STORAGE
AND
HYDROLOGIC
PROCESSES
Don M. Gray
P r o f e s s o r of A g r i c u l t u r a l Engineering
and Chairman, D i v i s i o n of Hydrology,
D. I . Norum
A s s i s t a n t P r o f e s s o r of A g r i c u l t u r a l E n g i n e e r i n g ,
U n i v e r s i t y of Saskatchewan, Saskatoon.
I N T R O D U C T I O N
I n many of t h e preceding papers mention was made of s t o r a g e
terms i n t h e Water Balance e q u a t i o n .
S t o r a g e , t h e r e f o r e , i s a common
denominator i n t h e d i s c u s s i o n s and we wish t o r e f l e c t on i t s r o l e a s
i t a f f e c t s c e r t a i n hydrologic processes with s p e c i f i c reference t o the
Water Balance Regime.
Considering t h e hyd,rologic c y c l e a s a continuum, one can
w r i t e a s i m p l e mass b a l a n c e o r c o n t i n u i t y e q u a t i o n t o d e s c r i b e t h e
system i n t h e form
i n which
I = t h e inflow r a t e ,
,
0 = the outflow r a t e ,
AS = change i n s t o r a g e volume which o c c u r s
w i t h i n some time i n c r e m e n t , A t .
From e q u a t i o n ( 1 ) i t i s o b v i o u s t h e i m p o r t a n c e of S t o r a g e i n t h e
h y d r o l o g i c s y s t e m i s t h a t c h a n g e s i n I o r 0 must b e b a l a n c e d by a
c o r r e s p o n d i n g change i n t h e t e r m & / A t .
C o w e r s e l y , any c h a n g e i n
/
b / A t must c a u s e a c o r r e s p o n d i n g change i n t h e d i f f e r e n c e I
-
0.
The
m a g n i t u d e of t h e changes i n t h e t e r m s , I and 0 , o f t h e h y d r o l o g i c
s y s t e m c a u s e d by a change i n
As/At
w i l l be discussed i n subsequent
paragraphs.
PRECIPITATION
S i n c e p r e c i p i t a t i o n u s u a l l y r e p r e s e n t s t h e m a j o r component
o f i n f l o w i n any s p a t i a l w a t e r b a l a n c e e v a l u a t i o n , i t i s o n l y appropr i a t e t h a t a somewhat q u a l i t a t i v e e v a l u a t i o n o f t h e i n t e r a c t i o n between
p r e c i p i t a t i o n and S t o r a g e ( w a t e r v a p o r i n t h e atmosphere) b e d i s c u s s e d .
I n t h e p r e c i p i t a t i o n p h a s e ( a t m o s p h e r i c p h a s e ) of t h e h y d r o l o g i c c y c l e
t h e r e o c c u r s v a p o r f l o w and v a p o r s t o r a g e , and c o n d e n s a t i o n o r f o r m a t i o n
o f p r e c i p i t a t i o n c r e a t e d by a change from v a p o r t o l i q u i d o r s o l i d s t a t e .
The p r i m a r y f a c t o r s needed t o p r o d u c e p r e c i p i t a t i o n a r e :
(a)
w a t e r i n e i t h e r vapor o r s o l i d ( i c e ) s t a t e ,
(b)
c o n d e n s a t i o n o r s u b l i m a t i o n n u c l e i , and
(c)
c o o l i n g mechanisms :
f o r example by a d i a b a t i c e x p a n s i o n
of a i r t h r o u g h l i f t i n g t o e l e v a t i o n s o f l o w e r p r e s s u r e
and t e m p e r a t u r e .
The m o i s t u r e c h a r g e c o n t a i n e d w i t h i n a v e r t i c a l column of
a i r o f u n i t c r o s s - s e c t i o n a l a r e a can b e e v a l u a t e d by -
i n which
W = moisture charge,
q = s p e c i f i c h u m i d i t y o f t h e column,
p
= mass d e n s i t y ,
g = a c c e l e r a t i o n of g r a v i t y
Pl,P2 = atmospheric p r e s s u r e s a t t h e top
and b o t t o m o f t h e columns.
I n theory t h i s moisture charge represents t h e p o t e n t i a l "Precipitable
Water"
-
even though i t i s r e c o g n i z e d t h a t n o n a t u r a l p r o c e s s w i l l
remove a l l w a t e r v a p o r from t h e a i r .
T a b l e s and nomographs p r e p a r e d
by F e r g u s o n ( 2 ) , P e t e r s o n (12) and t h e U.S. Weather Bureau (16) f a c i l i -
t a t e t h e c o m p u t a t i o n o f p r e c i p i t a b l e w a t e r from a t m o s p h e r i c measurements.
The amount of t h i s w a t e r r e l e a s e d by a p a r t i c u l a r storrn i s p r i m a r i l y
a f u n c t i o n o f t h e r a t e of a t m o s p h e r i c l i f t i n g - a p r o c e s s which i s
i n s u f f i c i e n t l y understood.
I t i s s u f f i c e t o c o n c l u d e however, t h a t f o r
s t o r m s h a v i n g t h e same m e c h a n i c a l e f f i c i e n c y ( i n which t h e r a t i o o f t h e
a c t u a l p r e c i p i t a t i o n , P t o t h e p r e c i p i t a b l e w a t e r i s c o n s t a n t : i . e . P/W =
c o n s t a n t ) t h a t t h e amount of p r e c i p i t a t i o n w i l l i n c r e a s e d i r e c t l y w i t h
t h e mass o f w a t e r i n s t o r a g e i n t h e atmosphere.
At t h e same t i m e , i t
must b e r a t i o n a l i z e d t h a t t h e r e i s a n u p p e r l i m i t of t h e r a t i o P/W.
R e g i o n a l l y , i t i s w e l l documented t h a t r e c o r d p r e c i p i t a t i o n
amounts h a v e b e e n r e c o r d e d a t l o c a t i o n s which a r e a d j a c e n t t o a d e q u a t e
- 4 m o i s t u r e s o u r c e s where w a t e r can b e o b t a i n e d through e v a p o r a t i o n t o
s u f f i c e t h e s t o r a g e c a p a c i t y o f t h e p r e v a i l i n g a i r mass.
Similarly,
under s e m i - a r j d c o n d i t i o n s , such a s t h o s e e n c o u n t e r e d on t h e Canadian
P r a i r i e s , i n t h e summer, l a r g e r p r e c i p i t a t i o n amounts a r e o f t e n
r e c o r d e d d u r i n g wet y e a r s o r shower a c t i v i t y o f t e n f o l l o w s r a i n y
periods.
The r e a s o n i s obvious inasmuch a s g r e a t e r e v a p o r a t i o n and
e v a p o t r a n s p i r a t i o n t a k e s p l a c e and' t h e atmosphere's c a p a c i t y f o r holdi n g w a t e r ( s t o r a g e ) i s g r e a t e s t i n summer.
The n e t e f f e c t i s a n
increase i n precipitation.
I n a n a t t e m p t t o summarize t h i s v e r y b r i e f d i s c u s s i o n of t h e
i n t e r a c t i o n between s t o r a g e and p r e c i p i t a t i o n r e f e r e n c e i s made t o t h e
Depth
-
D u r a t i o n Curves of Extreme R a i n f a l l s , (See F i g . 1 - t a k e n from
McKay ( 1 1 ) ) .
On t h i s f i g u r e two r e p r e s e n t a t i v e c u r v e s h a v e been drawn
t o show World and P r a i r i e c o n d i t i o n s .
Although i t would b e e r r o n e o u s
t o a t t a c h g r e a t numeric s i g n i f i c a n c e t o t h e v a l u e s g i v e n by t h e ' c u r v e s ;
t h e d i f f e r e n c e i n s h a p e s adopted by them a r e p e r t i n e n t t o t h e d i s c u s s i o n s .
(1)
For s h o r t - d u r a t i o n s t o r m s ( l e s s t h a n a p p r o x i m a t e l y 1 - h r . )
t h e s h a p e s and magnitudes of r a i n f a l l extremes g i v e n by t h e
two c u r v e s do n o t d i f f e r a p p r e c i a b l y .
That i s , r e g i o n a l
w a t e r s u p p l i e s i n s e m i - a r i d a r e a s may s u f f i c e t o s u p p l y
a d e q u a t e w a t e r ( t h r o u g h e v a p o r a t i o n and e v a p o t r a n s p i r a t i o n )
which can produce l o c a l i z e d c o n v e c t i v e s t o r m s which w i l l
produce p o i n t p r e c i p i t a t i o n amounts which compare f a v o r a b l y
w i t h World Extremes.
(2)
For l o n g e r d u r a t i o n s t o r m s ( g r e a t e r t h a n 1 - 2 h o u r s )
t h e two c u r v e s d i f f e r markedly i n b o t h t h e s h a p e s and v a l u e s .
E x p l a i n a b l y under P r a i r i e c o n d i t i o n s w a t e r i s n o t a v a i l a b l e
t o s u f f i c e t h e p o t e n t i a l s t o r a g e r e q u i r e m e n t s o r t h e atmosp h e r i c c a r r y i n g c a p a c i t y and t h u s t h e amounts of p r e c i p i t a t i o n s
w i l l b e lower t h a n World Extremes.
In these regards, the
l o c a t i o n s o f measurements of World Extremes a r e c l o s e t o t h e
ocean and u n p r o t e c t e d from l a r g e t o p o g r a p h i c d i v i d e s .
INFILTRATION
The p r o c e s s of i n f i l t r a t i o n i s by d e f i n i t i o n t h e e n t r y of
w a t e r i n t o t h e s o i l through t h e s o i l - a t m o s p h e r e i n t e r f a c e .
This process,
l i k e many o t h e r f l o w p r o c e s s e s i n porous media, i s governed by t h e
Richards s o i l m o i s t u r e d i f f u s i o n e q u a t i o n ,
i n which
9 = the volumetric moisture content,
k = the capillary conductivity,
and
@ = the t o t a l potential.
E q u a t i o n ( 3 ) , i s t h e c o n t i n u i t y e q u a t i o n f o r flow which h a s t h e f l u x ,
V, a t any p o i n t d e f i n e d by t h e Darcy e q u a t i o n ,
I t is e v i d e n t from e q u a t i o n (4) t h a t t h e f l u x a t any p o i n t i n a s o i l
system, including t h e s o i l s u r f a c e , is p r o p o r t i o n a l t o t h e h y d r a u l i c
o r c a p i l l a r y c o n d u c t i v i t y , k , and t h e t o t a l p o t e n t i a l g r a d i e n t , V@.
T h e r e f o r e , t h e i n f i l t r a t i o n p r o c e s s w i l l b e a f f e c t e d by any f a c t o r
which a f f e c t s e i t h e r o f t h e s e two q u a n t i t i e s ; one of t h e most importa n t of t h e s e i s t h e s o i l m o i s t u r e c o n t e n t .
One of t h e most s i g n i f i c a n t c o n t r i b u t i o n s t o u n d e r s t a n d i n g
i n f i l t r a t i o n and t h e c o n s e q u e n t e f f e c t of s o i l m o i s t u r e s t o r a g e on t h e
p r o c e s s was g i v e n by P h i l i p ( 1 3 ) .
H e presented t h e s o l u t i o n t o the
d i f f u s i o n e q u a t i o n ( e q u a t i o n 3 ) f o r one-dimensional v e r t i c a l i n f i l t r a t i o n i n t o a u n i f o r m , s e m i - i n f i n i t e medium, i n i t i a l l y a t a c o n s t a n t
moisture content.
P h i l i p ' s s o l u t i o n f o r t h e mass i n f i l t r a t i o n ,
Mf'
as a f u n c t i o n of t i m e , t , was
where t h e c o e f f i c i e n t s a can b e c a l c u l a t e d from t h e c a p i l l a r y conductm
i v i t y and c a p i l l a r y d i f f u s i v i ' t y c u w e s and t h e i n i t i a l m o i s t u r e c o n t e n t
of t h e s o i l .
The s i g n i f i c a n c e of P h i l i p ' s e q u a t i o n i s t h a t by t h e
r e l a t i o n s h i p s h e p r e s e n t e d , t h e q u a n t i t a t i v e e f f e c t of t h e i n i t i a l
m o i s t u r e c o n t e n t of t h e s o i l c a n b e e v a l u a t e d .
To e x e m p l i f y t h i s f a c t ,
t h e method w a s u s e d t o c a l c u l a t e t h e mass i n f i l t r a t i o n f o r a sandy loam
s o i l i n i t i a l l y a t two d i f f e r e n t m o i s t u r e l e v e l s ; 0 . 0 3 cm3/cm3 and 0 . 2 3
crn3/cm3.
el
The mass i n f i l t r a t i o n e q u a t i o n o b t a i n e d were:
= 0 . 0 3 cm3/cm3
where M i s i n c e n t i m e t e r s when t i s i n m i n u t e s .
f
It s h o u l d b e n o t e d
t h a t e q u a t i o n s (6) and (7) w i l l d i v e r g e from t h e t r u e s o l u t i o n when t
i s l a r g e as o n l y t h r e e t e r m s o f t h e i n f i n i t e s e r i e s h a v e b e e n u s e d .
The c o e f f i c i e n t s of t h e s e e q u a t i o n s r e f l e c t t h e r e l a t i v e
e f f e c t s t h a t t h e c a p i l l a r y and t h e g r a v i t a t i o n a l f o r c e s h a v e on t h e
i n f i l t r a t i o n process.
That is, t h e f i r s t term ( t
i s used t o
d e s c r i b e h o r i z o n t a l f l o w and t h u s t h e c o e f f i c i e n t a t t a c h e d t o t h i s t e r m
e v a l u a t e s t h e e f f e c t of c a p i l l a r y f o r c e s .
Similarly, the coefficients
a s s i g n e d t o t h e o t h e r t e r m s of t h e e q u a t i o n show t h e e f f e c t of g r a v i t y .
Thus, i t c a n b e o b s e r v e d i n e q u a t i o n s ( 6 ) and ( 7 ) t h a t t h e n e t e f f e c t
on t h e i n f i l t r a t i o n p r o c e s s of i n c r e a s i n g t h e i n i t i a l s o i l m o i s t u r e
c o n t e n t i s t o d e c r e a s e t h e e f f e c t of c a p i l l a r y f o r c e s and i n c r e a s e t h e
e f f e c t of g r a v i t y .
T h i s comes a b o u t b e c a u s e o f t h e d e c r e a s e i n c a p i l -
l a r y g r a d i e n t and t h e i n c r e a s e i n c a p i l l a r y c o n d u c t i v i t y of t h e s o i l
w i t h an i n c r e a s e i n moisture content.
As P h i l i p ' s s o l u t i o n a l s o g i v e s t h e m o i s t u r e c o n t e n t a s a
f u n c t i o n of d e p t h and t i m e , f o u r example " w e t t i n g f r o n t " p a t t e r n s h a v e
b e e n p l o t t e d i n F i g u r e 2.
These p a t t e r n s a r e f o r t h e two p r e v i o u s l y
assumed i n i t i a l m o i s t u r e c o n t e n t s , 0 . 0 3 and 0.23, and f o r two t i m e s ,
60 m i n u t e s and 240 m i n u t e s .
From F i g u r e 2 i t i s a p p a r e n t t h a t ,
(a)
Increasing t h e i n i t i a l s o i l moisture content increases
t h e v e l o c i t y a t which t h e w e t t i n g f r o n t moves b u t
d e c r e a s e s t h e i n f i l t r a t i o n r a t e , and
(b)
The i n i t i a l m o i s t u r e c o n t e n t of t h e s o i l a f f e c t s t h e
khape of t h e m o i s t u r e d i s t r i b u t i o n p r o f i l e , e s p e c i a l l y
a t s h o r t times a f t e r w e t t i n g .
T h i s l a t t e r c h a r a c t e r i s t i c can a l s o b e e x p l a i n e d by changes i n t h e
d i f f u s i v i t y and p o t e n t i a l g r a d i e n t a s s o c i a t e d w i t h t h e d i f f e r e n t i n i t i a l
s o i l moisture contents.
This i s , s i n c e a t t h e higher moisture content
t h e d i f f u s i v i t y i s more n e a r l y c o n s t a n t and t h e m o i s t u r e g r a d i e n t a c r o s s
t h e wet f r o n t i s s m a l l e r ; t h e wet f r o n t i s l e s s a b r u p t .
I n t h e example c a l c u l a t i o n , t h e q u a n t i t a t i v e e f f e c t s of
changes i n i n i t i a l s o i l m o i s t u r e on mass i n f i l t r a t i o n have been demonstrated.
However, t h e s e c a l c u l a t i o n s can o n l y b e completed when t h e
coefficients a
m
of e q u a t i o n (5) a r e e x p l i c i t l y known.
E v a l u a t i o n of
t h e s e c o e f f i c i e n t s i s u s u a l l y an arduous and d i f f i c u l t t a s k , hence, a s
a n a l t e r n a t i v e method f o r d e t e r n t i n i n g
t h e e f f e c t of t h e i n i t i a l s o i l
m o i s t u r e c o n t e n t on t h e i n f i l t r a t i o n r a t e , P h i l i p (14) s u g g e s t e d t h a t
f o r s h o r t times a f t e r i n f i l t r a t i o n has s t a r t e d , t h e i n f i l t r a t i o n r a t e
of s o i l , f , v a r i e s a p p r o x i m a t e l y a s t h e s q u a r e r o o t of t h e d i f f e r e n c e
between t h e s u r f a c e m o i s t u r e c o n t e n t , 8 and t h e i n i t i a l s o i l m o i s t u r e
n
c o n t e n t of t h e s o i l p r o f i l e 0
i
(capillary forces).
That i s ,
-
9
-
A f t e r l o n g t i m e s , the i n f i l t r a t i o n r a t e becomes i n d e p e n d e n t o f t h e
i n i t i a l m o i s t u r e c o n t e n t b e c a u s e t h e g r a d i e n t i n t h e upper r e g i o n
approaches u n i t y and t h e i n f i l t r a t i o n r a t e approaches t h e c a p i l l a r y
c o n d u c t i v i t y f o r t h e zone.
Holtan ( 4 ) r e p o r t e d a s i m i l a r approach t o
d e f i n e t h e i n f i l t r a t i o n r a t e of a s o i l a s a f u n c t i o n of t h e e x h a u s t i o n
of s o i l m o i s t u r e s t o r a g e .
i n which
The e x p r e s s i o n used was
S = p o t e n t i a l s o i l m o i s t u r e s t o r a g e volume o r
t h e v o l u m e t r i c d i f f e r e n c e between p o r e s a t u r a t i o n
and t h e 15-bar o r permanent w i l t i n g p e r c e n t a g e
f o r t h e s o i l zone above t h e c o n t r o l l a y e r .
f
C
= f i n a l c o n s t a n t r a t e of i n f i l t r a t i o n through t h e
c o n t r o l h o r i z o n , and
a,n = constants f o r a p a r t i c u l a r s o i l i n given condition
(according t o P h i l i p n = 112).
I n e q u a t i o n (9), t h e e f f e c t of i n c r e a s i n g t h e mass i n f i l t r a t i o n ,
Mf,
i s analogous t o i n c r e a s i n g t h e ( i n i t i a l ) s o i l m o i s t u r e c o n t e n t which
w i l l cause a decrease i n t h e i n f i l t r a t i o n r a t e .
A s p o i n t e d o u t by
H o l t a n , one i m p o r t a n t a s p e c t of e q u a t i o n (9) a s a p p l i e d t o h y d r o l o g i c
a n a l y s e s i s t h a t by s u b d i v i d i n g t h e s t o r a g e p o t e n t i a l i n t o t h e f r e e o r
g r a v i t a t i o n a l w a t e r volume and t h e c a p i l l a r y w a t e r volume t h e i n f i l t r a t i o n r e c o v e r y between r a i n p e r i o d s can be computed.
In this calculation
i t i s u s u a l l y assumed t h a t t h e f r e e w a t e r i s removed a t t h e r a t e of
g r a v i t y f l o w ( p e r h a p s f ) and t h a t t h e a v a i l a b l e w a t e r c a p a c i t y i s
C
d e p l e t e d a t a s l o w e r r a t e of e v a p o t r a n s p i r a t i o n .
Another f e a t u r e
shown i n t h e c q u a t i o i l i s t h a t when t h e mass i n f i l t r a t i o n , M f , e q u a l s
t h e moisture s t o r a g e , S, t h e i n f i l t r a t i o n r a t e is equal t o t h e t r a n s mission rate through t h e c o n t r o l l a y e r .
EVAPOTRANSPIRATION
E v a p o t r a n s p i r a t i o n i n c l u d e s t h e sums of volumes of w a t e r u s e d
i n e v a p o r a t i o n from t h e s o i l and t r a n s p i r a t i o n from t h e p l a n t .
With
t r a n s p i r a t i o n , a c o n t i n u o u s f l o w o f w a t e r o c c u r s from t h e s o i l m o i s t u r e
r e s e r v o i r t h r o u g h t h e s o i l and r o o t s o f t h e p l a n t s t o t h e l e a v e s where
i t i s t r a n s f o r m e d by s o l a r e n e r g y i n t o w a t e r v a p o r .
T h i s movement
t a k e s p l a c e due t o p o e n t i a l g r a d i e n t s ; t h e s u c t i o n g r a d i e n t ( n e g a t i v e
p r e s s u r e ) w i t h i n t h e s o i l and g r a d i e n t s of d i f f u s i o n p r e s s u r e w i t h i n
the plant.
H e r e i n , one i s c o n c e r n e d w i t h t h e i n t e r a c t i o n of S t o r a g e and
E v a p o t r a n s p i r a t i o n and t h u s , t h e f o l l o w i n g d i s c u s s i o n s w i l l b e l i m i t e d
t o r o l e of s o i l m o i s t u r e c o n t e n t i n t h e p r o c e s s .
Gardner (3) p r e s e n t e d
a mathematical s o l u t i o n t o t h e flow of w a t e r i n s o i l s d u r i n g t r a n s p i r a tion.
He found t h a t t h e s u c t i o n g r a d i e n t between r o o t and s o i l n e c e s s a r y
t o m a i n t a i n a g i v e n r a t e of w a t e r u p t a k e i s p r o p o r t i o n a l t o t h e p o t e n t i a l
e v a p o t r a n s p i r a t i o n demand and i n v e r s e l y p r o p o r t i o n a l t o t h e c a p i l l a r y
conductivity, k.
Inasmuch a s w i t h d e c r e a s i n g s o i l m o i s t u r e c o n t e n t
( s t a r t i n g a t a high i n i t i a l value), t h e capillary conductivity decreases
and s o i l s u c t i o n ,
Q, i n c r e a s e s v e r y r a p i d l y w i t h s o i l m o i s t u r e i t would
b e e x p e c t e d t h a t t h e e v a p o t r a n s p i r a t i o n r a t e would c o r r e s p o n d i n g l y
decrease.
P h i l i p (15) showed by a mathematical s o l u t i o n of t h e
e v a p o r a t i o n p r o c e s s , t h a t e v a p o r a t i o n from an i n i t i a l l y s a t u r a t e d s o i l
w i t h no u n d e r d r a i n a g e followed a c h a r a c t e r i s t i c p a t t e r n showing a cons t a n t r a t e phase followed by an e x p o n e n t i a l l y - s h a p e d f a l l i n g r a t e phase.
I n t h e t r a n s p i r a t i o n p r o c e s s , assuming a c o n s t a n t p o t e n t i a l
demand, a d e c r e a s e i n s o i l m o i s t u r e c o n t e n t (and subsequent changes i n k
and $) must b e compensated by an i n c r e a s e i n t h e t u r g o r p r e s s u r e d e f i c i t
within the plant.
This c a u s e s a d e c r e a s e i n t h e t u r g o r p r e s s u r e and
c l o s i n g of t h e s t o m a t a which r e s t r i c t s t h e flow of w a t e r i n p l a n t s causi n g t h e t r a n s p i r a t i o n r a t e t o decrease.
Thus, i t would b e expected t h a t
t r a n s p i r a t i o n r a t e s would d e c l i n e w i t h d e c r e a s i n g m o i s t u r e c o n t e n t and
t h a t t h e r a t e of d e c l i n e would b e a f u n c t i o n of s o i l p r o p e r t i e s and t h e
s o i l moisture-tension
relationship.
Denmead and Shaw (1) r e p o r t e d on an e x t e n s i v e s t u d y of t h e
r e l a t i o n s h i p between a c t u a l and p o t e n t i a l t r a n s p i r a t i o n r a t e s f o r c o r n
f o r d i f f e r e n t p o t e n t i a l demand c o n d i t i o n s .
a r e summarized i n F i g . 3 .
The f i n d i n g s of t h i s s t u d y
From t h e d a t a g i v e n i n t h e f i g u r e i t c a n b e
observed t h a t ; ( a ) under c o n d i t i o n s of h i g h p o t e n t i a l e v a p o t r a n s p i r a t i o n
demands, t h e a c t u a l t r a n s p i r a t i o n r a t e may b e c o n s i d e r a b l y l e s s t h a n t h e
p o t e n t i a l , even though t h e s o i l m o i s t u r e supply may b e c o n s i d e r e d
adequate,
(b) f o r low p o t e n t i a l e v a p o t r a n s p i r a t i o n r a t e s t h e a c t u a l
t r a n s p i r a t i o n r a t e w i l l e q u a l t h e p o t e n t i a l r a t e s down t o v e r y low
s o i l m o i s t u r e l e v e l s and
( c ) t h e w i d e r a n g e i n s h a p e s of t h e c u r v e s
s u g g e s t t h a t d i f f e r e n t models may b e u s e d w i t h e q u a l s u c c e s s i n modul a t e d s o i l m o i s t u r e budgets t o d e f i n e s o i l m o i s t u r e withdrawal depending
on t h e p r e v a i l i n g c o n d i t i o n s .
A n o t h e r i m p o r t a n t r o l e of t h e s o i l m o i s t u r e regime a s r e l a t e d
t o e v a p o t r a n s p i r a t i o n i s i t s e f f e c t on t h e r o o t - d i s t r i b u t i o n
t h e moisture withdrawal p a t t e r n .
and t h u s ,
I f t h e s o i l m o i s t u r e t e n s i o n i s uni-
f o r m t h r o u g h o u t t h e p r o f i l e , t h e n t h e r a t e of removal of w a t e r i s
p r o p o r t i o n a l t o t h e amount of r o o t a c t i v i t y .
Correspondingly, t h e
amount of r o o t growth i s h i g h l y c o r r e l a t e d w i t h s o i l m o i s t u r e ; t h e
w e t t e r t h e c o n d i t i o n s , t h e more p r o f u s e t h e r o o t s y s t e m i s a t s h a l l o w
depths.
D u r i n g t h e i n i t i a l p e r i o d s of e v a p o t r a n s p i r a t i o n f o l l o w i n g a
r a i n , s o i l m o i s t u r e i s removed p r i m a r i l y from t h e s u r f a c e l a y e r s and
progressively t o g r e a t e r depth.
Any d e f i c i e n c y i n r o o t development
w i t h d e p t h w i l l c o n s e q u e n t l y ' c a u s e t h e r a t i o of a c t u a l t o p o t e n t i a l
e v a p o t r a n s p i r a t i o n t o decrease w i t h time.
OVERLAND
FLOW
AND
GROUNDWATER
The e f f e c t of s t o r a g e , e i t h e r g r o u n d w a t e r , s u r f a c e , c h a n n e l
o r r e s e r v o i r s t o r a g e , on t h e o u t p u t h y d r o g r a p h of a b a s i n i s t o modera t e o r dampen t h e i n p u t f l u c t u a t i o n s .
Consideration h e r e w i l l be given
t o g r o u n d w a t e r and s u r f a c e s t o r a g e where s u r f a c e s t o r a g e i s t h e w a t e r
s t o r e d on t h e s u r f a c e w h i l e o v e r l a n d f l o w i s t a k i n g p l a c e . .
I z z a r d (6) d e v e l o p e d a d i m e n s i o n l e s s h y d r o g r a p h f o r o v e r l a n d
f l o w i n which a d i m e n s i o n l e s s o u t p u t , q / i , was p l o t t e d a g a i n s t a
d i m e n s i o n l e s s t i m e , t / t e , where:
i = steady s t a t e input ( i . e .
precipitation rate),
q = o u t p u t ( s u r f a c e d e p t h p e r u n i t of t i m e ) ,
t = t i m e , and
t e = t i m e a t which q / i = - 9 7 .
I n e f f e c t te i s t h e time r e q u i r e d f o r t h e s u r f a c e s t o r a g e t o reach a
maximum v a l u e .
t
e
E x p e r i m e n t s by I z z a r d showed t h a t f o r s o d c o n d i t i o n s
c o u l d b e e v a l u a t e d by
where
L = d i s t a n c e o f o v e r l a n d f l o w , and
S = surface slope.
The r e c e s s i o n p o r t i o n of t h e h y d r o g r a p h c a n b e p l o t t e d a s a
f u n c t i o n o f t h e same p a r a m e t e r s .
F i g u r e 4 shows t h e d i m e n s i o n l e s s
h y d r o g r a p h g i v e n by I z z a r d w i t h t h e r e c e s s i o n s t a r t i n g a t t / t
K r a i j e n h o f f v a n d e L e u r ( 7 ) , (8)
,
e
=
1.
(9) d e v e l o p e d and t e s t e d
a method f o r r o u t i n g g r o u n d w a t e r t h r o u g h s t o r a g e t o o u t p u t i n t o a
d i t c h o r channel.
S o l u t i o n o f t h e l i n e a r i z e d Boussine.sq e q u a t i o n f o r
t h i s two d i m e n s i o n a l s y s t e m r e s u l t e d i n t h e e q u a t i o n s f o r d i s c h a r g e a s :
where j i s known a s t h e r e s e r v o i r c o e f f i c i e n t and h a s t h e d i m e n s i o n of
time.
The p a r a m e t e r j c o n t a i n s a l l t h e h y d r o l o g i c p r o p e r t i e s o f t h e
s y s t e m and i s g i v e n by
where
p
=
active porosity o r specific yield,
L = h o r i z o n t a l l e n g t h of flow,
K = , h y d r a u l i c c o n d u c t i v i t y , and
D = d e p t h t o t h e impermeable l a y e r .
As t h e l i n e a r i z e d B o u s s i n e s q e q u a t i o n makes u s e o f t h e f a m i l i a r DupuitF o r c h e i m e r a s s u m p t i o n of h o r i z o n t a l f l o w , t h e Houghoudt (5) c o r r e c t i o n
f o r d e p t h t o impermeable l a y e r must b e u s e d t o o b t a i n a c o r r e c t e d D
value.
It c a n b e n o t e d from e q u a t i o n (11) t h a t f o r e i t h e r s m a l l
v a l u e s of t o r l a r g e v a l u e s of j , q / i a p p r o a c h e s z e r o , o r t h e o u t f l o w
approaches zero; conversely, i f j i s small o r t i s l a r g e q / i approaches
u n i t y and t h e i n p u t i s r o u t e d d i r e c t l y t h r o u g h t h e g r o u n d w a t e r s t o r a g e .
As e q u a t i o n (11) i s a s o l u t i o n of a l i n e a r 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 , t h e o u t p u t h y d r o g r a p h f o r any s t e p w i s e i n p u t c a n b e d e v e l o p e d
by s u p e r p o s i t i o n .
For example, i f t h e i n p u t was i f o r a t i m e p e r i o d b
-
15
-
and z e r o t h e r e a f t e r t h e o u t p u t would b e
4 = 8i
n
"'
for t
m
1
2 exp
C
= 1,3,5
...
( n 2 b / j ) exp ( - n 2 t / j )
n
(13)
b.
F i g u r e 4 c o n t a i n s t h e d i m e n s i o n l e s s a d v a n c e and r e c e s s i o n
h y d r o g r a p h f o r a c o n s t a n t i n p u t i up t o t i m e b = j and i
after.
=
0 there-
The c o n s t a n t i n p u t f o r t i m e b = j i s p r o b a b l y u n r e a l i s t i c as
j i s o f t e n q u i t e l a r g e u s u a l l y i n terms of d a y s ( s e e t h e f o l l o w i n g
example).
However, t h e i n i t i a l p o r t i o n of t h e o u t p u t h y d r o g r a p h can
b e u s e d f o r p u r p o s e s of s u p e r p o s i t i o n i n g a s e r i e s o f o u t p u t s f o r any
s t e p w i s e s e r i e s of i n p u t s .
From e q u a t i o n (13) i t i s a p p a r e n t t h a t when t becomes l a r g e
r e l a t i v e t o b , o n l y t h e f i r s t t e r m of t h e s e r i e s w i l l b e s i g n i f i c a n t .
C o n s e q u e n t l y , i f q / i v e r s u s t i s p l o t t e d on s e m i l o g a r i t h m i c g r a p h
p a p e r t h e s l o p e of t h e " t a i l ' : p o r t i o n ( t l a r g e ) w i l l b e - l / j .
This
a f f o r d s a r e l a t i v e l y s i m p l e method o f o b t a i n i n g t h e r e s e r v o i r c o e f f i c i e n t j w i t h o u t a c t u a l l y m e a s u r i n g i t s i n d i v i d u a l components.
I t s h o u l d b e n o t e d t h a t t h e g r o u n d w a t e r h y d r o g r a p h may b e
a l t e r e d somewhat b e c a u s e o f t h e l a g t i m e which o c c u r s between w a t e r
e n t e r i n g t h e s o i l s u r f a c e ( i n f i l t r a t i o n t a k e s p l a c e ) and w a t e r r e a c h ing the phreatic surface.
T h i s l a g may b e e s t i m a t e d from t h e i n f i l t r a -
t i o n c h a r a c t e r i s t i c s o f t h e s o i l as shown p r e v i o u s l y .
However, i n
most c a s e s i t w i l l b e n e g l i g i b l e compared t o t h e e f f e c t of j .
A
f u r t h e r c o m p l i c a t i o n may o c c u r when t h e l a n d i s s l o p i n g , a s K r a i j e n h o f f
v a n de L e u r ' s
(7) a n a l y s i s i s o n l y f o r l a n d w i t h z e r o s l o p e ; however,
L u t h i n and G u i t j e n s (10) h a v e shown t h a t t h e r e i s l i t t l e d i f f e r e n c e
between f l o w p a t t e r n s when t h e l a n d s l o p e s a s much a s 30% a s when t h e
land has zero slope.
Although t h e d i m e n s i o n l e s s h y d r o g r a p h s shown i n F i g u r e 4 do
i n d i c a t e t h e m o d e r a t i n g e f f e c t of g r o u n d w a t e r s t o r a g e compared t o
o v e r l a n d f l o w , t h i s e f f e c t c a n b e e x e m p l i f i e d much more e m p h a t i c a l l y
by comparing v a l u e s of t
e
and j f o r a t y p i c a l f i e l d c o n d i t i o n .
Suppose :
Then t
e
= 4 . 2 h r and j = 1626 h r o r 68 d a y s .
Consequently, t h e time
s c a l e s f o r t h e o v e r l a n d f l o w h y d r o g r a p h and t h e groundwater h y d r o g r a p h
would d i f f e r by a f a c t o r o f a p p r o x i m a t e l y 400.
-
R E F E R E N C E S
(1)
enme mead,
(2)
F e r g u s o n , H. L. A tephegram o v e r l a y f o r computing p r e c i p i t a b l e
w a t e r . Canada Dept. T r a n s p o r t , M e t e o r o l o g i c a l
Branch Tech. C i r c . 409, 1 2 pp. 1 9 6 2 .
(3)
G a r d n e r , W. R.
Some s t e a d y s t a t e s o l u t i o n s of t h e u n s a t u r a t e d
m o i s t u r e flow e q u a t i o n s w i t h a p p l i c a t i o n t o evaporat i o n from a w a t e r t a b l e . S o i l S c i . 85:228-232, 1958.
(4)
Holtan, H. N.
A concept f o r i n f i l t r a t i o n e s t i m a t e s i n watershed
e n g i n e e r i n g . USDA-ARS 41-51, Washington, D . C . , 1961.
(5)
Hooghoudt, S. B. B i j d r a g e n t o t d e k e n n i s v a n e e n i g e n a t u u r k u n d i g e
g r o o t h e d e n van den g r o n d . V e r s l a g . Landbouwk
Onderzoek. No. 46. 1 9 4 0 .
(6)
I z z a r d , C . F.
(7)
K r a i j e n h o f f v a n d e L e u r , D. A. A s t u d y o f non-steady g r o u n d w a t e r
flow w i t h s p e c i a l r e f e r e n c e t o a r e s e r v o i r c o e f f i c i e n t .
I n g e n i e u r 70:87-94, 1958.
(8)
K r a i j e n h o f f van d e L e u r , D. A. A s t u d y o f non-steady g r o u n d w a t e r
f l o w , I1 Computation methods f o r f l o w t o d r a i n s .
1962.
I n g e n i e u r 74:285-292,
(9)
K r a i j e n h o f f v a n d e L e u r , D. A. Some e f f e c t s o f t h e u n s a t u r a t e d
zone on n o n s t e a d y f r e e - s u r f a c e g r o u n d w a t e r f l o w a s
s t u d i e d i n a s c a l e d g r a n u l a r model. J . Geophy. Res.
67:4347-4362,
1962.
D. T. and R. H . Shaw. A v a i l a b i l i t y of s o i l w a t e r t o
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H y d r a u l i c s of r u n o f f from d e v e l o p e d s u r f a c e s .
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(10)
L u t h i n , J . N: and J . C . G u i t j e n s . T r a n s i e n t s o l u t i o n s f o r d r a i n a g e
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D r a i n Div. 93:IR3:43-51,
(11)
McKay, G. A .
(12)
P e t e r s o n , K. R. A p r e c i p i t a b l e w a t e r nomogran.
SOC. 42:2:119-121,
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P r e c i p i t a t i o n . S e c t i o n 11, I H D F a m i l i a r i z a t i o n Seminar
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R u l . h e r . Met.
(13)
P h i l i p , J . R.
Numerical s o l u t i o n of e q u a t i o n s o f t h e d i f f u s i o n
.
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(14)
P h i l i p , J . R.
The t h e o r y o f i n f i l t r a t i o n : 5 . I n f l u e n c e of
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(15)
P h i l i p , J . R.
The p h y s i c a l p r o p e r t i e s o f s o i l w a t e r movement
d u r i n g t h e i r r i g a t i o n c y c l e . 3 r d Congr. I r r i g . and
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