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Transient well-flow in an unconfined-confined aquifer system
Journal of Hydrology, 26 ( 1 9 7 5 ) 1 2 3 - - 1 4 0 © Elsevier Scientific P u b l i s h i n g C o m p a n y , A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s
TRANSIENT WELL-FLOW IN AN UNCONFINED--CONFINED AQUIFER SYSTEM
M.H. A B D U L K H A D E R a n d M.K. V E E R A N K U T T Y
Hydraulic Engineering Laboratory, Indian Institute of Technology, Madras (India) ( R e c e i v e d M a y 8, 1 9 7 4 ; a c c e p t e d N o v e m b e r 25, 1 9 7 4 )
ABSTRACT A b d u l K h a d e r , M.H. a n d V e e r a n k u t t y , M.K., 1975. T r a n s i e n t well-flow an u n c o n f i n e d - c o n f i n e d a q u i f e r s y s t e m . J. Hydrol., 26: 1 2 3 - - 1 4 0 . E q u a t i o n s for t r a n s i e n t flow t o a well p e n e t r a t i n g t w o aquifers - - a w a t e r t a b l e a q u i f e r o v e r l y i n g a c o n f i n e d a q u i f e r - - are derived. T w o cases are c o n s i d e r e d for analysis. In t h e first case flow t o w a r d s a well o f zero discharge w i t h d i f f e r e n t initial h e a d s is analysed. In t h e s e c o n d case t h e s o l u t i o n is o b t a i n e d for a well p e n e t r a t i n g t w o aquifers w i t h a cons t a n t discharge a n d identical initial heads. S c h a p e r y ' s m e t h o d o f i n v e r s i o n f o r Laplace t r a n s f o r m is used in t h e analysis. S o l u t i o n s o f t h e s e t w o cases are c o m b i n e d t o o b t a i n general s o l u t i o n s for d i f f e r e n t field c o n d i t i o n s . F r o m t h e e q u a t i o n s derived, it is also possible t o calculate t h e c o n t r i b u t i o n s f r o m individual a q u i f e r s t o t h e t o t a l discharge. I m p o r t a n t results of t h e analysis are given in tables a n d c h a r t s u s i n g t h e usual range o f values o f different parameters.
INTRODUCTION
The importance of groundwater in solving water supply problems in arid regions is increasingly felt in recent years because of its low initial cost and relatively good quality. Wells have been used as one of the important methods of development of groundwater. Wells are also used for drainage of agricultural lands, recharge of groundwater basins, relief of artesian pressure in mining areas and under dams and levees and control of salt water. In groundwater exploitation, it is quite c o m m o n that wells are drilled to penetrate into t w o or more aquifers having different hydraulic properties. Such aquifers which are connected only through the well may either be a system of confined aquifers or more commonly, a system in which the top one is a water table aquifer (Todd, 1959; Panchanathan et al., 1973). In Neyveli Lignite area in South India, mining operation was faced with a serious hydrological problem due to the existence of water under pressure in the artesian aquifers below the lignite bed. These aquifers are found to be a system of t w o or more layers separated by clay seams. In some areas these confined aquifers are overlain by water table aquifers also (Subramanyam, 1969).
124
In this paper the problem of flow towards a well penetrating a two-aquifer system with a water table aquifer at the t o p is analysed. Hydraulic characteristics of a multi-aquifer well are different from and more complex than those of a single aquifer well. The water level in a multiaquifer well lies between the highest and the lowest initial heads in the various aquifers even without pumping and there exists an internal flow. While pumping takes place discharging aquifers increase their flow into the well and the others decrease or even reverse their flow depending upon the rate and duration of pumping. Equations for single aquifer wells have been derived under the basic assumption of either constant discharge or constant head. If the discharge from each aquifer in a multi-aquifer well is controlled at a constant rate or if the head is maintained constant, single aquifer equations are applicable in this case also. But under actual field conditions there are periods of no withdrawal during which neither the head nor the discharge from each aquifer is constant. Moreover, practical means of accurately controlling the rate of flow from each aquifer are n o t available. Therefore it is necessary that independent analysis is carried o u t to derive equations for multiple aquifer wells. A simple steady state equation relating water level fluctuations in a nonpumping multi-aquifer well with changes in head in any one aquifer was derived b y Sokol (1963). Recently, a detailed study was carried out b y Papadopulos (1964, 1966) for unsteady flow into wells in a confined multiple aquifer system. Papadopulos (1966) derived equations for drawdown around the well using asymptotic solutions. For the case of a two-aquifer well penetrating one water table aquifer and a confined aquifer, Abdul Khader et al. (1973) have given a mathematical formulation. The aim of this paper is to present a complete solution for the problem of unsteady flow into a well penetrating a two-aquifer system with a phreatic aquifer at the top. Equations for drawdown around the well and for the discharge contributions from each aquifer are derived. The final solution is obtained using a method of inversion for Laplace transform suggested by Schapery (1962) for visco-elastic problems. Sternberg (1969, 1973) adopted this method for radial flow problems in groundwater and the results are found to be in good agreement with those obtained b y exact inversion. THEORETICAL FORMULATION
Two cases are considered for analysis, viz. a well of zero discharge with different initial heads and a well of constant discharge with identical initial heads. The solutions obtained in the t w o cases when superposed suitably, give the general solution for any of the field conditions generally encountered. The basic equations governing the flow of groundwater into wells are solved subject to appropriate initial and boundary conditions using the following assumptions: (a) For the b o u n d a r y condition describing the discharge, the well is idealised
125
as a sink. That is, instead of the actual cylindrical section for water entry, a hypothetical line is assumed. This simplification is frequently used in treating problems where the well radius is very small. Equations of flow towards wells in unconfined aquifers (Boulton, 1954), wells in leaky and non-leaky confined aquifers (Theis, 1935; Hantush and Jacob, 1955), partially penetrating wells (Hantush, 1957, 1961) and multi-aquifer wells (Papadopulos, 1964) have all been derived under this assumption. (b) In applying the boundary condition at the free surface in an unconfined aquifer, the lowering of the water table is assumed to be the same as changes in head along the static water table position where radial flow components are neglected (Boulton, 1954; Streltsova, 1973). (c) In the case of an unconfined aquifer the specific storage is assumed to be very small compared to the specific yield. It may be noted that recent studies by Neuman (1972) suggest that the compressibility of an unconfined aquifer is sometimes significant, especially in the early periods of flow, and that it cannot be completely ignored. However, in the present analysis, the contribution from aquifer compression in the unconfined aquifer is neglected (Boulton, 1954; Hantush, 1964). (d) In applying the boundary condition describing drawdown at the well face, the well losses are neglected. The head at the well face in any aquifer at any time (t > 0) is assumed to be equal to that in the well and identical to that of the other aquifers. Equations for drawdown around a multi-aquifer well (Papadopulos, 1964) were derived using this assumption. Abdul Khader et al. (1973) have also used the same assumption for the c o m m o n boundary condition at the well, in formulating the problem. (e) The aquifers are assumed to be separated by completely impermeable layers and are connected only through the well. Therefore leakage of any kind is not considered in the analysis. With the above assumptions, the initial and boundary value problem can be described as follows: Case 1: Well of zero discharge (Fig. 1) Equations for an unconfined aquifer (aquifer 1): 82sl
1 8sj
--+---
8r 2
~S 1
E
r 8r
b2sl +--=0
(I)
8z 2
~S 1
+ -- -~z k, 8t
= 01z = H,
(2)
~z
= 01z = o
(3)
s , (r, z, 0 )
= 0
(4)
s~ (~, z, t)
=0
(5)
~Sl
126 ~sl 2;rT~ r ~-r I = - O , ( t )
lim r-*O
(6)
I
for confined aquifer (aquifer 2): ~2s~
1 ~s2
1 ass
~r 2
r ~r
v ~t
s2 (r, 0)
=0
s~ (~, t)
=0
2~T2 r ~-r
lim r~O
(7) (8) (9)
= --Q2(t)
(10)
= H2 -- s2(r w, t)
(11)
=0
(12)
and at the well:
H1-
sl(r w, h w, t)
Q~(t) + Q2(t) in which: S1
82
H1 ,H2
kl,k2 e
T1
% b~ V
S~ rW
hw Q,(t)~ Q2(t)j
drawdown in unconfined aquifer at any distance r from the centre of the well, time t and height z measured from the b o t t o m of unconfined aquifer; = drawdown in the second aquifer at any distance r from the centre of the well and time t; = initial heads in first and second aquifers respectively measured from the b o t t o m of the first aquifer; = hydraulic conductivities of the first and second aquifers respectively; = specific yield of unconfined aquifer; = k ~ H l , transmissibility of unconfined aquifer; = k2 b2, transmissibility of second aquifer; = thickness of the second aquifer; = T2/$2, hydraulic diffusivity of second aquifer; = storage coefficient of second aquifer; = effective radius of the well; = water level in the well measured from b o t t o m of unconfined aquifer at any time t; = discharge contributions of first and second aquifers respectively at any time t. =
Me t h o d o f solution The differential equations 1 and 7 are solved using the boundary conditions in terms of the u n k n o w n discharge contributions, and then the two solutions are coupled using the c o m m o n boundary conditions. Laplace and Hankel transform techniques are adopted for solving the partial differential equations.
127
OROUNO LEVEL
s1 .
.
.
.
.
.
.
.
WATER TABLE
.
$2 UNCONFINED AQUIFER
I i
'l i
~
izl IAI
AQU,FER
~ONF,NEO
k2 ~///./
I ]
'
','
t
52
f
' H2
PIEZOMETRIC SURFACE FOR CONFINED AQUIFER
H,
I b
,'i . / / / / / ). / / / / / / / / / / .
./ / .../~///"///////////,j
"/////.'/////////~
Fig.1. D e f i n i t i o n s k e t c h for d r a w d o w n in a well o f zero discharge p e n e t r a t i n g t w o aquifers o f d i f f e r e n t initial heads.
By using Laplace transform with respect to 't' (Churchill, 1958) and initial conditions 4 and 8, the problem is expressed as: =0
(13)
= 01 z = H,
(14)
= 01z = 0
(15)
~,(oo, z,p)
=0
(16)
lira r-*0
-
--+ r ~r
r2 a~, - -
~z
~z2
e +
Pgl
- -
k,
~s~ ~z"
r
~2~-2 0r 2
-
-
0r )
I Os2 +
--
_ _
r ~r
s2 (oo, p) lim
r-*0
H, - -
P
l r ~s'2 f ~r
--g,(rw, h w, P)
~,(p) + ~ ( p )
2~ T,
(17)
p _ ------82
(18)
=0
(19)
u
= _ Q2(P_~) 21rT2
//2
(20)
= - - --g2(rw, p) P
(21)
=o
(22)
in which s,, s2 = Laplace transforms of s, and s: respectively, p = parameter of Laplace transformation, and QI(P), Q2(P) = Laplace transforms of Q,(t) and Q2(t) respectively.
128 Now, by using Hankel transformation with respect to r (Sneddon, 1957) and conditions 16 and 17, eqs.13--15 reduce to: m
|~/--/~
(
~z /
2=
0,(p) 21rT,
s, + - -
_
-~1 p~'
( -~z/
=0
(23)
= Ol z = H,
(24)
= 01z=0
(25)
where s I = Hankel transform of Sl,/3 = parameter for Hankel transformation. The general solution for eq.23 is: -
s, = A, cosh(~z) + A2 sinh(~z) +
1
Q,(p)
~
2~T,
(26)
where A, and A2 are constants. By using conditions 24 and 25, A, and A2 are evaluated and the consequent expression for s, is: s,
_Q,(P) /322~T,
[
1 --
p cosh(/~z) 1 p cosh(/]H,) + (k,/e)[J sinh(~H,)
(27)
Inversion of Hankel transform for eq.27 yields: ~
Q,(p) ~ Jo(~r) 11 pcosh(~z) 1 - 2~T, ~ p cosh(~H,) + (k,/e){3 sinh(~H,) d~
(28)
where J0 is the Bessel function of the first kind zero order. At the free surface putting z = h (variable height of upper boundary as shown in Fig.l), the expression for the free surface is obtained as:
~, _ O.,(p) ~f Jo({3r) [ 1 - p cosh(~h) ] 2~T, -~ p cosh(~H,) + (k,/e)~ sinh(~H,) 0
(29)
The unknown function Qt(P) in eq.29 can be evaluated only after obtaining the corresponding expression for s2 in the case of aquifer 2. The differential equation relating to aquifer 2 (eq.18) has the following solution:
s2 = B, Ko(~r) + B2Io(~r)
(30)
where B,, B2 = constants, Io, K0 = modified zero order Bessel functions of first and second kind, respectively, and ~ = (p/v) '/2. By using conditions 19 and 20 and the following properties of Bessel functions (Maclachlen, 1955), viz.: d Ko(~) = 0, I0(~) = ~, ~rr [Ko(~r)] = --~K,(;~r), and lim [xK,(x)] = 1, x--+o
129
eq.30 is reduced to: Q2(p) s2 - 2~T2 Ko(kr)
(31)
By using the common boundary conditions 21 and 22, Q,(p) and Q2(p) can be evaluated in terms of H, and H~. Therefore eqs.29 and 31 reduce to the following:
(H,--H2)}
J°(~r)II--
} d/~
p cosh(13H,) + (k,/e)~ sinh03H,)
/3 81 ~-
pcosh(~h)
O
(32)
P[f~ J°(~rw)~ I 1 --
p cosh (~hw) p cosh(~Ht) + (k,/e)~ sinh(~Hl) f d~ + T'
s2 =
(33)
--(H, -- H2)( TI/T2)Ko(Xr) J°(~rw)/3t
p cosh(13H,) + (k,/e)~ sinh03Hl)
L J0
d/~ + ~-~2K0(Xrw)
Eqs.32 and 33 on inversion give the drawdowns all around the well in the two aquifers. But the inverse Laplace transform for these equations are not readily available. By using the inversion integral formula these equations can be written as:
s, -
(H, -- H2) ~ c +i~ -2-~1 J fl (z)eZtdz c-- i~
s~ =
--(HI--H2) c+i~ 2~i f f2 (z)eZtdz
(34)
(35)
C--i¢¢
in which:
;J°(~r) o ~ f, (z) Z
11-
E0:'0,,rw,I f3
f2(z) = .
1-
z cosh(~h) I z cosh({3Hl)+(k,/e){3sinh(~H,) cl~
z cosh(f3Hl)+(kl/e){3 sinh03Hl)
f
dl3+
Zr
Ko(
w)
(T1/T2)Ko (x/z/vr) 13
z cosh(•H,) + (k,le)~ sinh03H,)
d/3 +--~2Ko(x/z/vrw)
c = any positive real constant such that the line x = c lies to the right of all singularities of eZtf,(z) and eZtf2(z), z = complex variable x + iy.
130 In view of the mathematical difficulty encountered in the direct evaluation of eqs.34 and 35, a m e t h o d suggested by Schapery (1962) is adopted for inversion of Laplace transform in these equations. This m e t h o d of inversion is based on the observation that if d [ p / ( p ) ] / d [ l o g ( p ) ] is a slowly varying function of log p then: 1 f( t ) ~
p-f(p )
p = -2t
Thus eqs.32 and 33 on inversion are reduced to the following:
s, =
s2 =
(H, --/-/2) • fl(T, pa, F)
B(T, a, Fw) + ( T , / T 2 ) K o ( v ~ --(H, -- H2) " (Ta/T2)Ko(px/2-u)
B(T, a, Fw) + (Ta/T2)Ko(~/2u)
(36)
(37)
where:
J°(aO) t c°sh Ocosh0 + TO sinh +T0Osinh0 --c°sh(FO) I
B( T,a,F) = f 0
a - rw , H,
U - rw2 4vt '
T-
2kit ell,
,
p-
r rw
,
F =, h , H,
Fw -
hw H1
By adopting the same m e t h o d for inversion of Laplace transform, the expression for circulation rate in the well is obtained as: 27rT,(HI - - H2)
Q,(t)
B(T, a, Fw) + (T,/T2)Ko(~2u-)
(38)
The function B(T, a, F) is computed for some practical ranges of values of T, a and F and given in Table II. The tables for modified zero order Bessel functions of the second kind are readily available.
Case 2: Well o f constant discharge with identical initial heads (Fig. 2) The head distribution around a well penetrating two aquifers, with a constant discharge and with identical initial heads in both aquifers, can also be described using eqs.1 to 10. However, at the well the following boundary conditions are applicable:
H o - - s , ( r w, h w, t)
= Ho--s2(r w, t)
(lla)
Ql(t) + Q2(t)
=Q
(12a)
in which Ho is the initial head in both the aquifers and Q the constant pumping rate.
131
lO
II
=1
---'-'-'--*-2~-- ~ 4
8ROUND LEVEL
II
x. i i
WATERTAaLEAND
~
PIEZOMETRICSURFACE
/
UNCONF(NEDAQUIFER
~
Ho
kI , E
l_ r ="1, i ]H
I i
~ \ X l
~//////////////////////2////A
I
~
U/7///////////////////////////,
II
I b
=',
CONFINEDAQUIFER
~ . ~" / r/ w ////////////////////////////// Fig.2. D e f i n i t i o n sketch for drawdown in a well of constant discharge penetrating two aquifers o f identical initial heads.
///////////////////'/////////////~'
By using Laplace transform with respect to t eqs.lla and 12a get reduced to:
Y~(rw, hw, p)
= ~(rw, p)
(21a) (22a)
QI(p) + Q2(p) = Q/p
As in case 1 the expressions for ~ and ~ are obtained and given by eqs.29 and 31. Using conditions 21a and 22a one can write:
Q,(P): Jo(3rw)l l _ 21rT1
/3
0
pc°sh(3hw)
p cosh(3H1) + (kl/e)3 sinh(3Hl) Q
"
QI(p)
fd3 K0(~r w) 2uT:
-
(T d T2)QKo(Xrw)
=
(39)
P cosh(3h w) fd~+ T1 g°(krw~ p cosh(flH1) + (kl/e)3 sinh(3H1)
P [ f~o J°(3rw) l Similarly:
J°(3rw) I 1 (~2(P) =
Qf0
3
p
J°(3rw) 1 -3
p
P c°sh(3hw) Id~ cosh(3H1) + (kl/e)3 sinh(3Hl)
P c°sh(3hw) cosh(3H1) + (kl/e)3 sinh(3H1)
d3 +
By substituting eqs.39 and 40, the eqs.29 and 31 are reduced to:
T1 Ko(krw)]
(4o)
132
T, Q--~ Ko(Xrw)foo Jo({3r) I 1
p cosh(~h) p
0
cosh(6H,)+ (k,/e)~ sinh(~H,)
}
dt~
(41) T1
p cosh(~H,) + (kl/e)~ sinh(~H1) d~
+ ~ K°(Xrw I
co
QKo(Xr)f 0
=
2~ T2p [ f ~
J°(/3rw) t 1 -P c°sh(6hw) ~3 t p cosh(~H,) + (k,/e)~ sinh(/3H1)
J°(~3rw) ~ -I
(42) _
P c°sh03hw) d/3 + Ko(),r w p cosh03H,) + (k,/e)~ sinh(/3H,) ~-:
Eqs.41 and 42 on inversion give the expressions for drawdown around a well penetrating two aquifers with identical initial heads with constant pumping rate Q. Inversion of Laplace transform in the above equations is not available in any of the standard tables. Therefore as in case 1 Schapery's m e t h o d is adopted for inversion. Thus eqs.41 and 42 are reduced to: T, Q-~, g0(x/2u)" B(T, p~, F) 2~T1 [B(T, ~, f w ) + (T,/T2)Ko(vf~-)]
(43)
Q " B( T, a, Fw) • go(p~/-2-u) s2 = 2~T2 [B(T, ~, Fw) + (T,/T2)go(x[2-u-)l
(44)
s,
in which the notations are as defined in case 1. Discharge from aquifers 1 and 2 are obtained by inverting eqs.39 and 40. That is: T, T--2Qgo(x/-2-u)
Q,(t) = B(T, ~, Fw) + ( T , / T 2 ) K o ( v / - ~
(45)
QB(T, ~, Fw) BiT, ~, F w) + ( T 1 / T 2 ) g o ( x [ - ~
(46)
Q2(t)
General solution for p u m p i n g wells with different initial heads The solutions obtained for the two cases are now combined linearly to get the general solutions for various field conditions. Two examples are given below. Ca) Solution for a well of constant discharge with different initial heads, assuming that pumping begins as soon as the well is completed, is as follows:
(H, -- H:) " B( T, pa, F) Sl =
B(T, ~, Fw) + (T~/T:)Ko(V~u)
+
--(H~ -- H:)( TJT:)Ko(p~/-~) S2
--
_ _
B(T, ~, Fw) + (T1/T2)Ko(V'-2-u)
Q(T,/T2)Ko(x/2-u) • B(T, p~, F) 2~rT,[B(T, cx, Fw) + (T,/T2)Ko(x/2-u)] QB(T, ~, Fw) . go(pVr-2-u)
+
2uT2[B(T, c~,Fw) + (T,/T2)Ko(x/-~]
(47)
(48)
133
Q,(t) =
(H, -- H2)2. T,
__ +
B(T. ~, Fw) + (V,/V2)Ko(v~-u)
Q( T,/T2)Ko(v/~) B(T, ~, Fw) + ( T , / T 2 ) K o ( v ~
--(H, --H2)2~T, QB(T, (~, Fw) Q:(t) = B(T, (~, Fw) + (V,/T2)Ko(VF~) + B(T, ~, Fw) + (T,/T2)Ko(~/-~
(49)
(50)
(b) Solution for a well penetrating t w o aquifers of different initial heads, the discharge being zero for a time t < to and the discharge being constant for time t > to, is as follows: The solution for the duration in which no pumping is done is the same as case 1 and eqs.36--38 are applicable. For the period t > to the equations are:
(H,- H2)B(T, pa, F) 8, =
S2 =
B(T, a, Fw) + (T,/T2)Ko(x/~-u) --(H, -- H2)( T,/7"2)Ko(px/2u) __+ B(T, a, Fw) + (T,/T2)Ko(x/-~
Q,(t) = Q:(t) = T*U* -
Q( T,/T2)Ko(~/ 2u* ) "B(T*, pa, F) +
2~TI[B(T*, a, Fw) + (T,/T2)Ko(2X/-2-~-)] Q - B(T*, a, Fw) • K o ( p ~ ) 2.T:[B(T*, a, Fw) + (T,/T2)Ko(2V~-~-)]
(H, -- H=)27rT, B(T, ~, Fw) + (T1/T2)go(x/2u)
Q( T,/T2)Ko(
+
B(T*, a, Fw) + (T1/T2)Ko(x/-2-~
--(H1 -- H2)2uT, B(T, ~, Fw) + (T,/T2)Ko(x/2-u)
2~/-2-~)
Q " B(T*, a, Fw) B(T*, a, Fw) + ( T , / T 2 ) K o ( ~
(51) (52)
(53) (54)
2k,(t -- to) ell, rw2
4v(t -- to) NUMERICAL COMPUTATIONS AND RESULTS
For various transmissibility ratios numerical computations were made, each time keeping the ratio of specific yield of unconfined aquifer to storage coefficient o f confined aquifer a constant. The sum of transmissibilities (T, + 7'2) was retained constant and the value of T,/T2 was varied. Graphs are presented for three values of T,/T2 (10, 1.0 and 0.1). Special subroutines were programmed for Bessel function and B(T, ~, F) function. The computations were carried o u t on IBM 3 7 0 / 1 5 5 digital c o m p u t e r at Indian Institute of Technology, Madras. Results were obtained separately for the t w o cases discussed. A generalised nondimensional time factor 7 (viz. [(T, + T2)t/(e + s~)rw 2] ) is used for plotting graphs showing time variations o f drawdown and discharge. Figs.3 and 4 show the time variations of water level in the well and the circulation rate (rate of flow from the aquifer of higher initial head to that having a lower head) for case 1. Fig.5 shows the d r a w d o w n distribution around the well for zero discharge. Figs.6 and 7 represent the time variations of water level in the well and of discharge contribution from unconfined aquifer for case 2. Fig.8 is the d r a w d o w n distribution around the well in case 2.
1.0
0.9
0.8
0,7
0.6
0.
--
J
!
:
"
-2
- -
3
!\--
2
/.
I
-
5
6
= 0,01
rw
H1
= zo
~/s2
zzz-t 1 / t 2 = o. 1
Z I - T 1 / T2
1. 0
0.05
0.0/,
0.03
0.02
0.01
o
~
~.
0.10
0.09
0,01~
0.07
_z 0.06
:~
÷
= lO
9
~
~
PARAMETERS z-t~/t~
7
/
Fig.4. Variation o f circulation rate with t i m e in well of zero discharge penetrating t w o aquifers o f different initial heads.
Fig.3. Variation o f water level with time in well of zero discharge penetrating t w o aquifers of different initial heads.
•
I
0.~
0
I
L o g { ( T 1 *T 2 ) t / ( l ~ + S 2 ) rw2 1 3 4
J
5
J
6
Log {(TI+T2) t /(~--+$2) rw2 }
J
B
HI
rw
(/s
m-tl/t
=
001
2 = ~o
2 = o. 1
ZZ- t 1 / T 2 = 1.0
I - T I / T 2 = 10
PARAMETERS
7
J
9
0.012 0.040 0.069 0.084 0.088 0.089 0.089 0.090 0.090 0.090
0.030 0.067 0.096 0.105 0.104 0.103 0.101 0.100 0.099 0.098
10 102 103 104 l0 s 106 10 ~ 108 109 101°
0.008 0.029 0.055 0.070 0.076 0.078 0.080 0.081 0.082 0.083
200.0
D r a w d o w n for case 1
20.0
/S 2
Dimension-~ __ less t i m e ~ ~.0
~ 0.084 0.040 0.025 0.018 0.015 0.012 0.010 0.009 0.008 0.007
2.0 0.046 0.028 0.200 0.015 0.013 0.011 0.009 0.008 0.007 0.007
20.0
C i r c u l a t i o n rate for case 1
0.029 0.021 0.016 0.013 0.011 0.010 0.008 0.007 0.007 0.006
200.0
Values o f d i m e n s i o n l e s s d r a w d o w n a n d discharge for d i f f e r e n t values o f
TABLE I 2 =
10.0 a n d
0.351 1.573 3.446 5.105 6.414 7.609 8.780 9.950 11.116 12.284
2.0 0.970 0.933 0.904 0.895 0.897 0.899 0.900 0.900 0.900 0.901
0.255 1.330 3.247 5.034 6.401 7.609 8.783 9.953 11.120 12.289
0.263 1.347 3.236 4.988 6.340 7.543 8.714 9.883 11.048 12.216
0.988 0.960 0.931 0.916 0.912 0.911 0.910 0.910 0.910 0.909
20.0
0.993 0.971 0.945 0.930 0.924 0.922 0.920 0.918 0.917 0.916
200.0
Discharge for case 2
0.01)
2.0
=
200.0
rw/H,
20.0
D r a w d o w n for case 2
e/S 2 (T,IT
~=~ C~ CJ~
136
1
10
30
50
r/r w 90
70
130
110
170
llt
0.2
_
150
. . . .
0.&
II
I 0.6 II
/!
-//
0.8
------
DRAWDOWN FOR AQUIFER 1
I-T1/T~-10
- -
DRAWDOWN FOR AQUIFER 2
{ T1 +T2) l / l (
* S2)rw 2 . 3 R108
t"s'i 0 "Ji'" °°'l m
L
1,0
HoS/T2.1.0 m-h/x2.o.1
Fig.5. Drawdown around well of zero discharge penetrating two aquifers of different initial heads.
Log t OL
'
{(T 1 +T2)t/(~*S2)r2 3
2
5
t 6
?
i
1 1
ii
] II ÷
~ . ~
I II
PARAMETERS
12V
I-T,/b II-T1/
= 1o T2 = 1 , 0
.z-T,/T~ oo.,
1/.
E/s2
:~o
• 0.01 16
I
HI
J
Ii
t
-f
Fig.6. Variation of water level with time in well of constant discharge penetrating two aquifers of identical initial heads.
137
TABLE II Values of the function B(T, c~, F) T ~
0.01
0.10
0.50
1.00
10.00
F=0.6 10 10: 103 104 10 s 104 107 108 109 101°
6.057 7.208 8.375 9.543 10.711 11.879 13.048 14.216 15.384 16.550
3.628 4.778 5.944 7.112 8.281 9.449 10.617 11.785 12.954 14.122
2.000 3.136 4.301 5.469 6.637 7.805 8.974 10.142 11.310 12.478
1.340 2.444 3.604 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.030 0.429 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
F=0.8 10 102 103 104 l0 s 106 107 108 109 10 l°
6.025 7.203 8.374 9.543 10.711 11.879 13.048 14.216 15.384 16.550
3.595 4.773 5.944 7.112 8.281 9.449 10.617 11.785 12.954 14.122
1.974 3.132 4.301 5.469 6.637 7.805 8.974 10.142 11.310 12.478
1.322 2.441 3.604 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.030 0.429 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
F = 1.0 10 102 103 104 l0 s 106 107 l0 s 109 10 l°
5.964 7.195 8.373 9.542 10.711 11.879 13.048 14.216 15.384 16.550
3.542 4.765 5.943 7.112 8.281 9.449 10.617 11.785 12.954 14.122
1.940 3.127 4.300 5.468 6.637 7.805 8.974 10.142 11.310 12.478
1.298 2.437 3.603 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.029 0.428 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
Computations were also made for various values o f the ratio e/S2 and the effect of this variation is shown in Table I. CONCLUSIONS
a
Complete analytical solution for the problem of flow into a well penetrating multi-aquifer system in which a water table aquifer lies above a confined
o
[
o
f~
0
o
g
r= N" L~
L~
)
~ "~
%
c~
~ ~
~
I T~ • T ~ / ~
~°
o
o
~°
~.
~0..,
~D
°
Q1 ( t ) / Q 0 tO
139
aquifer is obtained. The solution consists of equations which give the transient and spatial distribution of drawdown as well as variation of discharge contribution with time in respect of each aquifer. The graphs presented in Figs.3--8 can be directly used for practical situations for determining the head distributions in the aquifers and the discharge contributions. It can be seen from the graphs that the drawdown and discharge are largely influenced by the transmissibility ratios. The water level in the well lies nearer or farther from the initial head of a particular aquifer depending on whether transmissibility of that aquifer is larger or smaller. The results of this investigation are of great practical importance because they help to assess the desirability or otherwise of screening both the aquifers simultaneously. ACKNOWLEDGEMENTS
The authors are grateful to V. Sethuraman, Head of the Civil Engineering Department, for his interest in this work. They also gratefully acknowledge the cooperation and helpful suggestions offered by H. Suresh Rao, K. Elango and D. Ramadurgaiah of the Hydraulic Engineering Laboratory.
REFERENCES Abdul Khader, M.H., Elengo, K., Veerankutty, M.K. and Satyanandam, G., 1973. Studies on a Well penetrating a two aquifer system, Proc. Int. Symp. Dev. Groundwater Resour., Madras, 2, Pt. III: 43--52. Boulton, N.S., 1954. The drawdown of water table under nonsteady conditions near a pumping well in an unconfined formation. Proc. Inst. Civ. Eng. Lond., 3, Pt. III: 564--579. Churchill, R.V., 1958. Operational Mathematics. McGraw-Hill, New York, N.Y., 337 pp. Hantush, M.S., 1957. Non-steady flow to a well partially penetrating an infinite leaky aquifer. Proc. Iraqi Sci. Soc., 1: 10--19. Hantush, M.S., 1961. Drawdown around a partially penetrating well. Proc. Am. Soc. Cir. Eng., J. Hydraul. Div., 87 (HY4). Hantush, M.S., 1964. Hydraulics o f wells. In: Ven Te Chow (Editor), Advances in Hydroscience. Academic Press, New York, N.Y., 1: 281--432. Hantush, M.S. and Jacob, C.E., 1955. Non-steady flow in an infinite leaky aquifer. Trans. Am. Geophys. Union, 36: 95--100. McLachlen, N.W., 1955. Bessel Function for Engineers. Oxford Univ. Press, London, 239 pp. Neuman, S.P., 1972. Theory o f flow in unconfined aquifers considering delayed response of the water table, Water Resour. Res., 8(4): 1031--1045. Panchanathan, S., Loganathan, A. and Venkatraman, G., 1973. Study of aquifer systems in Cauvery Delta. Proc. Int. Symp. Dev. Groundwater Resour. Madras, 3(5): 83---95. Papadopulos, I.S., 1964. Non-steady flow to Multiaquifer Wells. Ph.D. Thesis, Civ. Eng. Dep. Princeton University, Princeton, N.J. Papadopulos, I.S., 1966. Non-steady flow to multiaquifer wells, J. Geophys. Res., 71: 4791--4797. Schapery, R.A., 1962. Approximate methods of transform inversion for viscoelastic stress analysis. Proc. 4th U.S. Nat. Congr. Appl. Mech., 1075--1085.
140
Sneddon, I.N., 1957. Elements of Partial Differential Equations, McGraw-Hill, N e w York, N.Y., 327 pp. Sokol, D., 1963. Position and fluctuations of water level in wells perforated in more than one aquifer. J. Geophys. Res., 68: 1079--1080. Sternberg, Y.M., 1969. S o m e approximate solutions of radial flow problems. J. Hydrol., 7: 158--166. Sternberg, Y.M., 1973. Theory and application o f skin effect concept to groundwater wells. Proc. Int. Syrup. Dev. Groundwater Resourc., Madras, 2(3): 23--32. Streltsova, T.D., 1973. Flow near a pumped well in an unconfined aquifer under nonsteady conditions. Water Resour. Res., 9(1): 227--235. Subramanyam, V., 1969. Geology and groundwater aspects of Neyveli Lignite Field, South Arcot District, Madras State, Mem. Geol. Surv. India, 9 4 : 3 1 6 pp. Theis, C.V., 1935. The relation between the lowering of piezometric surface and rate and duration of discharge of a well using groundwater storage. Trans, Am. Geophys. Un., 16: 519--524. Todd, D.K., 1959. Groundwater Hydrology. Wiley, New York, N.Y., 336 pp.
TRANSIENT WELL-FLOW IN AN UNCONFINED--CONFINED AQUIFER SYSTEM
M.H. A B D U L K H A D E R a n d M.K. V E E R A N K U T T Y
Hydraulic Engineering Laboratory, Indian Institute of Technology, Madras (India) ( R e c e i v e d M a y 8, 1 9 7 4 ; a c c e p t e d N o v e m b e r 25, 1 9 7 4 )
ABSTRACT A b d u l K h a d e r , M.H. a n d V e e r a n k u t t y , M.K., 1975. T r a n s i e n t well-flow an u n c o n f i n e d - c o n f i n e d a q u i f e r s y s t e m . J. Hydrol., 26: 1 2 3 - - 1 4 0 . E q u a t i o n s for t r a n s i e n t flow t o a well p e n e t r a t i n g t w o aquifers - - a w a t e r t a b l e a q u i f e r o v e r l y i n g a c o n f i n e d a q u i f e r - - are derived. T w o cases are c o n s i d e r e d for analysis. In t h e first case flow t o w a r d s a well o f zero discharge w i t h d i f f e r e n t initial h e a d s is analysed. In t h e s e c o n d case t h e s o l u t i o n is o b t a i n e d for a well p e n e t r a t i n g t w o aquifers w i t h a cons t a n t discharge a n d identical initial heads. S c h a p e r y ' s m e t h o d o f i n v e r s i o n f o r Laplace t r a n s f o r m is used in t h e analysis. S o l u t i o n s o f t h e s e t w o cases are c o m b i n e d t o o b t a i n general s o l u t i o n s for d i f f e r e n t field c o n d i t i o n s . F r o m t h e e q u a t i o n s derived, it is also possible t o calculate t h e c o n t r i b u t i o n s f r o m individual a q u i f e r s t o t h e t o t a l discharge. I m p o r t a n t results of t h e analysis are given in tables a n d c h a r t s u s i n g t h e usual range o f values o f different parameters.
INTRODUCTION
The importance of groundwater in solving water supply problems in arid regions is increasingly felt in recent years because of its low initial cost and relatively good quality. Wells have been used as one of the important methods of development of groundwater. Wells are also used for drainage of agricultural lands, recharge of groundwater basins, relief of artesian pressure in mining areas and under dams and levees and control of salt water. In groundwater exploitation, it is quite c o m m o n that wells are drilled to penetrate into t w o or more aquifers having different hydraulic properties. Such aquifers which are connected only through the well may either be a system of confined aquifers or more commonly, a system in which the top one is a water table aquifer (Todd, 1959; Panchanathan et al., 1973). In Neyveli Lignite area in South India, mining operation was faced with a serious hydrological problem due to the existence of water under pressure in the artesian aquifers below the lignite bed. These aquifers are found to be a system of t w o or more layers separated by clay seams. In some areas these confined aquifers are overlain by water table aquifers also (Subramanyam, 1969).
124
In this paper the problem of flow towards a well penetrating a two-aquifer system with a water table aquifer at the t o p is analysed. Hydraulic characteristics of a multi-aquifer well are different from and more complex than those of a single aquifer well. The water level in a multiaquifer well lies between the highest and the lowest initial heads in the various aquifers even without pumping and there exists an internal flow. While pumping takes place discharging aquifers increase their flow into the well and the others decrease or even reverse their flow depending upon the rate and duration of pumping. Equations for single aquifer wells have been derived under the basic assumption of either constant discharge or constant head. If the discharge from each aquifer in a multi-aquifer well is controlled at a constant rate or if the head is maintained constant, single aquifer equations are applicable in this case also. But under actual field conditions there are periods of no withdrawal during which neither the head nor the discharge from each aquifer is constant. Moreover, practical means of accurately controlling the rate of flow from each aquifer are n o t available. Therefore it is necessary that independent analysis is carried o u t to derive equations for multiple aquifer wells. A simple steady state equation relating water level fluctuations in a nonpumping multi-aquifer well with changes in head in any one aquifer was derived b y Sokol (1963). Recently, a detailed study was carried out b y Papadopulos (1964, 1966) for unsteady flow into wells in a confined multiple aquifer system. Papadopulos (1966) derived equations for drawdown around the well using asymptotic solutions. For the case of a two-aquifer well penetrating one water table aquifer and a confined aquifer, Abdul Khader et al. (1973) have given a mathematical formulation. The aim of this paper is to present a complete solution for the problem of unsteady flow into a well penetrating a two-aquifer system with a phreatic aquifer at the top. Equations for drawdown around the well and for the discharge contributions from each aquifer are derived. The final solution is obtained using a method of inversion for Laplace transform suggested by Schapery (1962) for visco-elastic problems. Sternberg (1969, 1973) adopted this method for radial flow problems in groundwater and the results are found to be in good agreement with those obtained b y exact inversion. THEORETICAL FORMULATION
Two cases are considered for analysis, viz. a well of zero discharge with different initial heads and a well of constant discharge with identical initial heads. The solutions obtained in the t w o cases when superposed suitably, give the general solution for any of the field conditions generally encountered. The basic equations governing the flow of groundwater into wells are solved subject to appropriate initial and boundary conditions using the following assumptions: (a) For the b o u n d a r y condition describing the discharge, the well is idealised
125
as a sink. That is, instead of the actual cylindrical section for water entry, a hypothetical line is assumed. This simplification is frequently used in treating problems where the well radius is very small. Equations of flow towards wells in unconfined aquifers (Boulton, 1954), wells in leaky and non-leaky confined aquifers (Theis, 1935; Hantush and Jacob, 1955), partially penetrating wells (Hantush, 1957, 1961) and multi-aquifer wells (Papadopulos, 1964) have all been derived under this assumption. (b) In applying the boundary condition at the free surface in an unconfined aquifer, the lowering of the water table is assumed to be the same as changes in head along the static water table position where radial flow components are neglected (Boulton, 1954; Streltsova, 1973). (c) In the case of an unconfined aquifer the specific storage is assumed to be very small compared to the specific yield. It may be noted that recent studies by Neuman (1972) suggest that the compressibility of an unconfined aquifer is sometimes significant, especially in the early periods of flow, and that it cannot be completely ignored. However, in the present analysis, the contribution from aquifer compression in the unconfined aquifer is neglected (Boulton, 1954; Hantush, 1964). (d) In applying the boundary condition describing drawdown at the well face, the well losses are neglected. The head at the well face in any aquifer at any time (t > 0) is assumed to be equal to that in the well and identical to that of the other aquifers. Equations for drawdown around a multi-aquifer well (Papadopulos, 1964) were derived using this assumption. Abdul Khader et al. (1973) have also used the same assumption for the c o m m o n boundary condition at the well, in formulating the problem. (e) The aquifers are assumed to be separated by completely impermeable layers and are connected only through the well. Therefore leakage of any kind is not considered in the analysis. With the above assumptions, the initial and boundary value problem can be described as follows: Case 1: Well of zero discharge (Fig. 1) Equations for an unconfined aquifer (aquifer 1): 82sl
1 8sj
--+---
8r 2
~S 1
E
r 8r
b2sl +--=0
(I)
8z 2
~S 1
+ -- -~z k, 8t
= 01z = H,
(2)
~z
= 01z = o
(3)
s , (r, z, 0 )
= 0
(4)
s~ (~, z, t)
=0
(5)
~Sl
126 ~sl 2;rT~ r ~-r I = - O , ( t )
lim r-*O
(6)
I
for confined aquifer (aquifer 2): ~2s~
1 ~s2
1 ass
~r 2
r ~r
v ~t
s2 (r, 0)
=0
s~ (~, t)
=0
2~T2 r ~-r
lim r~O
(7) (8) (9)
= --Q2(t)
(10)
= H2 -- s2(r w, t)
(11)
=0
(12)
and at the well:
H1-
sl(r w, h w, t)
Q~(t) + Q2(t) in which: S1
82
H1 ,H2
kl,k2 e
T1
% b~ V
S~ rW
hw Q,(t)~ Q2(t)j
drawdown in unconfined aquifer at any distance r from the centre of the well, time t and height z measured from the b o t t o m of unconfined aquifer; = drawdown in the second aquifer at any distance r from the centre of the well and time t; = initial heads in first and second aquifers respectively measured from the b o t t o m of the first aquifer; = hydraulic conductivities of the first and second aquifers respectively; = specific yield of unconfined aquifer; = k ~ H l , transmissibility of unconfined aquifer; = k2 b2, transmissibility of second aquifer; = thickness of the second aquifer; = T2/$2, hydraulic diffusivity of second aquifer; = storage coefficient of second aquifer; = effective radius of the well; = water level in the well measured from b o t t o m of unconfined aquifer at any time t; = discharge contributions of first and second aquifers respectively at any time t. =
Me t h o d o f solution The differential equations 1 and 7 are solved using the boundary conditions in terms of the u n k n o w n discharge contributions, and then the two solutions are coupled using the c o m m o n boundary conditions. Laplace and Hankel transform techniques are adopted for solving the partial differential equations.
127
OROUNO LEVEL
s1 .
.
.
.
.
.
.
.
WATER TABLE
.
$2 UNCONFINED AQUIFER
I i
'l i
~
izl IAI
AQU,FER
~ONF,NEO
k2 ~///./
I ]
'
','
t
52
f
' H2
PIEZOMETRIC SURFACE FOR CONFINED AQUIFER
H,
I b
,'i . / / / / / ). / / / / / / / / / / .
./ / .../~///"///////////,j
"/////.'/////////~
Fig.1. D e f i n i t i o n s k e t c h for d r a w d o w n in a well o f zero discharge p e n e t r a t i n g t w o aquifers o f d i f f e r e n t initial heads.
By using Laplace transform with respect to 't' (Churchill, 1958) and initial conditions 4 and 8, the problem is expressed as: =0
(13)
= 01 z = H,
(14)
= 01z = 0
(15)
~,(oo, z,p)
=0
(16)
lira r-*0
-
--+ r ~r
r2 a~, - -
~z
~z2
e +
Pgl
- -
k,
~s~ ~z"
r
~2~-2 0r 2
-
-
0r )
I Os2 +
--
_ _
r ~r
s2 (oo, p) lim
r-*0
H, - -
P
l r ~s'2 f ~r
--g,(rw, h w, P)
~,(p) + ~ ( p )
2~ T,
(17)
p _ ------82
(18)
=0
(19)
u
= _ Q2(P_~) 21rT2
//2
(20)
= - - --g2(rw, p) P
(21)
=o
(22)
in which s,, s2 = Laplace transforms of s, and s: respectively, p = parameter of Laplace transformation, and QI(P), Q2(P) = Laplace transforms of Q,(t) and Q2(t) respectively.
128 Now, by using Hankel transformation with respect to r (Sneddon, 1957) and conditions 16 and 17, eqs.13--15 reduce to: m
|~/--/~
(
~z /
2=
0,(p) 21rT,
s, + - -
_
-~1 p~'
( -~z/
=0
(23)
= Ol z = H,
(24)
= 01z=0
(25)
where s I = Hankel transform of Sl,/3 = parameter for Hankel transformation. The general solution for eq.23 is: -
s, = A, cosh(~z) + A2 sinh(~z) +
1
Q,(p)
~
2~T,
(26)
where A, and A2 are constants. By using conditions 24 and 25, A, and A2 are evaluated and the consequent expression for s, is: s,
_Q,(P) /322~T,
[
1 --
p cosh(/~z) 1 p cosh(/]H,) + (k,/e)[J sinh(~H,)
(27)
Inversion of Hankel transform for eq.27 yields: ~
Q,(p) ~ Jo(~r) 11 pcosh(~z) 1 - 2~T, ~ p cosh(~H,) + (k,/e){3 sinh(~H,) d~
(28)
where J0 is the Bessel function of the first kind zero order. At the free surface putting z = h (variable height of upper boundary as shown in Fig.l), the expression for the free surface is obtained as:
~, _ O.,(p) ~f Jo({3r) [ 1 - p cosh(~h) ] 2~T, -~ p cosh(~H,) + (k,/e)~ sinh(~H,) 0
(29)
The unknown function Qt(P) in eq.29 can be evaluated only after obtaining the corresponding expression for s2 in the case of aquifer 2. The differential equation relating to aquifer 2 (eq.18) has the following solution:
s2 = B, Ko(~r) + B2Io(~r)
(30)
where B,, B2 = constants, Io, K0 = modified zero order Bessel functions of first and second kind, respectively, and ~ = (p/v) '/2. By using conditions 19 and 20 and the following properties of Bessel functions (Maclachlen, 1955), viz.: d Ko(~) = 0, I0(~) = ~, ~rr [Ko(~r)] = --~K,(;~r), and lim [xK,(x)] = 1, x--+o
129
eq.30 is reduced to: Q2(p) s2 - 2~T2 Ko(kr)
(31)
By using the common boundary conditions 21 and 22, Q,(p) and Q2(p) can be evaluated in terms of H, and H~. Therefore eqs.29 and 31 reduce to the following:
(H,--H2)}
J°(~r)II--
} d/~
p cosh(13H,) + (k,/e)~ sinh03H,)
/3 81 ~-
pcosh(~h)
O
(32)
P[f~ J°(~rw)~ I 1 --
p cosh (~hw) p cosh(~Ht) + (k,/e)~ sinh(~Hl) f d~ + T'
s2 =
(33)
--(H, -- H2)( TI/T2)Ko(Xr) J°(~rw)/3t
p cosh(13H,) + (k,/e)~ sinh03Hl)
L J0
d/~ + ~-~2K0(Xrw)
Eqs.32 and 33 on inversion give the drawdowns all around the well in the two aquifers. But the inverse Laplace transform for these equations are not readily available. By using the inversion integral formula these equations can be written as:
s, -
(H, -- H2) ~ c +i~ -2-~1 J fl (z)eZtdz c-- i~
s~ =
--(HI--H2) c+i~ 2~i f f2 (z)eZtdz
(34)
(35)
C--i¢¢
in which:
;J°(~r) o ~ f, (z) Z
11-
E0:'0,,rw,I f3
f2(z) = .
1-
z cosh(~h) I z cosh({3Hl)+(k,/e){3sinh(~H,) cl~
z cosh(f3Hl)+(kl/e){3 sinh03Hl)
f
dl3+
Zr
Ko(
w)
(T1/T2)Ko (x/z/vr) 13
z cosh(•H,) + (k,le)~ sinh03H,)
d/3 +--~2Ko(x/z/vrw)
c = any positive real constant such that the line x = c lies to the right of all singularities of eZtf,(z) and eZtf2(z), z = complex variable x + iy.
130 In view of the mathematical difficulty encountered in the direct evaluation of eqs.34 and 35, a m e t h o d suggested by Schapery (1962) is adopted for inversion of Laplace transform in these equations. This m e t h o d of inversion is based on the observation that if d [ p / ( p ) ] / d [ l o g ( p ) ] is a slowly varying function of log p then: 1 f( t ) ~
p-f(p )
p = -2t
Thus eqs.32 and 33 on inversion are reduced to the following:
s, =
s2 =
(H, --/-/2) • fl(T, pa, F)
B(T, a, Fw) + ( T , / T 2 ) K o ( v ~ --(H, -- H2) " (Ta/T2)Ko(px/2-u)
B(T, a, Fw) + (Ta/T2)Ko(~/2u)
(36)
(37)
where:
J°(aO) t c°sh Ocosh0 + TO sinh +T0Osinh0 --c°sh(FO) I
B( T,a,F) = f 0
a - rw , H,
U - rw2 4vt '
T-
2kit ell,
,
p-
r rw
,
F =, h , H,
Fw -
hw H1
By adopting the same m e t h o d for inversion of Laplace transform, the expression for circulation rate in the well is obtained as: 27rT,(HI - - H2)
Q,(t)
B(T, a, Fw) + (T,/T2)Ko(~2u-)
(38)
The function B(T, a, F) is computed for some practical ranges of values of T, a and F and given in Table II. The tables for modified zero order Bessel functions of the second kind are readily available.
Case 2: Well o f constant discharge with identical initial heads (Fig. 2) The head distribution around a well penetrating two aquifers, with a constant discharge and with identical initial heads in both aquifers, can also be described using eqs.1 to 10. However, at the well the following boundary conditions are applicable:
H o - - s , ( r w, h w, t)
= Ho--s2(r w, t)
(lla)
Ql(t) + Q2(t)
=Q
(12a)
in which Ho is the initial head in both the aquifers and Q the constant pumping rate.
131
lO
II
=1
---'-'-'--*-2~-- ~ 4
8ROUND LEVEL
II
x. i i
WATERTAaLEAND
~
PIEZOMETRICSURFACE
/
UNCONF(NEDAQUIFER
~
Ho
kI , E
l_ r ="1, i ]H
I i
~ \ X l
~//////////////////////2////A
I
~
U/7///////////////////////////,
II
I b
=',
CONFINEDAQUIFER
~ . ~" / r/ w ////////////////////////////// Fig.2. D e f i n i t i o n sketch for drawdown in a well of constant discharge penetrating two aquifers o f identical initial heads.
///////////////////'/////////////~'
By using Laplace transform with respect to t eqs.lla and 12a get reduced to:
Y~(rw, hw, p)
= ~(rw, p)
(21a) (22a)
QI(p) + Q2(p) = Q/p
As in case 1 the expressions for ~ and ~ are obtained and given by eqs.29 and 31. Using conditions 21a and 22a one can write:
Q,(P): Jo(3rw)l l _ 21rT1
/3
0
pc°sh(3hw)
p cosh(3H1) + (kl/e)3 sinh(3Hl) Q
"
QI(p)
fd3 K0(~r w) 2uT:
-
(T d T2)QKo(Xrw)
=
(39)
P cosh(3h w) fd~+ T1 g°(krw~ p cosh(flH1) + (kl/e)3 sinh(3H1)
P [ f~o J°(3rw) l Similarly:
J°(3rw) I 1 (~2(P) =
Qf0
3
p
J°(3rw) 1 -3
p
P c°sh(3hw) Id~ cosh(3H1) + (kl/e)3 sinh(3Hl)
P c°sh(3hw) cosh(3H1) + (kl/e)3 sinh(3H1)
d3 +
By substituting eqs.39 and 40, the eqs.29 and 31 are reduced to:
T1 Ko(krw)]
(4o)
132
T, Q--~ Ko(Xrw)foo Jo({3r) I 1
p cosh(~h) p
0
cosh(6H,)+ (k,/e)~ sinh(~H,)
}
dt~
(41) T1
p cosh(~H,) + (kl/e)~ sinh(~H1) d~
+ ~ K°(Xrw I
co
QKo(Xr)f 0
=
2~ T2p [ f ~
J°(/3rw) t 1 -P c°sh(6hw) ~3 t p cosh(~H,) + (k,/e)~ sinh(/3H1)
J°(~3rw) ~ -I
(42) _
P c°sh03hw) d/3 + Ko(),r w p cosh03H,) + (k,/e)~ sinh(/3H,) ~-:
Eqs.41 and 42 on inversion give the expressions for drawdown around a well penetrating two aquifers with identical initial heads with constant pumping rate Q. Inversion of Laplace transform in the above equations is not available in any of the standard tables. Therefore as in case 1 Schapery's m e t h o d is adopted for inversion. Thus eqs.41 and 42 are reduced to: T, Q-~, g0(x/2u)" B(T, p~, F) 2~T1 [B(T, ~, f w ) + (T,/T2)Ko(vf~-)]
(43)
Q " B( T, a, Fw) • go(p~/-2-u) s2 = 2~T2 [B(T, ~, Fw) + (T,/T2)go(x[2-u-)l
(44)
s,
in which the notations are as defined in case 1. Discharge from aquifers 1 and 2 are obtained by inverting eqs.39 and 40. That is: T, T--2Qgo(x/-2-u)
Q,(t) = B(T, ~, Fw) + ( T , / T 2 ) K o ( v / - ~
(45)
QB(T, ~, Fw) BiT, ~, F w) + ( T 1 / T 2 ) g o ( x [ - ~
(46)
Q2(t)
General solution for p u m p i n g wells with different initial heads The solutions obtained for the two cases are now combined linearly to get the general solutions for various field conditions. Two examples are given below. Ca) Solution for a well of constant discharge with different initial heads, assuming that pumping begins as soon as the well is completed, is as follows:
(H, -- H:) " B( T, pa, F) Sl =
B(T, ~, Fw) + (T~/T:)Ko(V~u)
+
--(H~ -- H:)( TJT:)Ko(p~/-~) S2
--
_ _
B(T, ~, Fw) + (T1/T2)Ko(V'-2-u)
Q(T,/T2)Ko(x/2-u) • B(T, p~, F) 2~rT,[B(T, cx, Fw) + (T,/T2)Ko(x/2-u)] QB(T, ~, Fw) . go(pVr-2-u)
+
2uT2[B(T, c~,Fw) + (T,/T2)Ko(x/-~]
(47)
(48)
133
Q,(t) =
(H, -- H2)2. T,
__ +
B(T. ~, Fw) + (V,/V2)Ko(v~-u)
Q( T,/T2)Ko(v/~) B(T, ~, Fw) + ( T , / T 2 ) K o ( v ~
--(H, --H2)2~T, QB(T, (~, Fw) Q:(t) = B(T, (~, Fw) + (V,/T2)Ko(VF~) + B(T, ~, Fw) + (T,/T2)Ko(~/-~
(49)
(50)
(b) Solution for a well penetrating t w o aquifers of different initial heads, the discharge being zero for a time t < to and the discharge being constant for time t > to, is as follows: The solution for the duration in which no pumping is done is the same as case 1 and eqs.36--38 are applicable. For the period t > to the equations are:
(H,- H2)B(T, pa, F) 8, =
S2 =
B(T, a, Fw) + (T,/T2)Ko(x/~-u) --(H, -- H2)( T,/7"2)Ko(px/2u) __+ B(T, a, Fw) + (T,/T2)Ko(x/-~
Q,(t) = Q:(t) = T*U* -
Q( T,/T2)Ko(~/ 2u* ) "B(T*, pa, F) +
2~TI[B(T*, a, Fw) + (T,/T2)Ko(2X/-2-~-)] Q - B(T*, a, Fw) • K o ( p ~ ) 2.T:[B(T*, a, Fw) + (T,/T2)Ko(2V~-~-)]
(H, -- H=)27rT, B(T, ~, Fw) + (T1/T2)go(x/2u)
Q( T,/T2)Ko(
+
B(T*, a, Fw) + (T1/T2)Ko(x/-2-~
--(H1 -- H2)2uT, B(T, ~, Fw) + (T,/T2)Ko(x/2-u)
2~/-2-~)
Q " B(T*, a, Fw) B(T*, a, Fw) + ( T , / T 2 ) K o ( ~
(51) (52)
(53) (54)
2k,(t -- to) ell, rw2
4v(t -- to) NUMERICAL COMPUTATIONS AND RESULTS
For various transmissibility ratios numerical computations were made, each time keeping the ratio of specific yield of unconfined aquifer to storage coefficient o f confined aquifer a constant. The sum of transmissibilities (T, + 7'2) was retained constant and the value of T,/T2 was varied. Graphs are presented for three values of T,/T2 (10, 1.0 and 0.1). Special subroutines were programmed for Bessel function and B(T, ~, F) function. The computations were carried o u t on IBM 3 7 0 / 1 5 5 digital c o m p u t e r at Indian Institute of Technology, Madras. Results were obtained separately for the t w o cases discussed. A generalised nondimensional time factor 7 (viz. [(T, + T2)t/(e + s~)rw 2] ) is used for plotting graphs showing time variations o f drawdown and discharge. Figs.3 and 4 show the time variations of water level in the well and the circulation rate (rate of flow from the aquifer of higher initial head to that having a lower head) for case 1. Fig.5 shows the d r a w d o w n distribution around the well for zero discharge. Figs.6 and 7 represent the time variations of water level in the well and of discharge contribution from unconfined aquifer for case 2. Fig.8 is the d r a w d o w n distribution around the well in case 2.
1.0
0.9
0.8
0,7
0.6
0.
--
J
!
:
"
-2
- -
3
!\--
2
/.
I
-
5
6
= 0,01
rw
H1
= zo
~/s2
zzz-t 1 / t 2 = o. 1
Z I - T 1 / T2
1. 0
0.05
0.0/,
0.03
0.02
0.01
o
~
~.
0.10
0.09
0,01~
0.07
_z 0.06
:~
÷
= lO
9
~
~
PARAMETERS z-t~/t~
7
/
Fig.4. Variation o f circulation rate with t i m e in well of zero discharge penetrating t w o aquifers o f different initial heads.
Fig.3. Variation o f water level with time in well of zero discharge penetrating t w o aquifers of different initial heads.
•
I
0.~
0
I
L o g { ( T 1 *T 2 ) t / ( l ~ + S 2 ) rw2 1 3 4
J
5
J
6
Log {(TI+T2) t /(~--+$2) rw2 }
J
B
HI
rw
(/s
m-tl/t
=
001
2 = ~o
2 = o. 1
ZZ- t 1 / T 2 = 1.0
I - T I / T 2 = 10
PARAMETERS
7
J
9
0.012 0.040 0.069 0.084 0.088 0.089 0.089 0.090 0.090 0.090
0.030 0.067 0.096 0.105 0.104 0.103 0.101 0.100 0.099 0.098
10 102 103 104 l0 s 106 10 ~ 108 109 101°
0.008 0.029 0.055 0.070 0.076 0.078 0.080 0.081 0.082 0.083
200.0
D r a w d o w n for case 1
20.0
/S 2
Dimension-~ __ less t i m e ~ ~.0
~ 0.084 0.040 0.025 0.018 0.015 0.012 0.010 0.009 0.008 0.007
2.0 0.046 0.028 0.200 0.015 0.013 0.011 0.009 0.008 0.007 0.007
20.0
C i r c u l a t i o n rate for case 1
0.029 0.021 0.016 0.013 0.011 0.010 0.008 0.007 0.007 0.006
200.0
Values o f d i m e n s i o n l e s s d r a w d o w n a n d discharge for d i f f e r e n t values o f
TABLE I 2 =
10.0 a n d
0.351 1.573 3.446 5.105 6.414 7.609 8.780 9.950 11.116 12.284
2.0 0.970 0.933 0.904 0.895 0.897 0.899 0.900 0.900 0.900 0.901
0.255 1.330 3.247 5.034 6.401 7.609 8.783 9.953 11.120 12.289
0.263 1.347 3.236 4.988 6.340 7.543 8.714 9.883 11.048 12.216
0.988 0.960 0.931 0.916 0.912 0.911 0.910 0.910 0.910 0.909
20.0
0.993 0.971 0.945 0.930 0.924 0.922 0.920 0.918 0.917 0.916
200.0
Discharge for case 2
0.01)
2.0
=
200.0
rw/H,
20.0
D r a w d o w n for case 2
e/S 2 (T,IT
~=~ C~ CJ~
136
1
10
30
50
r/r w 90
70
130
110
170
llt
0.2
_
150
. . . .
0.&
II
I 0.6 II
/!
-//
0.8
------
DRAWDOWN FOR AQUIFER 1
I-T1/T~-10
- -
DRAWDOWN FOR AQUIFER 2
{ T1 +T2) l / l (
* S2)rw 2 . 3 R108
t"s'i 0 "Ji'" °°'l m
L
1,0
HoS/T2.1.0 m-h/x2.o.1
Fig.5. Drawdown around well of zero discharge penetrating two aquifers of different initial heads.
Log t OL
'
{(T 1 +T2)t/(~*S2)r2 3
2
5
t 6
?
i
1 1
ii
] II ÷
~ . ~
I II
PARAMETERS
12V
I-T,/b II-T1/
= 1o T2 = 1 , 0
.z-T,/T~ oo.,
1/.
E/s2
:~o
• 0.01 16
I
HI
J
Ii
t
-f
Fig.6. Variation of water level with time in well of constant discharge penetrating two aquifers of identical initial heads.
137
TABLE II Values of the function B(T, c~, F) T ~
0.01
0.10
0.50
1.00
10.00
F=0.6 10 10: 103 104 10 s 104 107 108 109 101°
6.057 7.208 8.375 9.543 10.711 11.879 13.048 14.216 15.384 16.550
3.628 4.778 5.944 7.112 8.281 9.449 10.617 11.785 12.954 14.122
2.000 3.136 4.301 5.469 6.637 7.805 8.974 10.142 11.310 12.478
1.340 2.444 3.604 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.030 0.429 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
F=0.8 10 102 103 104 l0 s 106 107 108 109 10 l°
6.025 7.203 8.374 9.543 10.711 11.879 13.048 14.216 15.384 16.550
3.595 4.773 5.944 7.112 8.281 9.449 10.617 11.785 12.954 14.122
1.974 3.132 4.301 5.469 6.637 7.805 8.974 10.142 11.310 12.478
1.322 2.441 3.604 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.030 0.429 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
F = 1.0 10 102 103 104 l0 s 106 107 l0 s 109 10 l°
5.964 7.195 8.373 9.542 10.711 11.879 13.048 14.216 15.384 16.550
3.542 4.765 5.943 7.112 8.281 9.449 10.617 11.785 12.954 14.122
1.940 3.127 4.300 5.468 6.637 7.805 8.974 10.142 11.310 12.478
1.298 2.437 3.603 4.771 5.939 7.107 8.276 9.444 10.612 11.780
0.029 0.428 1.345 2.464 3.625 4.792 5.960 7.128 8.297 9.465
Computations were also made for various values o f the ratio e/S2 and the effect of this variation is shown in Table I. CONCLUSIONS
a
Complete analytical solution for the problem of flow into a well penetrating multi-aquifer system in which a water table aquifer lies above a confined
o
[
o
f~
0
o
g
r= N" L~
L~
)
~ "~
%
c~
~ ~
~
I T~ • T ~ / ~
~°
o
o
~°
~.
~0..,
~D
°
Q1 ( t ) / Q 0 tO
139
aquifer is obtained. The solution consists of equations which give the transient and spatial distribution of drawdown as well as variation of discharge contribution with time in respect of each aquifer. The graphs presented in Figs.3--8 can be directly used for practical situations for determining the head distributions in the aquifers and the discharge contributions. It can be seen from the graphs that the drawdown and discharge are largely influenced by the transmissibility ratios. The water level in the well lies nearer or farther from the initial head of a particular aquifer depending on whether transmissibility of that aquifer is larger or smaller. The results of this investigation are of great practical importance because they help to assess the desirability or otherwise of screening both the aquifers simultaneously. ACKNOWLEDGEMENTS
The authors are grateful to V. Sethuraman, Head of the Civil Engineering Department, for his interest in this work. They also gratefully acknowledge the cooperation and helpful suggestions offered by H. Suresh Rao, K. Elango and D. Ramadurgaiah of the Hydraulic Engineering Laboratory.
REFERENCES Abdul Khader, M.H., Elengo, K., Veerankutty, M.K. and Satyanandam, G., 1973. Studies on a Well penetrating a two aquifer system, Proc. Int. Symp. Dev. Groundwater Resour., Madras, 2, Pt. III: 43--52. Boulton, N.S., 1954. The drawdown of water table under nonsteady conditions near a pumping well in an unconfined formation. Proc. Inst. Civ. Eng. Lond., 3, Pt. III: 564--579. Churchill, R.V., 1958. Operational Mathematics. McGraw-Hill, New York, N.Y., 337 pp. Hantush, M.S., 1957. Non-steady flow to a well partially penetrating an infinite leaky aquifer. Proc. Iraqi Sci. Soc., 1: 10--19. Hantush, M.S., 1961. Drawdown around a partially penetrating well. Proc. Am. Soc. Cir. Eng., J. Hydraul. Div., 87 (HY4). Hantush, M.S., 1964. Hydraulics o f wells. In: Ven Te Chow (Editor), Advances in Hydroscience. Academic Press, New York, N.Y., 1: 281--432. Hantush, M.S. and Jacob, C.E., 1955. Non-steady flow in an infinite leaky aquifer. Trans. Am. Geophys. Union, 36: 95--100. McLachlen, N.W., 1955. Bessel Function for Engineers. Oxford Univ. Press, London, 239 pp. Neuman, S.P., 1972. Theory o f flow in unconfined aquifers considering delayed response of the water table, Water Resour. Res., 8(4): 1031--1045. Panchanathan, S., Loganathan, A. and Venkatraman, G., 1973. Study of aquifer systems in Cauvery Delta. Proc. Int. Symp. Dev. Groundwater Resour. Madras, 3(5): 83---95. Papadopulos, I.S., 1964. Non-steady flow to Multiaquifer Wells. Ph.D. Thesis, Civ. Eng. Dep. Princeton University, Princeton, N.J. Papadopulos, I.S., 1966. Non-steady flow to multiaquifer wells, J. Geophys. Res., 71: 4791--4797. Schapery, R.A., 1962. Approximate methods of transform inversion for viscoelastic stress analysis. Proc. 4th U.S. Nat. Congr. Appl. Mech., 1075--1085.
140
Sneddon, I.N., 1957. Elements of Partial Differential Equations, McGraw-Hill, N e w York, N.Y., 327 pp. Sokol, D., 1963. Position and fluctuations of water level in wells perforated in more than one aquifer. J. Geophys. Res., 68: 1079--1080. Sternberg, Y.M., 1969. S o m e approximate solutions of radial flow problems. J. Hydrol., 7: 158--166. Sternberg, Y.M., 1973. Theory and application o f skin effect concept to groundwater wells. Proc. Int. Syrup. Dev. Groundwater Resourc., Madras, 2(3): 23--32. Streltsova, T.D., 1973. Flow near a pumped well in an unconfined aquifer under nonsteady conditions. Water Resour. Res., 9(1): 227--235. Subramanyam, V., 1969. Geology and groundwater aspects of Neyveli Lignite Field, South Arcot District, Madras State, Mem. Geol. Surv. India, 9 4 : 3 1 6 pp. Theis, C.V., 1935. The relation between the lowering of piezometric surface and rate and duration of discharge of a well using groundwater storage. Trans, Am. Geophys. Un., 16: 519--524. Todd, D.K., 1959. Groundwater Hydrology. Wiley, New York, N.Y., 336 pp.