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Annual operation of a coastal groundwater basin at a prescribed reliability level
Journal of Hydrology, 31 (1976) 97--118
97
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
A N N U A L OPERATION OF A COASTAL GROUNDWATER BASIN AT A PRESCRIBED RELIABILITY LEVEL
Y. BACHMAT
Hydrological Service, Jerusalem (Israel) (Received September 10, 1975; accepted for publication December 15, 1975)
ABSTRACT Bachmat, Y., 1976. Annual operation of a coastal groundwater basin at a prescribed reliability level. J. Hydrol., 31: 97--118. Short-term operation of a groundwater basin is preferably based on a hydrological operating rule. Such a rule is derived for the gross annual pumpage from a single-cell coastal groundwater basin at a prescribed reliability level of supply. The annual pumpage given by the rule consists of a fixed part, irrespective of the actual storage, and of a flexible part which is proportional to the prevailing actual storage above sea level. The rule rests upon the physical characteristics of the basin, its groundwater balance, the statistical parameters of a random normally distributed natural replenishment and design variables. The dependence of the parameters of the operating rule on the operational and initial storage, as well as on the parameters of the groundwater system is analyzed. A procedure for deriving the operating rule is formulated. The application of the procedure is illustrated on the coastal groundwater basin in Israel. Possible extensions of the rule are suggested.
INTRODUCTION
Long-term management models of water resources are usually composed of large-scale elements and embrace a multitude of aspects (e.g. economical, hydrological, ecological, political etc.). As a result these models bear a " h e a v y " load of uncertainty which is often difficult to express in terms of reliability of o u t c o m e and risk involved. At the same time short-range operations (e.g., annual pumpage from a groundwater basin) require that the probability of a failure, which m a y result in an irreversible damage to the water resource or to the water consumers, be as small as possible (Bachmat, 1974). In view of the aforementioned situation, and taking into account the cautiousness of the decision makers, it seems appropriate to have a separate evaluation of the annual exploitable yield o f a natural water reservoir which rests primarily on its physical characteristics. The objective of such an evaluation is to provide a rule of annual pumpage which ensures a given level of reliability of supply while securing the capability of the reservoir to serve as a water resource for a sufficiently long period of time.
98
The present paper deals with determining such a rule for a coastal groundwater basin. At this stage the problem is handled on a basin-wide scale and is concerned with a linear rule. A similar problem was treated in the past b y Langbein (1958) with reference to the design of a surface storage reservoir. Here Langbein's approach has been modified and extended to suit the particular features of a groundwater basin. STATEMENT OF THE PROBLEM
Consider a coastal groundwater basin which under natural conditions is replenished b y rainfall and drained b y flow to the sea. It is planned to p u m p water from the basin for consumption in the area overlying the basin. A fraction, ~, of the consumed water returns back to the basin. It is required to determine a rule of long-term annual pumpage from the basin subject to the following conditions: (1) The annual volume of pumpage, P, will consist of t w o parts: P
= Pmin +
Ap
where Pmin is a fixed guaranteed volume, that can be p u m p e d every year regardless of the actual storage of groundwater in the basin, whereas ~ P is a flexible volume that can be p u m p e d on t o p of Pmin and is a linear function of the actual storage at the beginning of the year. (2) The probability of a failure to provide the designed volume of annual pumpage should not exceed a given level, 1 - 0. DYNAMIC MODEL
A schematic layout of the system is presented in Fig. 1. The basin has a rectangular shape in the horizontal plane with sizes B and L along the coastline and perpendicular to it, respectively, D is the uniform depth o f the basin below sea level. K and n are the uniform and steady hydraulic conductivity and (effective) porosity of the basin, respectively. R denotes rainfall and O is o u t f l o w to the sea. The annual balance o f groundwater storage in the basin is given by:
U i - Ui-1 = N i - P i - Oi + ~ci
(1)
where:
Ui = the groundwater storage (total volume of groundwater) in the basin at the end o f time interval number i Ni = the total volume of natural replenishment during the ith time interval Pi = the total volume o f pumpage during the ith time interval Oi = the volume o f o u t f l o w to the sea during the ith time interval Ci = the total volume of water consumed in the area overlying the basin during the ith interval of time = the coefficient of return flow; 0 ~ ~ ~ 1
99
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.
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Cr0s6-section Fig. 1. Schematic model of the groundwater system.
Henceforth, the time interval h t = ti - ti-1 will be a year. Let the natural replenishment be given by: 0<~<1
Ni = a R i ;
(2)
where Ri is the volume of annual rainfall in the ith year, and ~ is a constant. Following the statement of the problem, the rule o f annual pumpage can be presented in the form: Pi = Pmin + k(Ui-z - Umin)
(3)
where Pmin and k are constants to be determined, and Umi n is the storage below the sea level. The annual flow to the sea can be approximated by a quasi-steady state model based on the Darcy law and on Dupuit's approximation, as follows: Oi = K
(HL)i - 0
(H L + D)i + D
L
2
B " A t ~- T
(I'lL)i
L
B " At
(4)
where the overbar denotes a mean annual value, and: T =- K r (HL)i t 2
+ D] ~KD;
(HL) i < < 2 9
(4a)
A linear approximation of the water table across the basin yields: (HL)i 2
=-Hi = ( Ui-l + Ui 2
U m i n ) (nBL) -I
(5)
100
where Hi is the basinwide annum average of the groundwater level above the sea level in the ith year. Hence:
T°At Oi ~ 6(Ui-~ + U i - 2 U m i n ) ;
~ --
> 0
nL:
(6)
By the definition o f the flow model 25 cannot exceed 1. Hence: 0 < ~ < 0.5
(6a)
By the statement of the problem:
Ci - Pi
(7)
Inserting eqs. 2--7 into eq. 1 yields:
Ui = A ( k ) Vi_ 1 + B Ri + C
(8)
where: A(k)
=
[ 1 - ~ - ( 1 - / 3 ) k ] .(1+~5)-'; A(k)
B = e(1+8) -I C
=
[(1
-
(9)
~) (kUmin - Pmin) + 25Umin ] (1+~) -~
Eq. 8 is a recursion formula for Ui and can be expressed in terms of the series of annual rainfalls between the beginning of a selected first year and the end of the ith year. The resulting form of eq. 8 is:
i Ui = A i ( k ) U o + B ~
1 -
[ A ( k ) l i-j R j +
J=l
1-A
Ai(k) (k)
C
(10)
STATISTICAL MODEL
The annual rainfall, Ri, is a stochastic variable. Then, b y eqs. 10 and 3, the storage, Ui, and the annual pumpage, Pi, are stochastic variables as well. Let R i have the same expectation, # R , and the same variance, oh, for any i. Also, let Ri and Rj be independent for any i ~ j. By eq. 10, the expectation of the storage at any time i (#Ui), the variance of the storage at any time i a2 ( Ui)' and the covariance b e t w e e n Ui and Ui-l, are given by:
~Ui = Ai(k) Uo +
1 - Ai(k)
1-A
1 - A2i(k) 2
aUi
_
1 - A 2 (k)
B:
(k) oh
cov(Ui, Ui-,) = A ( k )a2Ui_l
(B# R + C) (11)
101 For i -~ ~o, eqs. 11 reduce to:
P u = Uvoo = (BUR +C) [ 1 - A ( k ) ] 2 ° = B2O2R 0 b =-- Ouo
[l-A2(k)
-'
(12)
]-]
It can be shown that PU and o~] are equal to the average value of the annual storage and o f its variance over an indefinitely large number of years. By eqs. 11 and 12:
U U i - Uu = A i ( k ) ( U o - UU) o 2Ui - o:U = - A 2 i ( k ) O2U
(13)
cov(Ui, Ui-1) = A [1 - A 2(i-1)] o}/ i.e., the expectation and the variance of the storage at any finite point in time differ from their values at infinity b y an a m o u n t which diminishes exponentially in time. The variance of the storage at any finite point in time is always smaller than that at infinity. The same holds for the covariance COY(Wi, Wi-l). The expectation and variance of the annual pumpage follow from eqs. 3 and 13, yielding:
Upi = Pmin + k [ U u - Umin + (U0 - u u ) A i - l ( k ) ] o~ i = le2OUi_,2
= [l_A2(i_O(k) ]
k2o2U
(14)
For i -* 0% and by eqs. 12, 14 and 1 :
Up ~ Up.
= Pmin + k ( u U - Umin) -- (aUR - U O . ) k2B: --
(1-3)-' (15)
k2°b = 1 - A 2 ( k )
Hence:
Upi
=
lap + k A i-1 (k)
(Uo - UU)
-,
(16)
°2pi = [ 1 - A 2(i-') (k)] o~ = [ 1 - A 2(i-') (k)] k2S2o2R [1 - A2(k)] i.e., the variance of the pumpage at any finite point in time is smaller than its value at infinity, while the gap between them is diminishing exponentially in time. The parameters of the annual storage and the annual pumpage solely depend on those of the annual rainfall. In the present model it will be assumed that the annual rainfall is normally distributed. In view of eqs. 10
102
and 3, Ui and Pi will be normally distributed as well. The annual o u t f l o w to the sea, as expressed b y eq. 6 is also a stochastic variable. Hence, taking the expectation and variance of Oi in eq. 6 and employing the relations of eqs. 13 yields: I~O~ = ~ [ { A i - ' ( k ) + A i ( k ) } ( U o - I.LU) + 2(/~ U - Umin)]
(17)
2 = ~2 [ 2 - A2i-I ( k ) - A2(i-1) (k)l o~ ( I + A ) OOi
The expectation of the annual outflow decreases in time as the effect of the initial storage decreases. However, the variance o f the outflow increases as time goes on. For i -* ~, eqs. 17 reduce to: ;~o = ~O~ = 2 8 ( ~ u - U m i n ) 2 = 252ob ( I + A ) a~) = OO~
(18)
It can be shown that/~O is the long-term mean annual outflow. By eqs. 17 and 18: ~ 0 i ffi I~0
1 +
A j-I + A i
2
Uo - ~t U
~u
Umin
] (19)
A2(i -1) + A2i-I
CONFIDENCE INTERVAL FOR THE ANNUAL PUMPAGE
The rule of annual pumpage from a groundwater basin has to be based on a given desired level of confidence that the promised supply will actually be provided. By the statement of the problem this level of confidence is 0. We therefore require that: 0 = Pr ( I e i - ~ p i l
(20)
<<'Zo Opi)
where Zo is a n u m b e r of standard deviations corresponding to 0. Hence: (Pi)min = Pmin = b~pi- Zo Opi
(20a)
which, by virtue o f eqs. 14 and 16, and b y the definition of B in eq. 9, leads to:
/ ~ U - Umin -- Z0
l'
l _ A 2 ( i - 1 ) ( k ) 2 ~o___RR _ A i _ l ( k ) _ U o _ l ~ U 1 A2(k) " 1+5
(21)
103 Let the operational storage of the basin, U , , be defined by: U. = Umax- Umin = 2(/~U - Umin)
(22)
where Umax is the desirable upper bound of the storage. Inserting eq. 22 into eq. 21 and rearranging terms yields: 1
U, = 2Zo r
" a°Rl+3 -2Ai-'(k) ( U ° - P U ) - - F
A2(k)
(23)
The right-hand side of eq. 23 determines the operational storage capacity which is required in order to satisfy the desired confidence level, 0. It shows that this capacity depends on the standard deviation of the natural replenishment, a aR, on the natural outflow coefficient, 8; on the initial storage, U0; on the confidence level, 0; and on the coefficient of the annual pumpage, k. In practice, however, the storage capacity of a groundwater basin may be given and one may have to determine the value of k which meets the required reliability of supply. To this end it is necessary to solve eq. 23 for k for each particular value of i, thus obtaining a series of values of k(i). Another approach is that adopted in the present work, where one looks for a fixed value of h which satisfies the confidence level, 8, at all times. Since the operational storage has to be sufficiently large at all times, we seek to replace the right-hand side of eq. 23 by its maximum value. It can be shown that if U0 > ~ u this maximum is attained for i -~ ~, whereas if U0 < ~zu the maximum is attained for a finite value of i. Indeed, differentiating the right-hand side of eq. 23, F, with respect to i, as if it were continuous, one obtains: ( fori~
Uo~u
if
F -- F I T I ~
1 + (G/-I) 2 2 In A(k)
for i
or
U0 - Umin ~ U./2
otherwise
where:
2ZoaOR G -- ( 1 + 5 ) [ l _ A 2 ( k ) ] ½
and
I = 2(Uo-PU)
In the first case Fmax is obtained directly from the right-hand side of eq. 23, yielding: U, = [1
-
2Zo A2(k)]~
~oR 1 +8
U o - Umin for
u,
> 1/2
(24)
In the second case i of F m a x depends on A(k) which is not k n o w n a priori.
104 Here Fmax has to be found by "trial and error" with regard to A(k), or one can adopt the approximate relation: 2 Zo
=
v.
~ OR
[1-A~(k)]½
1+~
+ 2 ( p v - Uo)
Vmin
Uo -
for
u.
(25)
< 1/2
Combining the relations (24) and (25) we require that for any i: 2Zo 2 l [1-A (k)]~
~o R
] U, -- 2 Z o S u
for
U0-
=
Vmin ~ 1/2 (26)
U,
1+8
2 (U0 - Vmin) otherwise PARAMETERSOF ANNUALPUMPAGE The rule of annual pumpage is determined by two parameters, Pmin and k. Inserting the expression of A ( k ) from the identity (9) into eq. 26 (1) and solving the latter for k, yields: [1
-
8
-
x/(l+6) 2
-
( 2 Z o ~ O R / U . ) 2] (1-6)-'
?L_ for U°-u:min 1> 1/2 (27)
k = [1
5
X/(1+8) 2
[ Z o ~ a R / ( U o - Umin)] 2] (1-6)-' otherwise
By eqs. 15(1) and 22: Pmin = Pp - k V , / 2
Pmin I> 0
(28)
By eqs. 15(1), 18 and 22: t~p = ( ~ R - ~ U . ) (1 - ~)-'
(29)
Hence:
Pmin
- 1 -/3
1 -/~ +
u.
(30)
The rule of annual pumpage can now be presented in several forms:
I Pmin + k ( U i - 1 P/ =
-
Umin); Pmin~
Pi ~
Pp + k ( U i - ~ - I ~ U ) ;
pp-kU./2~Pi~Pp
aUR - ~ U .
(ni-,
1 - f3
+ knLB
-
~);
Pmin + k U .
U~7 -
+kU./2
(31)
U. 2 nLB
A graphical presentation of the parameters k (1-6) and Pmin/l~p case U0 - Umin/> U./2 is given in Fig. 2.
for the
105
96
Pmin//Up = 1 - k ( 1 - 0 ) u
k ( 1 - 6 ) = 1 - d - ~ N - 2 Z e o~d R / U x
u = 1/2 U x / [ p p ( I - B ) ]
d - T . a t / ( n I~)
72
68 64 60 56 52 48
.a:
32 28 24
16 12
,.o
o.s
0.6
0.4
Pmin/pp
0.2
0.0
0.2
0.4
.CL
0.6
0.8
1.0
Fig. 2. Graphical presentation of:
g(l-~)= f(8,•) and
P m i n / ~ p ffiF
(I-~),\ ~ / j
PARAMETERS AND CONFIDENCE INTERVAL OF THE NATURAL OUTFLOW
By eqs. 18 and 22: = Uo/U,
(32)
i.e.,~ is the fraction of the operational storage which constitutes the average annual loss to the sea. It m a y also be interpreted as the frequency of renewal of groundwater in the basin above the sea level.
106
By definition (see eq. 6), 8 solely depends on geometric and hydraulic characteristics of the basin. Also, by eqs. 18 and 26(1): 1
oo
_
~,o
[9, l {-:AZ~-)J
" (Z , ~ ) U .
= V-~z~
(33)
Hence, if the annual pumpage is provided at a confidence level, O, then by eq. 26 (1), the 0-confidence limits of the annual outflow to the sea are:
PO + Ze°o
8 U. [1 ± ~/(I+A)/2J
or
(33a)
The following discussion is concerned with the case of eq. 27 (1). A similar analysis can be conducted considering the case of eq. 27 (2). ANALYSIS OF THE OPERATIONAL PARAMETERS
The U, parameter
The feasible range of the operational storage is determined by three types of constraints: (1) Minimum and maximum admissible groundwater levels in the area overlying the basin. These are determined by the topography of the area, and by requirements concerning land use, standards of admissible water quality and protection of environment and facilities. Let the maximum operational storage capacity obeying these requirements be: U**, i.e., U** =
(34)
max (Uma x - Umin)
(2) The desired average depth o f seawater intrusion (L,). The steady position of the toe of the interface between fresh water and seawater in a singlelayered groundwater basin of the type considered in the present paper is illustrated in Fig. 3.
( taR )I
Sea Level 6s -'
uu~e, a. ; W=ter i
~
~'~" o
',
~ ~ I ~'" Fresh Water 7IIIIII,~/.-L'L;C./-,L~IIIIIIIIIIII'II Fig. 3. Steady interface in a coastal aquifer.
107 An approximate expression for LI, based on the hydraulic t h e o r y of steady groundwater flow is given b y (Bear, 1972):
eL1 LI
1 +*? KD2/(2Qs);
2Qs
,?2
*?
- 7s 7f
(35)
-
where ?f and ?s are the specific weights of freshwater and seawater, respectively;e is the intensity of the natural replenishment (dimensions L T -~) and Qs is the discharge of fresh groundwater to the sea per unit length of coastline. In the present work:
a(PR)l ,a#R eLl = ~ = B'At B'At Qs D
-
L1 L
#o B.~t
(36)
Umin
nBL
Inserting the relations (4(1)), (6), (32) and (36) into eq. 35 and considering the fact that *? >> 1, one obtains:
LI
Urain(O~#R
L I ) -1
L
*?U,
L
2-
8U,
(37)
or:
U. ~
Vmin
+
2*?(L~/L)
~#R
(L~/L)
28
(37a)
The desired LI can be obtained as an optimal solution o f the problem of maximizing the net benefit from seawater intrusion. Eq. 37a can be also interpreted as a relation b e t w e e n the mean o u t f l o w to the sea (8 U , ) and the depth of seawater intrusion. It can be shown that the loss o f groundwater to the sea can be brought to a minimum at the point: 1
L1/L = [8 Umin/(*?apR) ] ~ (3) Constraints on the operational parameters. By the definition of the problem k and Pmin must be non-negative. According to eq. 27 (1) the requirement h/> 0 leads to the constraint: 48 ~<~22 -
2ZeaaR ) 2
u,
•
< (1+8) 2
(38)
108
Inserting the expression for k from eq. 27 into the expression for Pmin, given by eq. 30, and requiring Pmin/> 0, yields the constraint:
( ZooRl2 PR
~<
/
1+5
U, - 1
(39)
~PR
By eqs. 38 and 39:
aUR
( 1+
Zo.oR
-
<~ U, <~ - -
(40)
The totality of the above constraints imposed on U. leads to a selection of U. according to the rule: U.= m a x ) staR
(
Z°OR),
l 1+6 1 +
min
U**, L
+ '
26
L,/L
(41)
The Zo parameter The confidence level, 0, implies that during an infinitely long period of time one m a y expect the undesired event Ui < Vmin or Ui > Vmax once in every (1 - 0) -l years, (1 - 0) -l being the recurrence interval of a failure to provide the promised supply (Riggs, 1961). In practice, however, one plans the annual operation of a basin subject to the requirement that with a given probability, P, there will be no failure or any prescribed number of failures (k) during a given finite number of consecutive years (n). By the binomial distribution and, considering a confidence level o f annual supply, 0, one has:
Pk = (~) (I - O)k 0 n-k
(42)
If no failure is required during the first no consecutive years: k = 0,
and
wherefrom:
P-P0=0n°
(42a)
0 =n~/ Po =- 0 o
(43)
Eq. 42 can serve as the simplest relation for determining 0, and consequently Ze, by prescribed values of no and Po. By eq. 42, 0 is an increasing function of both P0 and no and it is always larger than P0. Another constraint is imposed on Ze by the inequalities (38) and (39), yielding: _1 _1
6 2 U, <~ Zo <~ _
U, - 1
_
o~o R
ot PR
UR aR
(44)
109
By eqs. 43 and 44, once U, was selected, the rule for selecting Zo is given by: 1
= max
Zo
t
!
- - ,
rain
Oo,
- -
t~ o R
U.-1
(45)
at laR
The k parameter
By eqs. 27 and 30: 0 <~ k ~< 2/l °~laR
~i)\ (1-13) - I
\u.
=
2lap/U,
(46)
Also, by eq. 27: 0k d5 +
dk
d~ +
--
ozo
o8
dZo +
do R +
dU.
dU,
(47)
Now: Ok
--
[1 + 1/~/1-
= -
{~21(1 + 8 ) ) 2] (1 - ~)-i < 0
(48)
08
Ok --
ap
=
Ok Oa
[I-,S-
~/(I+8)
2-~2]
(l-~)-2=k/(l-p)>0
~2 ( 1 + ~ ) ~ / ( 1 + 6 ) 2 - a 2 > 0;
0~2
~2 ---> OZo Z0
0~2
~2 ---> OoR oR
0;
(49)
(50)
~2 = 2 Z o a a R / U .
0;
0~2 ~2 . . . . < 0 OU, U,
(51)
Thus, the coefficient of the flexible portion o f the annual pumpage, k, is an increasing function of ~, 0 and o R , yet decreasing with 8 and U , . T h e Pmin p a r a m e t e r
Pmin is the fixed part of the :annual pumpage. By eqs. 28 and 46:
(52)
0 ~ Pmin <~ lap
By eq. 30: _U,(
~Pmin
- - -
a5
1-/3
aPmin
a,6
1
.
alap . a~
+
U. ak . 2 a.6
Ok)
2~8
-
,Up 1-,G
.
-U. [ 1+8 ] 12(1-~) ~/(1 +8)2- ~22 kU,/2
Pmin
1-,6
1-,6
i>0
> 0
(53)
(54)
110
By eqs. 30, 27, 50 and 51: aPmi n
~
k
aU,
1-~
2
U. ak 2
~2 : ]
3~2: + ( l + a ) : [ 1 - x/1
au,
2(l_~)x/(l+a)2_n
2
>0
(55)
In conclusion, both k and Pmin are increasing as the coefficient of return flow increases. On the other hand, b y eq. 28, Pmin is linearly increasing as the mean replenishment increases and decreasing as k increases. As a result, any increase of the operational storage, U,, and of the transfer coefficient a contribute to the transfer of pumpage from the flexible part to the fixed part, whereas any increase of the confidence level, 0, and of the standard deviation of the natural replenishment contribute to an increase of the flexible part of the annual pumpage. f o r k -* kmi n = O,
Pmin ~ (Pmin)max
for k-+ kmax - 2#p U, '
auR -au,
-
1-
Pmin ~ (Pmin)min
=
~.
#p
0
The Pi parameter The total annual pumpage, Pi, as given in eq. 31 can n o w be analyzed b y testing the effect of the operational parameters and of the storage on it. Thus: a Pmin
aP i
aa
-
aa
+ (Ui-
ak U m i n ) -a~ -
=
'
--1
-
(Ui
-
Umin)
1 %/1- 1 n
(1%--~)
Hence: O,
• (1-/3)-' (56)
2
1+
l
~2 2 - ~
(57)
> O,
ou.
fl
U,/2
for Ui - Umin
aPi a8
aPi
2 +1
aPmin
ou.
otherwise
ak
+ (Ui- Umin)-
_--
__
(.a
+
k)
+ ( U / - Umi n - U . / 2 ) -
ak
av.
111 Hence:
OPi
(58)
0U,
1+./~ > 0,
~2 2 1+5
otherwise
SAMPLING EFFECT The mean and standard deviation of the annual rainfall, PR and aR, are actually u n k n o w n population parameters, which have to be estimated from sample data. Let _~ and SR be the sample mean and standard deviation of the annual volume of rainfall and let N be the sample size. One possibility is to employ point estimates (Spiegel, 1961):
(59)
/
N °R = SR "~N - 1
Another possibility is to employ interval estimates of the above parameters and calculate k and Pmin for the confidence limits to obtain a range for Pi as a function of the confidence level, 0. The interval estimates are given b y (Spiegel, 1961):
=
t ~ + Zo SR / x/r-N,
forN>
30
(60)
PR _~ + t(N-') + 0/2 • SR/V/-N-1, -
aR
= /N~
"0.$
SR ( l + Z o
forN<
x/2N),
30
forN;~30
(61) f - ~ . (N-I) SR V IV/Xo.s+o/2 < o R < SR %/-N/x (N-') O.s-o/2
'
forN<
30
where Z0 is the number of standard deviations of a normal distribution on each side of the mean corresponding to the confidence level, 0. t(N-~) 0.s +e/2 is the equivalent of Ze for a t-distribution at a number of degrees of freedom v = N - I .
2(N-I)
_ 2(N-0
Xo.s +0/2 and ×0.5 -0/2 are values of chi-square corresponding
to the 0-1evel confidence limits of the standard deviation.
112
Estimate #R, o'R (59) Assess at,/% 6, U**,Umin (6), (34)
I
I Design L,/L i I oes'~° °°, ~°
I
i I ~et ~°. ~° (~) Determine
I
U, (41) ]
J Adjust U, I= NO
No 4 Compute k ~ Y e s (27 (2))
.o ~i Adju,t zo
.oj
Compute (27 (1,2))k I Compute I Pmin (30)
i Determine
Oi (31)
l
End
I [
Fig. 4. Diagram presenting steps involved in generating the linear rule of annual pumpage from a coastal groundwater basin. Numbers in parentheses indicate equation numbers. PROCEDURE FOR DETERMINING THE RULE OF ANNUAL OPERATION
The diagram of Fig. 4 presents the sequence of steps involved in generating the linear rule of annual pumpage from a coastal groundwater basin. The numbers in the parentheses indicate numbers of equations. The procedure will n o w be illustrated on the coastal groundwater basin in Israel, south of Mt. Carmel. The configuration of the basin in the horizontal plane can be represented schematically by a rectangle extending B = 114 km along the coastline at an average width of L = 15 km. Under natural conditions
113
the basin is fed by replenishment f r o m rainfall and drained by o u t f l o w t o the Mediterranean Sea. Data pertaining to t he probl em at hand are as follows: ~R - / ~ = 920 MCM/year based on N = 46 years o f record in 11 raingauging stations (see eq. 59); MCM = million cubic meters
aR =SR ~ N N 1
= 240 4~__~ ~ 245 MCM/year (seeeq. 59)
D = 60 average d e p t h o f the basin below sea level in meters a ~ 1/3 coefficient of natural replenishment ~ 1/4 coefficient of r et ur n flow T = 103 m2/day equivalent transmissivity n = 1/5 representative effective por os i t y Vmi n = nLBD = 20.5" 103 MCM U** = 1 6 . 2 . 1 0 3 MCM m a x i m u m admissible operational storage between min imu m and m a x i m u m allowable water levels Uo = 3 - 1 0 3 MCM + Umin T°A t
5 -
103. 360
-
nL 2
1/5 × (15.103) 2
= 8" 10 -3
L~ = 1.5 km designed e x p e c t e d de pt h of seawater intrusion inland. Selecting no = 5; Po = 0.9935 yields by 43: ao = 0.9987;
Zo O = 3
By eq. 41:
v.
= max [A, rain (U**, B, C)]
where: A -
B --
a/JR
(
2^2
1+
Ze'*aR
W
= 1.7" 103 MCM
=
2.7"10 3MCM
Umin C
=
2~L~IL
+
O~R
L1
2~
L
.... ~ 4.8"103 MCM
By eq. 41: U, = 2 . 7 . 1 0 3 MCM However, this does n o t satisfy eq. 37a. As the designed value o f L1 is binding, we select: U, = C = 4 .8 " 103 MCM Now: Uo-
Umin = 3 " 1 0 3 M C M >
U./2
114
Hence, we have to check, whether value of U, we find:
eq. 44 is satisfied.
With the finally selected
Hence 20 has to be adjusted. Let ZB =’ 6. Inserting this value into the expression of B above would satisfy both eqs. 37a and 41. Likewise it satisfies eq. 44. The next step would be to compute k by eq. 27. The resulting value is: k = 0.009 whereas, by eq. 30: Pmin = 408.9 - 72.9 = 336 MCM Hence, by eq. 31(l): Pi = 336 + 0.009 (Vi-1 - Umln) or, by eq. 31(3): Pi = 358 + 3.1 (Ri-1 - 7.0); GENERALIZED
SUPPLY
336 MCM < Pi < 380 MCM
POLICY
So far the consumption of water in the region overlying the basin was presumed to be equal to the pumpage from the basin. In practice, however, the demand for water in the region may exceed the permissible pumpage from the underlying basin. Under such conditions one may consider a more general policy of supply, such as: Ci=Pi+Ii;
Ii> 0;
Ci = f(i)
Where i refers to ith year, C is the allocated (predetermined) an import from outside the region for direct consumption. becomes:
(62)
supply and I is With eq. 62, eq. 10
i
U = ai(k)Uo + C j=l
where :
aim’(k) (bRj + cj)
(63)
115
a(k) = ( 1 - 5 - k ) ( 1 + 5 ) -1 b
= a (I +6)-'
cj
=
(64)
[ ( k + 25 ) Umi n - Pmin + ~ Cj ]
(1+6)-'
The expectation of eq. 63 yields:
1 -a i #ui = aiUo + - - b p 1-a
i R + ~ ai-Jcj
(65a)
j=l m
/~U -= VU. = bPR/(1-a) + lira ~
am-J cj
(65b)
.i=1 Hence: m
i
am-Jcj - ~ j--1
ai-Jcj)
(66)
j=l
Also, by eqs. 3 and 66: UPi = Pmin + k I~tU - Umin + (Uo -
~-a ]ai_l(k)_ Gi_l(k) 1
(67)
where: m Gi(k ) - lira ~ am-Y(k)cjm~o~ j=l
i
~_j ai-J(k)cj j=l
By analogy with eq. 11 and since cj is deterministic:
: °ui
-----
2 1-a2i(k) b2a2R; 03--oU~ =b 2 aR2 [ 1 - a : ( k ) ]
-1
l_aS(k)
(68)
By eqs. 3 and 68: 2 2 Opi = k 2 Ovi_,
=
[1-a2(i-')(k)] k2o~r=[1-a2(i-')(k)] k 2" °
[1-a:(k)]
As eq. 20 must hold, we obtain, by virtue of eqs. 67 and 69:
(69)
116 1
I I-a2(i-O(k)l ~ PU
-
Umin
-
=
l - a 2 (k)
go IUo
-
J
°~PR (1 +~) (-l-a)
aaR 1+6
+ G i - , ( k )
I ai-'(k)
-
(70)
Introducing the operational storage, as defined in eq. 22, and demanding that the operational storage be sufficient to satisfy eq. 70 at any i, we obtain:
U.
=
2Z0
' [1-a2(k)] : "
for U0 ~>
aoR 1+6
m + lim ~ a m - l ( k ) c j
m-~o.
j=l
OlPR (1+6) ( l - a )
(71a) m
U.
=
2Zo O~aR , .-lim ~ a m - i ( k ) c j + [1-a2(k)]~ 1+6 + m.-*~ j=l
for U0 <
O~pR (1 +6) ( l - a )
°~l'lR - Vo (l+6)[1-a(k)]
(71b)
Each of the eqs. (71) can be solved for k, according to the comparative value of Uo. By eq. 67, eq. 28 still holds. Also, by eqs. 1, 6 and 62: n
Up = a P R - 6 U * + ~ l i m n.--~ c~
(72)
~ f(i)/n i=1
Hence: Pmin = a#R + ~ lim n--~oo
~ f(i)/ni=1
6 +
U,
(73)
and, finally, with eqs. 3 and 62: [i = Ci - Pi = f ( i ) - Pmin - k ( U i - , - Umin)
(74)
SUMMARY AND CONCLUSIONS A rule was derived for the total annual pumpage from a coastal groundwater basin subject to a prescribed reliability level of supply. The rule contains a fixed part (= firm supply) and a variable part which is proportional to the actual groundwater storage above the sea level. The basin is replenished naturally by a fixed fraction of a random and normally distributed rainfall,
117 and -- artificially -- by a fixed fraction of the volume of water consumed in the area overlying the basin. The basin is drained b y pumpage and b y o u t f l o w to the sea, the latter being proportional to the actual storage above the sea level. T w o alternatives of consumption were considered. In the first alternative, which takes up the major part of this paper, the consumption is taken to be equal to the pumpage. It was found that the parameters of the rule depend, among others, on the ratio between the initial storage and the long-term mean storage, b o t h above the sea level. Most of the paper analyzes the case when this ratio exceeds 1, although reference is made to the opposite case as well. A certain operational storage is needed in order to satisfy the prescribed reliability level. However, some factors, -- namely, the proportionality of the natural o u t f l o w to the storage above the sea level, and the m a x i m u m admissible groundwater level and the desired depth of seawater intrusion -- set bounds on the possible range of the operational storage. This situation is characteristic for a coastal groundwater basin, as opposed to the case of a surface storage reservoir. Within this range any increase of the operational storage increases the fixed part of the annual pumpage at the expense of the flexible part. A similar effect has a composite dimensionless hydrological parameter, 8, which represents the rate of renewal of the groundwater storage above the sea level. On the other hand, given the operational storage, any increase of the reliability level of supply and of the variance of the natural replenishment bring a b o u t an increase of the variable (flexible) part of the pumpage and a reduction of the fixed part. As to the effect o f other factors it was f o u n d that the fixed part of the annual pumpage is a linear increasing function of the mean annual replenishment, whereas the total annual pumpage increases with the coefficient of return flow. The analysis of the parameters of the operational rule is followed b y a procedure for determining these parameters. The application of the procedure is illustrated by an example on the coastal groundwater basin in Israel. Finally, an extension is made to the case of a predetermined consumption, including the possibility of importing water from outside the basin. The results of this paper can be further extended to any groundwater basin which is drained b y any natural outlet (e.g., a spring or a river) introducing a proper modification of the o u t f l o w function. Another worthwhile effort is to generalize the operating rule to a multicell basin. ACKNOWLEDGEMENT This work is a part of a study on the annual operation of the coastal aquifers c o n d u c t e d at the Hydrological Service o f the State o f Israel. The
118
author is indebted to the Director of the Hydrological Service for his permission to publish this work. Thanks are also due to Dr. T. Herman for his valuable comments. REFERENCES Bachmat, Y., 1974. A pragmatic approach to the annual operation of a coastal aquifer. In: Water in Israel. State of Israel, Ministry of Agriculture, Water Commission. Bear, J., 1972. Dynamics of Fluids in Porous Media. American Elsevier, New York, 764 pp. Langbein, W.B., 1958. Queuing theory and water storage. J. Hydraul. Div., Am. Soc. Cir. Eng., HY5: 1--24. Riggs, H.C., 1961. Frequency of natural events. J. Hydraul. Div., Am. Soc. Civ. Eng., HYI: 15--26. Spiegel, M.R., 1961. Statistics. Schaum's Outline Series, McGraw-Hill, New York, N.Y., 359 pp.
97
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
A N N U A L OPERATION OF A COASTAL GROUNDWATER BASIN AT A PRESCRIBED RELIABILITY LEVEL
Y. BACHMAT
Hydrological Service, Jerusalem (Israel) (Received September 10, 1975; accepted for publication December 15, 1975)
ABSTRACT Bachmat, Y., 1976. Annual operation of a coastal groundwater basin at a prescribed reliability level. J. Hydrol., 31: 97--118. Short-term operation of a groundwater basin is preferably based on a hydrological operating rule. Such a rule is derived for the gross annual pumpage from a single-cell coastal groundwater basin at a prescribed reliability level of supply. The annual pumpage given by the rule consists of a fixed part, irrespective of the actual storage, and of a flexible part which is proportional to the prevailing actual storage above sea level. The rule rests upon the physical characteristics of the basin, its groundwater balance, the statistical parameters of a random normally distributed natural replenishment and design variables. The dependence of the parameters of the operating rule on the operational and initial storage, as well as on the parameters of the groundwater system is analyzed. A procedure for deriving the operating rule is formulated. The application of the procedure is illustrated on the coastal groundwater basin in Israel. Possible extensions of the rule are suggested.
INTRODUCTION
Long-term management models of water resources are usually composed of large-scale elements and embrace a multitude of aspects (e.g. economical, hydrological, ecological, political etc.). As a result these models bear a " h e a v y " load of uncertainty which is often difficult to express in terms of reliability of o u t c o m e and risk involved. At the same time short-range operations (e.g., annual pumpage from a groundwater basin) require that the probability of a failure, which m a y result in an irreversible damage to the water resource or to the water consumers, be as small as possible (Bachmat, 1974). In view of the aforementioned situation, and taking into account the cautiousness of the decision makers, it seems appropriate to have a separate evaluation of the annual exploitable yield o f a natural water reservoir which rests primarily on its physical characteristics. The objective of such an evaluation is to provide a rule of annual pumpage which ensures a given level of reliability of supply while securing the capability of the reservoir to serve as a water resource for a sufficiently long period of time.
98
The present paper deals with determining such a rule for a coastal groundwater basin. At this stage the problem is handled on a basin-wide scale and is concerned with a linear rule. A similar problem was treated in the past b y Langbein (1958) with reference to the design of a surface storage reservoir. Here Langbein's approach has been modified and extended to suit the particular features of a groundwater basin. STATEMENT OF THE PROBLEM
Consider a coastal groundwater basin which under natural conditions is replenished b y rainfall and drained b y flow to the sea. It is planned to p u m p water from the basin for consumption in the area overlying the basin. A fraction, ~, of the consumed water returns back to the basin. It is required to determine a rule of long-term annual pumpage from the basin subject to the following conditions: (1) The annual volume of pumpage, P, will consist of t w o parts: P
= Pmin +
Ap
where Pmin is a fixed guaranteed volume, that can be p u m p e d every year regardless of the actual storage of groundwater in the basin, whereas ~ P is a flexible volume that can be p u m p e d on t o p of Pmin and is a linear function of the actual storage at the beginning of the year. (2) The probability of a failure to provide the designed volume of annual pumpage should not exceed a given level, 1 - 0. DYNAMIC MODEL
A schematic layout of the system is presented in Fig. 1. The basin has a rectangular shape in the horizontal plane with sizes B and L along the coastline and perpendicular to it, respectively, D is the uniform depth o f the basin below sea level. K and n are the uniform and steady hydraulic conductivity and (effective) porosity of the basin, respectively. R denotes rainfall and O is o u t f l o w to the sea. The annual balance o f groundwater storage in the basin is given by:
U i - Ui-1 = N i - P i - Oi + ~ci
(1)
where:
Ui = the groundwater storage (total volume of groundwater) in the basin at the end o f time interval number i Ni = the total volume of natural replenishment during the ith time interval Pi = the total volume o f pumpage during the ith time interval Oi = the volume o f o u t f l o w to the sea during the ith time interval Ci = the total volume of water consumed in the area overlying the basin during the ith interval of time = the coefficient of return flow; 0 ~ ~ ~ 1
99
.[......... i ........ il w
F
r
F
w
e
i
w
•
Top
¢
J
r
r
r
w
'
View
R
INL
,
e
w
w
w
e
w
w w .
w w .
w .
w
.
r
~
.
~
Cr0s6-section Fig. 1. Schematic model of the groundwater system.
Henceforth, the time interval h t = ti - ti-1 will be a year. Let the natural replenishment be given by: 0<~<1
Ni = a R i ;
(2)
where Ri is the volume of annual rainfall in the ith year, and ~ is a constant. Following the statement of the problem, the rule o f annual pumpage can be presented in the form: Pi = Pmin + k(Ui-z - Umin)
(3)
where Pmin and k are constants to be determined, and Umi n is the storage below the sea level. The annual flow to the sea can be approximated by a quasi-steady state model based on the Darcy law and on Dupuit's approximation, as follows: Oi = K
(HL)i - 0
(H L + D)i + D
L
2
B " A t ~- T
(I'lL)i
L
B " At
(4)
where the overbar denotes a mean annual value, and: T =- K r (HL)i t 2
+ D] ~KD;
(HL) i < < 2 9
(4a)
A linear approximation of the water table across the basin yields: (HL)i 2
=-Hi = ( Ui-l + Ui 2
U m i n ) (nBL) -I
(5)
100
where Hi is the basinwide annum average of the groundwater level above the sea level in the ith year. Hence:
T°At Oi ~ 6(Ui-~ + U i - 2 U m i n ) ;
~ --
> 0
nL:
(6)
By the definition o f the flow model 25 cannot exceed 1. Hence: 0 < ~ < 0.5
(6a)
By the statement of the problem:
Ci - Pi
(7)
Inserting eqs. 2--7 into eq. 1 yields:
Ui = A ( k ) Vi_ 1 + B Ri + C
(8)
where: A(k)
=
[ 1 - ~ - ( 1 - / 3 ) k ] .(1+~5)-'; A(k)
B = e(1+8) -I C
=
[(1
-
(9)
~) (kUmin - Pmin) + 25Umin ] (1+~) -~
Eq. 8 is a recursion formula for Ui and can be expressed in terms of the series of annual rainfalls between the beginning of a selected first year and the end of the ith year. The resulting form of eq. 8 is:
i Ui = A i ( k ) U o + B ~
1 -
[ A ( k ) l i-j R j +
J=l
1-A
Ai(k) (k)
C
(10)
STATISTICAL MODEL
The annual rainfall, Ri, is a stochastic variable. Then, b y eqs. 10 and 3, the storage, Ui, and the annual pumpage, Pi, are stochastic variables as well. Let R i have the same expectation, # R , and the same variance, oh, for any i. Also, let Ri and Rj be independent for any i ~ j. By eq. 10, the expectation of the storage at any time i (#Ui), the variance of the storage at any time i a2 ( Ui)' and the covariance b e t w e e n Ui and Ui-l, are given by:
~Ui = Ai(k) Uo +
1 - Ai(k)
1-A
1 - A2i(k) 2
aUi
_
1 - A 2 (k)
B:
(k) oh
cov(Ui, Ui-,) = A ( k )a2Ui_l
(B# R + C) (11)
101 For i -~ ~o, eqs. 11 reduce to:
P u = Uvoo = (BUR +C) [ 1 - A ( k ) ] 2 ° = B2O2R 0 b =-- Ouo
[l-A2(k)
-'
(12)
]-]
It can be shown that PU and o~] are equal to the average value of the annual storage and o f its variance over an indefinitely large number of years. By eqs. 11 and 12:
U U i - Uu = A i ( k ) ( U o - UU) o 2Ui - o:U = - A 2 i ( k ) O2U
(13)
cov(Ui, Ui-1) = A [1 - A 2(i-1)] o}/ i.e., the expectation and the variance of the storage at any finite point in time differ from their values at infinity b y an a m o u n t which diminishes exponentially in time. The variance of the storage at any finite point in time is always smaller than that at infinity. The same holds for the covariance COY(Wi, Wi-l). The expectation and variance of the annual pumpage follow from eqs. 3 and 13, yielding:
Upi = Pmin + k [ U u - Umin + (U0 - u u ) A i - l ( k ) ] o~ i = le2OUi_,2
= [l_A2(i_O(k) ]
k2o2U
(14)
For i -* 0% and by eqs. 12, 14 and 1 :
Up ~ Up.
= Pmin + k ( u U - Umin) -- (aUR - U O . ) k2B: --
(1-3)-' (15)
k2°b = 1 - A 2 ( k )
Hence:
Upi
=
lap + k A i-1 (k)
(Uo - UU)
-,
(16)
°2pi = [ 1 - A 2(i-') (k)] o~ = [ 1 - A 2(i-') (k)] k2S2o2R [1 - A2(k)] i.e., the variance of the pumpage at any finite point in time is smaller than its value at infinity, while the gap between them is diminishing exponentially in time. The parameters of the annual storage and the annual pumpage solely depend on those of the annual rainfall. In the present model it will be assumed that the annual rainfall is normally distributed. In view of eqs. 10
102
and 3, Ui and Pi will be normally distributed as well. The annual o u t f l o w to the sea, as expressed b y eq. 6 is also a stochastic variable. Hence, taking the expectation and variance of Oi in eq. 6 and employing the relations of eqs. 13 yields: I~O~ = ~ [ { A i - ' ( k ) + A i ( k ) } ( U o - I.LU) + 2(/~ U - Umin)]
(17)
2 = ~2 [ 2 - A2i-I ( k ) - A2(i-1) (k)l o~ ( I + A ) OOi
The expectation of the annual outflow decreases in time as the effect of the initial storage decreases. However, the variance o f the outflow increases as time goes on. For i -* ~, eqs. 17 reduce to: ;~o = ~O~ = 2 8 ( ~ u - U m i n ) 2 = 252ob ( I + A ) a~) = OO~
(18)
It can be shown that/~O is the long-term mean annual outflow. By eqs. 17 and 18: ~ 0 i ffi I~0
1 +
A j-I + A i
2
Uo - ~t U
~u
Umin
] (19)
A2(i -1) + A2i-I
CONFIDENCE INTERVAL FOR THE ANNUAL PUMPAGE
The rule of annual pumpage from a groundwater basin has to be based on a given desired level of confidence that the promised supply will actually be provided. By the statement of the problem this level of confidence is 0. We therefore require that: 0 = Pr ( I e i - ~ p i l
(20)
<<'Zo Opi)
where Zo is a n u m b e r of standard deviations corresponding to 0. Hence: (Pi)min = Pmin = b~pi- Zo Opi
(20a)
which, by virtue o f eqs. 14 and 16, and b y the definition of B in eq. 9, leads to:
/ ~ U - Umin -- Z0
l'
l _ A 2 ( i - 1 ) ( k ) 2 ~o___RR _ A i _ l ( k ) _ U o _ l ~ U 1 A2(k) " 1+5
(21)
103 Let the operational storage of the basin, U , , be defined by: U. = Umax- Umin = 2(/~U - Umin)
(22)
where Umax is the desirable upper bound of the storage. Inserting eq. 22 into eq. 21 and rearranging terms yields: 1
U, = 2Zo r
" a°Rl+3 -2Ai-'(k) ( U ° - P U ) - - F
A2(k)
(23)
The right-hand side of eq. 23 determines the operational storage capacity which is required in order to satisfy the desired confidence level, 0. It shows that this capacity depends on the standard deviation of the natural replenishment, a aR, on the natural outflow coefficient, 8; on the initial storage, U0; on the confidence level, 0; and on the coefficient of the annual pumpage, k. In practice, however, the storage capacity of a groundwater basin may be given and one may have to determine the value of k which meets the required reliability of supply. To this end it is necessary to solve eq. 23 for k for each particular value of i, thus obtaining a series of values of k(i). Another approach is that adopted in the present work, where one looks for a fixed value of h which satisfies the confidence level, 8, at all times. Since the operational storage has to be sufficiently large at all times, we seek to replace the right-hand side of eq. 23 by its maximum value. It can be shown that if U0 > ~ u this maximum is attained for i -~ ~, whereas if U0 < ~zu the maximum is attained for a finite value of i. Indeed, differentiating the right-hand side of eq. 23, F, with respect to i, as if it were continuous, one obtains: ( fori~
Uo~u
if
F -- F I T I ~
1 + (G/-I) 2 2 In A(k)
for i
or
U0 - Umin ~ U./2
otherwise
where:
2ZoaOR G -- ( 1 + 5 ) [ l _ A 2 ( k ) ] ½
and
I = 2(Uo-PU)
In the first case Fmax is obtained directly from the right-hand side of eq. 23, yielding: U, = [1
-
2Zo A2(k)]~
~oR 1 +8
U o - Umin for
u,
> 1/2
(24)
In the second case i of F m a x depends on A(k) which is not k n o w n a priori.
104 Here Fmax has to be found by "trial and error" with regard to A(k), or one can adopt the approximate relation: 2 Zo
=
v.
~ OR
[1-A~(k)]½
1+~
+ 2 ( p v - Uo)
Vmin
Uo -
for
u.
(25)
< 1/2
Combining the relations (24) and (25) we require that for any i: 2Zo 2 l [1-A (k)]~
~o R
] U, -- 2 Z o S u
for
U0-
=
Vmin ~ 1/2 (26)
U,
1+8
2 (U0 - Vmin) otherwise PARAMETERSOF ANNUALPUMPAGE The rule of annual pumpage is determined by two parameters, Pmin and k. Inserting the expression of A ( k ) from the identity (9) into eq. 26 (1) and solving the latter for k, yields: [1
-
8
-
x/(l+6) 2
-
( 2 Z o ~ O R / U . ) 2] (1-6)-'
?L_ for U°-u:min 1> 1/2 (27)
k = [1
5
X/(1+8) 2
[ Z o ~ a R / ( U o - Umin)] 2] (1-6)-' otherwise
By eqs. 15(1) and 22: Pmin = Pp - k V , / 2
Pmin I> 0
(28)
By eqs. 15(1), 18 and 22: t~p = ( ~ R - ~ U . ) (1 - ~)-'
(29)
Hence:
Pmin
- 1 -/3
1 -/~ +
u.
(30)
The rule of annual pumpage can now be presented in several forms:
I Pmin + k ( U i - 1 P/ =
-
Umin); Pmin~
Pi ~
Pp + k ( U i - ~ - I ~ U ) ;
pp-kU./2~Pi~Pp
aUR - ~ U .
(ni-,
1 - f3
+ knLB
-
~);
Pmin + k U .
U~7 -
+kU./2
(31)
U. 2 nLB
A graphical presentation of the parameters k (1-6) and Pmin/l~p case U0 - Umin/> U./2 is given in Fig. 2.
for the
105
96
Pmin//Up = 1 - k ( 1 - 0 ) u
k ( 1 - 6 ) = 1 - d - ~ N - 2 Z e o~d R / U x
u = 1/2 U x / [ p p ( I - B ) ]
d - T . a t / ( n I~)
72
68 64 60 56 52 48
.a:
32 28 24
16 12
,.o
o.s
0.6
0.4
Pmin/pp
0.2
0.0
0.2
0.4
.CL
0.6
0.8
1.0
Fig. 2. Graphical presentation of:
g(l-~)= f(8,•) and
P m i n / ~ p ffiF
(I-~),\ ~ / j
PARAMETERS AND CONFIDENCE INTERVAL OF THE NATURAL OUTFLOW
By eqs. 18 and 22: = Uo/U,
(32)
i.e.,~ is the fraction of the operational storage which constitutes the average annual loss to the sea. It m a y also be interpreted as the frequency of renewal of groundwater in the basin above the sea level.
106
By definition (see eq. 6), 8 solely depends on geometric and hydraulic characteristics of the basin. Also, by eqs. 18 and 26(1): 1
oo
_
~,o
[9, l {-:AZ~-)J
" (Z , ~ ) U .
= V-~z~
(33)
Hence, if the annual pumpage is provided at a confidence level, O, then by eq. 26 (1), the 0-confidence limits of the annual outflow to the sea are:
PO + Ze°o
8 U. [1 ± ~/(I+A)/2J
or
(33a)
The following discussion is concerned with the case of eq. 27 (1). A similar analysis can be conducted considering the case of eq. 27 (2). ANALYSIS OF THE OPERATIONAL PARAMETERS
The U, parameter
The feasible range of the operational storage is determined by three types of constraints: (1) Minimum and maximum admissible groundwater levels in the area overlying the basin. These are determined by the topography of the area, and by requirements concerning land use, standards of admissible water quality and protection of environment and facilities. Let the maximum operational storage capacity obeying these requirements be: U**, i.e., U** =
(34)
max (Uma x - Umin)
(2) The desired average depth o f seawater intrusion (L,). The steady position of the toe of the interface between fresh water and seawater in a singlelayered groundwater basin of the type considered in the present paper is illustrated in Fig. 3.
( taR )I
Sea Level 6s -'
uu~e, a. ; W=ter i
~
~'~" o
',
~ ~ I ~'" Fresh Water 7IIIIII,~/.-L'L;C./-,L~IIIIIIIIIIII'II Fig. 3. Steady interface in a coastal aquifer.
107 An approximate expression for LI, based on the hydraulic t h e o r y of steady groundwater flow is given b y (Bear, 1972):
eL1 LI
1 +*? KD2/(2Qs);
2Qs
,?2
*?
- 7s 7f
(35)
-
where ?f and ?s are the specific weights of freshwater and seawater, respectively;e is the intensity of the natural replenishment (dimensions L T -~) and Qs is the discharge of fresh groundwater to the sea per unit length of coastline. In the present work:
a(PR)l ,a#R eLl = ~ = B'At B'At Qs D
-
L1 L
#o B.~t
(36)
Umin
nBL
Inserting the relations (4(1)), (6), (32) and (36) into eq. 35 and considering the fact that *? >> 1, one obtains:
LI
Urain(O~#R
L I ) -1
L
*?U,
L
2-
8U,
(37)
or:
U. ~
Vmin
+
2*?(L~/L)
~#R
(L~/L)
28
(37a)
The desired LI can be obtained as an optimal solution o f the problem of maximizing the net benefit from seawater intrusion. Eq. 37a can be also interpreted as a relation b e t w e e n the mean o u t f l o w to the sea (8 U , ) and the depth of seawater intrusion. It can be shown that the loss o f groundwater to the sea can be brought to a minimum at the point: 1
L1/L = [8 Umin/(*?apR) ] ~ (3) Constraints on the operational parameters. By the definition of the problem k and Pmin must be non-negative. According to eq. 27 (1) the requirement h/> 0 leads to the constraint: 48 ~<~22 -
2ZeaaR ) 2
u,
•
< (1+8) 2
(38)
108
Inserting the expression for k from eq. 27 into the expression for Pmin, given by eq. 30, and requiring Pmin/> 0, yields the constraint:
( ZooRl2 PR
~<
/
1+5
U, - 1
(39)
~PR
By eqs. 38 and 39:
aUR
( 1+
Zo.oR
-
<~ U, <~ - -
(40)
The totality of the above constraints imposed on U. leads to a selection of U. according to the rule: U.= m a x ) staR
(
Z°OR),
l 1+6 1 +
min
U**, L
+ '
26
L,/L
(41)
The Zo parameter The confidence level, 0, implies that during an infinitely long period of time one m a y expect the undesired event Ui < Vmin or Ui > Vmax once in every (1 - 0) -l years, (1 - 0) -l being the recurrence interval of a failure to provide the promised supply (Riggs, 1961). In practice, however, one plans the annual operation of a basin subject to the requirement that with a given probability, P, there will be no failure or any prescribed number of failures (k) during a given finite number of consecutive years (n). By the binomial distribution and, considering a confidence level o f annual supply, 0, one has:
Pk = (~) (I - O)k 0 n-k
(42)
If no failure is required during the first no consecutive years: k = 0,
and
wherefrom:
P-P0=0n°
(42a)
0 =n~/ Po =- 0 o
(43)
Eq. 42 can serve as the simplest relation for determining 0, and consequently Ze, by prescribed values of no and Po. By eq. 42, 0 is an increasing function of both P0 and no and it is always larger than P0. Another constraint is imposed on Ze by the inequalities (38) and (39), yielding: _1 _1
6 2 U, <~ Zo <~ _
U, - 1
_
o~o R
ot PR
UR aR
(44)
109
By eqs. 43 and 44, once U, was selected, the rule for selecting Zo is given by: 1
= max
Zo
t
!
- - ,
rain
Oo,
- -
t~ o R
U.-1
(45)
at laR
The k parameter
By eqs. 27 and 30: 0 <~ k ~< 2/l °~laR
~i)\ (1-13) - I
\u.
=
2lap/U,
(46)
Also, by eq. 27: 0k d5 +
dk
d~ +
--
ozo
o8
dZo +
do R +
dU.
dU,
(47)
Now: Ok
--
[1 + 1/~/1-
= -
{~21(1 + 8 ) ) 2] (1 - ~)-i < 0
(48)
08
Ok --
ap
=
Ok Oa
[I-,S-
~/(I+8)
2-~2]
(l-~)-2=k/(l-p)>0
~2 ( 1 + ~ ) ~ / ( 1 + 6 ) 2 - a 2 > 0;
0~2
~2 ---> OZo Z0
0~2
~2 ---> OoR oR
0;
(49)
(50)
~2 = 2 Z o a a R / U .
0;
0~2 ~2 . . . . < 0 OU, U,
(51)
Thus, the coefficient of the flexible portion o f the annual pumpage, k, is an increasing function of ~, 0 and o R , yet decreasing with 8 and U , . T h e Pmin p a r a m e t e r
Pmin is the fixed part of the :annual pumpage. By eqs. 28 and 46:
(52)
0 ~ Pmin <~ lap
By eq. 30: _U,(
~Pmin
- - -
a5
1-/3
aPmin
a,6
1
.
alap . a~
+
U. ak . 2 a.6
Ok)
2~8
-
,Up 1-,G
.
-U. [ 1+8 ] 12(1-~) ~/(1 +8)2- ~22 kU,/2
Pmin
1-,6
1-,6
i>0
> 0
(53)
(54)
110
By eqs. 30, 27, 50 and 51: aPmi n
~
k
aU,
1-~
2
U. ak 2
~2 : ]
3~2: + ( l + a ) : [ 1 - x/1
au,
2(l_~)x/(l+a)2_n
2
>0
(55)
In conclusion, both k and Pmin are increasing as the coefficient of return flow increases. On the other hand, b y eq. 28, Pmin is linearly increasing as the mean replenishment increases and decreasing as k increases. As a result, any increase of the operational storage, U,, and of the transfer coefficient a contribute to the transfer of pumpage from the flexible part to the fixed part, whereas any increase of the confidence level, 0, and of the standard deviation of the natural replenishment contribute to an increase of the flexible part of the annual pumpage. f o r k -* kmi n = O,
Pmin ~ (Pmin)max
for k-+ kmax - 2#p U, '
auR -au,
-
1-
Pmin ~ (Pmin)min
=
~.
#p
0
The Pi parameter The total annual pumpage, Pi, as given in eq. 31 can n o w be analyzed b y testing the effect of the operational parameters and of the storage on it. Thus: a Pmin
aP i
aa
-
aa
+ (Ui-
ak U m i n ) -a~ -
=
'
--1
-
(Ui
-
Umin)
1 %/1- 1 n
(1%--~)
Hence: O,
• (1-/3)-' (56)
2
1+
l
~2 2 - ~
(57)
> O,
ou.
fl
U,/2
for Ui - Umin
aPi a8
aPi
2 +1
aPmin
ou.
otherwise
ak
+ (Ui- Umin)-
_--
__
(.a
+
k)
+ ( U / - Umi n - U . / 2 ) -
ak
av.
111 Hence:
OPi
(58)
0U,
1+./~ > 0,
~2 2 1+5
otherwise
SAMPLING EFFECT The mean and standard deviation of the annual rainfall, PR and aR, are actually u n k n o w n population parameters, which have to be estimated from sample data. Let _~ and SR be the sample mean and standard deviation of the annual volume of rainfall and let N be the sample size. One possibility is to employ point estimates (Spiegel, 1961):
(59)
/
N °R = SR "~N - 1
Another possibility is to employ interval estimates of the above parameters and calculate k and Pmin for the confidence limits to obtain a range for Pi as a function of the confidence level, 0. The interval estimates are given b y (Spiegel, 1961):
=
t ~ + Zo SR / x/r-N,
forN>
30
(60)
PR _~ + t(N-') + 0/2 • SR/V/-N-1, -
aR
= /N~
"0.$
SR ( l + Z o
forN<
x/2N),
30
forN;~30
(61) f - ~ . (N-I) SR V IV/Xo.s+o/2 < o R < SR %/-N/x (N-') O.s-o/2
'
forN<
30
where Z0 is the number of standard deviations of a normal distribution on each side of the mean corresponding to the confidence level, 0. t(N-~) 0.s +e/2 is the equivalent of Ze for a t-distribution at a number of degrees of freedom v = N - I .
2(N-I)
_ 2(N-0
Xo.s +0/2 and ×0.5 -0/2 are values of chi-square corresponding
to the 0-1evel confidence limits of the standard deviation.
112
Estimate #R, o'R (59) Assess at,/% 6, U**,Umin (6), (34)
I
I Design L,/L i I oes'~° °°, ~°
I
i I ~et ~°. ~° (~) Determine
I
U, (41) ]
J Adjust U, I= NO
No 4 Compute k ~ Y e s (27 (2))
.o ~i Adju,t zo
.oj
Compute (27 (1,2))k I Compute I Pmin (30)
i Determine
Oi (31)
l
End
I [
Fig. 4. Diagram presenting steps involved in generating the linear rule of annual pumpage from a coastal groundwater basin. Numbers in parentheses indicate equation numbers. PROCEDURE FOR DETERMINING THE RULE OF ANNUAL OPERATION
The diagram of Fig. 4 presents the sequence of steps involved in generating the linear rule of annual pumpage from a coastal groundwater basin. The numbers in the parentheses indicate numbers of equations. The procedure will n o w be illustrated on the coastal groundwater basin in Israel, south of Mt. Carmel. The configuration of the basin in the horizontal plane can be represented schematically by a rectangle extending B = 114 km along the coastline at an average width of L = 15 km. Under natural conditions
113
the basin is fed by replenishment f r o m rainfall and drained by o u t f l o w t o the Mediterranean Sea. Data pertaining to t he probl em at hand are as follows: ~R - / ~ = 920 MCM/year based on N = 46 years o f record in 11 raingauging stations (see eq. 59); MCM = million cubic meters
aR =SR ~ N N 1
= 240 4~__~ ~ 245 MCM/year (seeeq. 59)
D = 60 average d e p t h o f the basin below sea level in meters a ~ 1/3 coefficient of natural replenishment ~ 1/4 coefficient of r et ur n flow T = 103 m2/day equivalent transmissivity n = 1/5 representative effective por os i t y Vmi n = nLBD = 20.5" 103 MCM U** = 1 6 . 2 . 1 0 3 MCM m a x i m u m admissible operational storage between min imu m and m a x i m u m allowable water levels Uo = 3 - 1 0 3 MCM + Umin T°A t
5 -
103. 360
-
nL 2
1/5 × (15.103) 2
= 8" 10 -3
L~ = 1.5 km designed e x p e c t e d de pt h of seawater intrusion inland. Selecting no = 5; Po = 0.9935 yields by 43: ao = 0.9987;
Zo O = 3
By eq. 41:
v.
= max [A, rain (U**, B, C)]
where: A -
B --
a/JR
(
2^2
1+
Ze'*aR
W
= 1.7" 103 MCM
=
2.7"10 3MCM
Umin C
=
2~L~IL
+
O~R
L1
2~
L
.... ~ 4.8"103 MCM
By eq. 41: U, = 2 . 7 . 1 0 3 MCM However, this does n o t satisfy eq. 37a. As the designed value o f L1 is binding, we select: U, = C = 4 .8 " 103 MCM Now: Uo-
Umin = 3 " 1 0 3 M C M >
U./2
114
Hence, we have to check, whether value of U, we find:
eq. 44 is satisfied.
With the finally selected
Hence 20 has to be adjusted. Let ZB =’ 6. Inserting this value into the expression of B above would satisfy both eqs. 37a and 41. Likewise it satisfies eq. 44. The next step would be to compute k by eq. 27. The resulting value is: k = 0.009 whereas, by eq. 30: Pmin = 408.9 - 72.9 = 336 MCM Hence, by eq. 31(l): Pi = 336 + 0.009 (Vi-1 - Umln) or, by eq. 31(3): Pi = 358 + 3.1 (Ri-1 - 7.0); GENERALIZED
SUPPLY
336 MCM < Pi < 380 MCM
POLICY
So far the consumption of water in the region overlying the basin was presumed to be equal to the pumpage from the basin. In practice, however, the demand for water in the region may exceed the permissible pumpage from the underlying basin. Under such conditions one may consider a more general policy of supply, such as: Ci=Pi+Ii;
Ii> 0;
Ci = f(i)
Where i refers to ith year, C is the allocated (predetermined) an import from outside the region for direct consumption. becomes:
(62)
supply and I is With eq. 62, eq. 10
i
U = ai(k)Uo + C j=l
where :
aim’(k) (bRj + cj)
(63)
115
a(k) = ( 1 - 5 - k ) ( 1 + 5 ) -1 b
= a (I +6)-'
cj
=
(64)
[ ( k + 25 ) Umi n - Pmin + ~ Cj ]
(1+6)-'
The expectation of eq. 63 yields:
1 -a i #ui = aiUo + - - b p 1-a
i R + ~ ai-Jcj
(65a)
j=l m
/~U -= VU. = bPR/(1-a) + lira ~
am-J cj
(65b)
.i=1 Hence: m
i
am-Jcj - ~ j--1
ai-Jcj)
(66)
j=l
Also, by eqs. 3 and 66: UPi = Pmin + k I~tU - Umin + (Uo -
~-a ]ai_l(k)_ Gi_l(k) 1
(67)
where: m Gi(k ) - lira ~ am-Y(k)cjm~o~ j=l
i
~_j ai-J(k)cj j=l
By analogy with eq. 11 and since cj is deterministic:
: °ui
-----
2 1-a2i(k) b2a2R; 03--oU~ =b 2 aR2 [ 1 - a : ( k ) ]
-1
l_aS(k)
(68)
By eqs. 3 and 68: 2 2 Opi = k 2 Ovi_,
=
[1-a2(i-')(k)] k2o~r=[1-a2(i-')(k)] k 2" °
[1-a:(k)]
As eq. 20 must hold, we obtain, by virtue of eqs. 67 and 69:
(69)
116 1
I I-a2(i-O(k)l ~ PU
-
Umin
-
=
l - a 2 (k)
go IUo
-
J
°~PR (1 +~) (-l-a)
aaR 1+6
+ G i - , ( k )
I ai-'(k)
-
(70)
Introducing the operational storage, as defined in eq. 22, and demanding that the operational storage be sufficient to satisfy eq. 70 at any i, we obtain:
U.
=
2Z0
' [1-a2(k)] : "
for U0 ~>
aoR 1+6
m + lim ~ a m - l ( k ) c j
m-~o.
j=l
OlPR (1+6) ( l - a )
(71a) m
U.
=
2Zo O~aR , .-lim ~ a m - i ( k ) c j + [1-a2(k)]~ 1+6 + m.-*~ j=l
for U0 <
O~pR (1 +6) ( l - a )
°~l'lR - Vo (l+6)[1-a(k)]
(71b)
Each of the eqs. (71) can be solved for k, according to the comparative value of Uo. By eq. 67, eq. 28 still holds. Also, by eqs. 1, 6 and 62: n
Up = a P R - 6 U * + ~ l i m n.--~ c~
(72)
~ f(i)/n i=1
Hence: Pmin = a#R + ~ lim n--~oo
~ f(i)/ni=1
6 +
U,
(73)
and, finally, with eqs. 3 and 62: [i = Ci - Pi = f ( i ) - Pmin - k ( U i - , - Umin)
(74)
SUMMARY AND CONCLUSIONS A rule was derived for the total annual pumpage from a coastal groundwater basin subject to a prescribed reliability level of supply. The rule contains a fixed part (= firm supply) and a variable part which is proportional to the actual groundwater storage above the sea level. The basin is replenished naturally by a fixed fraction of a random and normally distributed rainfall,
117 and -- artificially -- by a fixed fraction of the volume of water consumed in the area overlying the basin. The basin is drained b y pumpage and b y o u t f l o w to the sea, the latter being proportional to the actual storage above the sea level. T w o alternatives of consumption were considered. In the first alternative, which takes up the major part of this paper, the consumption is taken to be equal to the pumpage. It was found that the parameters of the rule depend, among others, on the ratio between the initial storage and the long-term mean storage, b o t h above the sea level. Most of the paper analyzes the case when this ratio exceeds 1, although reference is made to the opposite case as well. A certain operational storage is needed in order to satisfy the prescribed reliability level. However, some factors, -- namely, the proportionality of the natural o u t f l o w to the storage above the sea level, and the m a x i m u m admissible groundwater level and the desired depth of seawater intrusion -- set bounds on the possible range of the operational storage. This situation is characteristic for a coastal groundwater basin, as opposed to the case of a surface storage reservoir. Within this range any increase of the operational storage increases the fixed part of the annual pumpage at the expense of the flexible part. A similar effect has a composite dimensionless hydrological parameter, 8, which represents the rate of renewal of the groundwater storage above the sea level. On the other hand, given the operational storage, any increase of the reliability level of supply and of the variance of the natural replenishment bring a b o u t an increase of the variable (flexible) part of the pumpage and a reduction of the fixed part. As to the effect o f other factors it was f o u n d that the fixed part of the annual pumpage is a linear increasing function of the mean annual replenishment, whereas the total annual pumpage increases with the coefficient of return flow. The analysis of the parameters of the operational rule is followed b y a procedure for determining these parameters. The application of the procedure is illustrated by an example on the coastal groundwater basin in Israel. Finally, an extension is made to the case of a predetermined consumption, including the possibility of importing water from outside the basin. The results of this paper can be further extended to any groundwater basin which is drained b y any natural outlet (e.g., a spring or a river) introducing a proper modification of the o u t f l o w function. Another worthwhile effort is to generalize the operating rule to a multicell basin. ACKNOWLEDGEMENT This work is a part of a study on the annual operation of the coastal aquifers c o n d u c t e d at the Hydrological Service o f the State o f Israel. The
118
author is indebted to the Director of the Hydrological Service for his permission to publish this work. Thanks are also due to Dr. T. Herman for his valuable comments. REFERENCES Bachmat, Y., 1974. A pragmatic approach to the annual operation of a coastal aquifer. In: Water in Israel. State of Israel, Ministry of Agriculture, Water Commission. Bear, J., 1972. Dynamics of Fluids in Porous Media. American Elsevier, New York, 764 pp. Langbein, W.B., 1958. Queuing theory and water storage. J. Hydraul. Div., Am. Soc. Cir. Eng., HY5: 1--24. Riggs, H.C., 1961. Frequency of natural events. J. Hydraul. Div., Am. Soc. Civ. Eng., HYI: 15--26. Spiegel, M.R., 1961. Statistics. Schaum's Outline Series, McGraw-Hill, New York, N.Y., 359 pp.