e r s e d is p e r s iv ity in
a d d itio n to th e lo n gitu d is a ld is p e r s iv ity T h e r e is a p a u c ity o f d a ta in th e ıite r a tu r e o n
th e r e la tio n s h ip s o f lo n gitu d in a lto tr a n s v e r s e d is p e r s iv 让他s F T o m th e fe w fie ld s tu d
ie s a v a ila b le ,
CtL " LT " in th e r a n ge o f 6 to 2 0 iA nd e r s o n 197 9 ,
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d is p e r s iv ity r a tio s b a s e d o n fie ld s tu d ie s a r e b a s e d o n fittin g th e d iffu s iu i国团团図
m o d e l o f d is p e r s io n to c a s e s w h e r e it m ig h t n o t b e a p p lic a b le

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Ex A M pLE A so ilc o lu m n is 40 c m lo n g It is se t u p s o th a t d e io n iz e d w a te r ís flo w in g ıiìro uÈpRo BLEM a t a n a v e r a ge lin e a r v e lo c ity o f 1 35 × 10 2
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a n d a fte r 4 000 s e c o n d s ,
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FıG U R E 2 9 D im e n s io n le s s typ e c u r v e s
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5 8 C o p y rig ht b A m e ric a n G e o p hys ic a lU n io n
Re s e a rc h 16 , n o (19 80 ):14 5
fo r the c o n tin u o u s in je c tio n o fa tra c e r in to a o n e di
to th e r a te o f tr a n s p o r t b y h y d r o d y n a m ic d is p e r s io n F o r p e c le t n u m b e r 1,
th e fix e d
s te p ħın c tio n a n d th e c o n tin u o u s in je c tio n fu n c tio n giv e qu ite d iffe r e n t r e s u lts,
w h e r e
m id w a y b e tw e e n th e o th e r tw o T h is figu r e s u gg e s ts th a t fo r p e c le t n u m b e r s te s s th a n
a b o u t 10 ,
th e e x a c t s o lu tio n s n e e d to b e c o n s id e r e d ,
w h e r e a s fo r p e c le t n u m be r sg r e a te r th a n 10 ,
th e a p p r o x im a te s o lu tio n is p r o b a b ly a c c e p ta b le ,
e sp e c ia lly a s th eP e c le t n u m b e r a p p r o a c h e s 100 T his P e c le t n u m b e r in c r e a s e s w ith flo w p a th le n gtha s a d v e c tiv e tr a n sp o r t b e c o m e s m o r e d o m in a n t o v e r d is p e r s iv e tr a n sp o r t T h u s fo rm a s s tr a n s p o r t n e a r th e in le t b o u n d a r y it is im p o r ta n t to u s e th e c o rr e c t e q u a tio n ,
b u ta s o n e g o e s a w a y fr o m th e in le t b o u n d a r % it is le s s im p o r ta n t th a t th e c o rr e c t fo r m
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o f c h lo r id e ín th e b r in e is 157 5 m g L 1
T h e flo w in th e a q u ife r th a t r e c e iv e s th e
b r in e is e s s e n tia lly o n e d im e n s io n a l a n d h a s th e fo llo w in g c h a r a c te r is tic s
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m s
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m
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ta n c e 12 5 m fr o m th e le a k in g v a t 0 50 ye a r s a fte r th e le a k b e g a n
W e w ill a p p ly e q u a tio n 2 2 5 to th is p r o b le m N o te th a t th is is a n a pp r o x im a te
s o lu tio n a s th e s e c o n d te rm o f e q u a tio n 2 2 3 h a s b e e n d r o p p e d
c =
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th e c o n c e n tr a tio n
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If a s lu g o f c o n ta m in a tio n is in s ta n ta n e o u s ly in je c te d in to a u n ifo r m ,
o n e d im e n s io n a l
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it w illp a s s th r o u gh th e a q u ife r a s a p u ls e w ith a p e a k c o n c e n tr a tio n ,
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a t s o m e tim e a fte r in je c tio n ,
im a x
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(S a u ty 19 80 )is in d im e n s io n le s s fo r m
C R (tR ,
P e ) = . . ( (1 tR Y (2 2 8 )
w ith
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EX A M P LE
P R O B LEM
A s s u m e a D o f 1 X 10 9
m
2
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to giv e a D *
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10 0 y r X 36 5 d a /y r x 144 0 m in /d a X 60 s e c /m in - 3 15 X 109
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(į _ e r fc 1 99 = 0 0 05
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15. Scanned by CamScanner
EX A M PLE D is s o lv e d n itr a te in a c o n c e n tr a tio n o f 18 0 m glL is b e in g a d v e c te d w ith flo w in g
PRO B LEM gr o u n d w a te ľ a t a v e lo c ity o f 0 33 1 m /d a y in a n a q u ife r w ith a p o r o s ity o f 0 2 2 5
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n itr a te in to th e s tr e a m if th e a q u ife r is 1 80 m th ic k a n d 123 m w id e w h e r e it
d is c h a r ge s in to th e s tr e a m ?
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3

16. Scanned by CamScanner
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