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Transient groundwater flow in an aquifer-aquitard system in response to water level changes in rivers or canals
Journal of Hydrology, 133 (1992) 233-257
233
Elsevier Science Publishers B.V., Amsterdam [3]
Transient groundwater flow in an aquifer-aquitard system in response to water level changes in rivers or canals W.Z. Zhang Wuhan University of Hydraulic and Electric Engineering, Department of Irrigation and Drainage, Wuhan, Hubei, People's Republic of China (Received 30 January 1990; revision accepted 24 August 1991)
ABSTRACT Zhang, W.Z., 1992. Tiansient groundwater flow in an aquifer-aquitard system in response to water level changes in rivers or canals. J. Hydrol., 133:233-257 Previous authors have studied unsteady flow in aquifer-aquitard systems with negligible storativity in the confined aquifer layer. Formulae have lyze~ developed for groundwater level in the aquifer (or aquitard) and no adequate tables and graphs have been provided. In this paper, solutions for transient flow in an aquifer-aquitard system that consider storativity in a :onfined layer in response to both abrupt and uniform water level changes, as well as steady seepage rate frc,m rivers (or canals), are develo0e~i, Analytical formulae for groundwater levels both in the aquifer a:~d .quitard, and for seepage rate and total amount of water percolated from rivers, have been derived. To facilitate computation, graphs for the special functions involved in the proposed equations have been prepared.
INTRODUCTION
In many plain areas, rivers flow through alluvial deposits comprising shallow water-bearing strata that consist mainly of two layers: an upper s?lty loam or clay aquitard, and a lower sand or gravel aquifer. The study of groundwater flow in a stream-aquifer system is of theoretical and practical interest: (1) to evaluate the influence of,'iver stages on the groundwater regime in nearby areas, (2) to predict the change of groundwater level and the range of waterlogging induced by water storage in reservoirs in plain areas, and (3) to assess the recharge of rivers to aquifers. Formulae, tables and graphs for unsteady groundwater flow in a homogeneous aquifer have been given by Zhang (1955, 1959, 1983), Averianov (1957) and Ferris et al. (1962). Verigin and Shestakov (1954) and Shestakov (1963) have investigated unsteady flow in the case of negfigible storativity in a lower confined aquifer. However, fcrmulae were given for an upper (or lower) layer only, and the tables for
0022-1694[92[$05.00
© 1992 - - Elsevier Science Publishers B.V. All rights reserved
234
W.Z.
NOTATION
B
D gl, g2 G1E, G I L G2E, G2L ho hz h2 /-/m, H,,, , /40 kl, k2 Dl In l tli 2
MI, M2 MIE, M I L M2E, M2L P P q~ q0
Q $1 s, SIE, SIL S2E, S2L I Ira, l,): 1 l, t T2 Vm"
Vm -- I
vo V
leakage factor, B = ~/k2m2m~/k~ [L] function in eqr~. (25) gl = # 2 ( r / - 1)//fl(q- 1 ) + 1. g2 = / t 2 ( q - l)/B"(r/- 1 ) functions in eqns. (53) and (55) functions in eqns. (54) and (56) initial water table height [L] water table he;~ght in aquitard [L] piezometric head in aquifer [L] change of water level in river at t = t,, and t = t,,,_, [L] change of water level in river at time t = 0 [L] hydraulic conductivity of aquitard and aquifer [LT -1] ordinal number of occurrence of water level change in river saturated thickness of aquitard [L] thickness of aquifer [L] functions in eqns. (30) and (31) functions in eqns. (36) and (38) functions in eqns. (37) and (39) variable of Laplace transform for early time solutions variable of Laplace transform for late time solutions seepage from canal [L-'T-'] seepage rate at x = 0 [L-'T ~] function in eqn. (24) water table change in aquitard [L] piezometric head change in aquifer [L] functions in eqns. (17') and (18') functions in eqns. (21) and (22) time [T] time of occurrence of ruth and m - lth water level change [T] dimensionless time. 7 = ~2t. t ' = ~lt transmissivity of aquifer [L2T -~] rate of change of water level at t = t,., and t = t,,,_ ~ [LT- ~] rate of uniform change of water level at t = 0 [LT ~] total volume of percolated water [L: L ~]
Greek letters ~, = k l l m j l ~ j ~2 ~;* = k)Imj
q ttl It2
specific yield of aquitard storativity of aquifer
ZHANG
235
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
special functions presented in their papers are not sufficient ¢or practical use. In the present work, solutions for unsteady flow in an aquifer-aquitard system that consider storativity in a confined layer are developed: formulae and tables for early and late time are also given. THE FUNDAMENTAL EQUATIONS FOR TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
The aquifer-aquitard system studied is shown in Fig. 1(a). The thickness of the aquitard below the groundwater table is ml, its hydraulic conductivity is kl and specific yield/zl. The thickness of the lower aquifer is m2, its hydraulic conductivity is k2 and elastic storativity P2. The aquifer is underlain by a horizontal impervious layer. As the hydraulic conductivity in the aquitard k~ is far less than that in the aquifer (kl '~ k2), in practice, we can assume that the flow direction is vertical in the upper layer and horizontal in the lower layer (Bear, 1972). When the changes of groundwater level are far less than the initial saturated thickness ofaquitard, for simplicity rnl can be taken as the average thickness of the upper layer without introducing notable error. Owing to the pressure difference in aquifer and aquitard, the intensity of leakage e from the former to the latter can be expressed as kl e
-
(h2
--
hI)
mi
where h2, h I denote the elevations of the piezometric head and water table in aquifer and aquitard respectively. Since the intensity of recharge should be equal to the change of storage in aquitard per unit timex we have the following flow equation for the upper layer: t~hl
kl (h 2
Pl ~t
= ~5 i
jfl//
(1)
,
~Z
.."
/.t~ • / ' ,"
112 "'-//2
..../
I.
ha
The aquifcr-aquitard system.
" t ,~f.tt /i. i~, i . . . i~ "-l~-~"/~i'~'~.,,~l ~t ~//'t/ ;t"
O"
///////J/i//"J/1/////////.(a),/////,/i..t/iiiil Fig.
ts
." ." / ,
" / /
v/
I
h I)
/ / /'.///,.'"
i '-
--
m-"~
l ll
i
ll 'J~Z
l~.O
//l/~lillf lll/lli///I/il/i//l/iTllllllii/Jll ] l fb)
236
w.z. ZHANG
In the confined layer, the change of flow rate per unit distance in the x direction Oq
Cq(k2m2 -~)
~X
~X
82h, =
- - k 2 m 2 63x 2
the intensity of leakage loss to the aquitard kl (h2 - hi) ml
and the change of storage in aquifer per unit time 8h2 #2 c3t
should keep balance. Thus, we have t;3h2
#2 8t
=
632h2 k2m2 ¢3x2
k,
ml
(h 2 -- hi )
(2)
Let kl
=
k2m 2 =
8*,
ml
T2
where 7"2 is the transmissivity of the aquifer, and sl
=
hi - ho,
s2 =
h2 - ho
where s~ and s2 denote the changes of water table and piezometric head in aquitard and aquifer respectively. Substituting ~*, T2, s~ and s2 into (1) and (2), we obtain ~s 1
#1 c~t = ~*(s2 - s~)
(1')
#2 -~-Os2 =
(2')
T2 O2s--2-~2Ox - ~*(s2 - s,)
TRANSIENT G R O U N D W A T E R FLOW IN THE CASE OF A B R U P T WATER LEVEL CHANGES
Derivation o f formulae f o r groundwater levels
When river stage fluctuates accerding to a hydrograph consisting of stepwise abrupt changes, as shown in Fig. 2(a), the water level in the river at any
237
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
S~J s2co,t~
I
sz~o.t,
Hm
T
H~
?t
,--t =
trn
(a)
-
(b)
Fig. 2. Step-wise changes on hydrograph of river stage.
time t > t,. can be expressed as o/
S2(0, t) = E (Hm -- H.,_I)
t > tm,
(3)
0
W h e n 0 < t < fi, only a single abrupt water level change exists, so that s2(0, t) =
H0
(3')
If the initial groundwater levels in both aquifer and aquitard are equal to ho before river stage changes, then the initial conditions will be
s2(x, O) =
0
(4)
s,(x,O)
0
(5)
=
In the case o f a semi-infinite aquifer, we have s 2 ( ~ , t)
=
0
(6)
s l ( ~ , t)
=
0
(7)
Introducing dimensionless variables/" = ~2t and £ = x/~ into (1') and (2'), we obtain (r/ -
1) Osl &t =
63S2
C~2S2
3t': =
022
s2-
(8)
sl
(s2 - sl)
(9)
where kl o¢, "
=
, m i/h
.ut -P2
/~l + =
~2 --
/~2
1
=
q--
1,
238
w.z. Z H A n G
k m 2m
P =
,i-
Applying Laplace transformation to (8) and (9), we obtain
(rl
-
1)pgl
pS2
=
g2
d292 dx2
=
-
(lO)
gi
(li)
(32 -- $1)
From (10), we obtain $2
g~
(to')
( q - l)p + 1
Substituting (10') into (11), we have
d2s2 dg 2
(0 - 1)P2 + tlP = 0 (r/-
(ll')
l)p + 1
The general solution for (11') is
'~2
=
CI
exp
2p + ~ q
+ c, e x p [ - 2 " L
1 1
~ q p + ~ 1 q-I
1
]
(12)
Determining the constants ct, c2 from the boundary conditions (6) and (3'), we have q
r/l p + ~ I/-1
Ho
g2(5c, p) =
- - exp P
s~(.~, p)
p[(q-
(13)
No
1)p + 11 exp
_~
~/
..... f-P+n-1
(14)
239
TRANSIENT GROUNDWATER FLOW IN J~N AQU1FER-AQUITARD SYSTEM
Similarly, by introducing the dimensionless variables F = ~lt and ~l =
k t/mlp I, another group of Laplace transform solutions can be obtained: g,( 2, P) "
gl(2,
Hoexp[_2
-
P
P(P+ q)
/,
,_,,,q-
I)(P + l ) j
P) - p(p + 1) exp - 2
-]
(13')
(14')
~- l)(P + 1)
With the aid of the inverse theorem of Laplace transformation, we obtain the solutions for water levels:
s,(x, t) = H° - ~1{
2 [ f es xi np ( 2- gu l\ F/ q) - g l ) d U o u
t/v',/.z~
(15)
7,,_ I
U
•,/nln
sin(2u~/rl--g')du no (1 - 2 { f exp ( - g , [ ) u[l + u2(q - 1)1
s2(x, t)
1~
0
1t,7--i
-
~
exp
(-g2"i')
sin (2ux/rg~2- q ) } ) u[u2(r / _ 1) -- 1] du
(16)
~;~/q- I
where
u2(q- 1) gl
u2(q- 1)
1 + u2(q- 1)'
g2 = uZ(q_ 1 ) - 1
Approximate solutionsfor groundwaler levels Approximate solutions ]'or aquitard From the theory of Laplace transformation, at an early time 7 < corresponds to p > 20 (in this case p + 1 ~ p), so that (14) becomes gt = ( q _
H°
l)p2
exp
(-Scxfp + 1)
After inverse transformation, we obtain sl(2,[) -
(q_
1) 2
I + 5- e x , ( 2 ) erfc
+(1-~-)exp(-.~)erfc(2 ~_
+
vG-)]
(17)
240
w.z. ZHANG
At a late time, when f > 20/q, P < r//20, ~I + P '~ q, then (14') becomes H° gm =
exp ~ - .~
p ( p + 1)
+
1
After inverse transformation, we obtain
sl(x, t) = Ito 1 - yrr
~exp u
1 +
u:
du
(18)
In (I7), 7 = k~t/m~#2; in (18), T = k~t/m~tq. Equatior~s (17) and (18) can be rewritten as
s~(x, t) . . . . . . if_o_. S ! E r/-I
(i7')
s,,(x, t) = 11o" S I L
(18')
where S!E
= ~
S1L
=
1 + ~_ exp(2) erfc
1 --2
~z
i~sin~Uexp(~ u
--~_+ 2x/t
u2
1 + u2
0
du
o
On the basis of (18), graphs for S1L have been prepared and are shown in Fig. 3.
Approximate solutions for aqu~l'er Similarly, from (13) and (14) we obtain approximate solutions for water levels in an aquifer: r <
, s2(x, t) = Ho 2 exp (2) erfc
F > -q- , s2(x, t) = Ho 1 - - rt
~--~/~_+ -x/t
u(i + u 2) exp
1 + u2
du
(20)
TRANSIENT GROUNDWATER
Z
3 4 5678~t~I
F L O W IN A N
~
AQUIFER-AQUITARD S Y S T E M
3 4 56?~9~
2
3
4 5678~10
2
3
241
a 5 6 789JDz
2
3
,~ 5 5 7 8 9 0
~D 8 6 5 4
3
.,J f-O
ool 8 6 5 4
o0Ol
0(
d
~
4
D O I 0~*~
"t Fig. 3. Type curves for function SIL.
or
7< n~-~,s'(x, t) = Ho" S2E
(21)
/=> ~,20 s,(x,~ q
(22)
t) =
Ho" S 2 L
where SZE -- l[exp (2)erfc (2w//. + / ~ _ ) + exp (-2)erfc S2L = 1 - ~2 i u(~sin£u + 7,2)exp (
2x/?
1 +u2u" t~ d ,
Type curves for S2E (type A curve) and S2L (type B curve) are given in Fig. 4.
\
t
'%~,,
\
I
I
!
\ \
G" " ~
i....
~ ~
i~~ "---.. i ¸~,
~:iii
~L
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
243
The seepage rate and the total amount of percolation water from river
From Darcy's law, at x = 0 qo =
c~s2 - 7" ~x
(23)
Os2/Ox can be obtained from Os2/CY, b'" i-~werse transformation. Substituting Cs2/&x into (23), we have qo =
T2no
T
Q
(24)
where O(r/,7)
=
~2 i 1 +x / u' ~2 -( r- g/ -,
2 I/~i-'~ ~4~
1) exp ( - g t 7) du
~
-
-i u 2 ( r / -
r/
1)-
1
exp (--g27) du
The total amount of seepage water from the river is
(25)
V = IqHo'B'D
where O(r/,i-)
=
~
o u2(r/ -- 1)
[1 -- exp (--g~ T)] du
f u 2 ( r / - 1) ~,~/,7-I
_
TRANSIENT GROUNDWATER FLOW IN THE CASE OF UNIFORM WATER LEVEL CHANGES
Derivation of formulae for groundwater levels
When the water level in a river or canal rises or falls with a uniform rate, its hydrograph is as shown in Fig. 2. The water level at any time t > tm can be expressed by m
s2(0, t) = ~ (v,, - vm)(t - tin)
(26)
0
whenO< s2(O, t) =
t < h, rot
(2 L~
254
W.Z. ZHANG
curve. The value of 2 0 l he selected type A curve and the abscissas o f any m a t c h point t, i are n,,ted. From t" ~
0~21
we have
c¢2
=
lit
k l /tlrl '
~
I ]2 2
from 2
=
x/8,
e
=
we obtain
a2 =
T/~2
=
0~2 B2
The field.data curve is then m o v e d horizontally to the right until it is m a t c h e d to a type B curve of the same 2 value as that o b t a i n e d in m a t c h i n g the early time data. In the second position the c o r r e s p o n d i n g abscissas o f any m a t c h point t, Zare noted. Since t = ~ t, we obtain e~ = ~t. From ~2 -
kl
and ~
mi l1~.
=
kl
m: i~!
we have q - 1
-
" eL
-
- 1 ~2
W h e n seepage rate q0 from the river at any given time t~ is available, we can estimate #1 f r o m
qo(fi)
=
f~,BvoD(tl, fi)
or
]d I
-~-
qo(~) BvoDOl, tl )
From q
/-ll
--
+
we have ]-'/1 t~2
-
t~2
TRANS'IEL~F G R O U N D W A T E R F L O W IN A N A Q U I F E R - A t d u I T A R D SYSTEM
255
F r o m T = a2p2, T can be determined. F r o m 9 2
Tm 1
=
kl
we have k, ml
T2 B2,
T ~ m
or kl
T h u s all the parameters
k,/m~(or k~ ), #~,
T ( o r k?.m2), k2) and p_, are obtained.
Example A two-layered succession consisting o f an upper silty loam aquitard 13.13 m thick (below initial water table) a n d a lower sand aquifer of 30.0 m thick is used as an example. A river fully penetrates the a q u i t a r d and extends to the aq~dfer layer. Before storage of water; in the river channel the flow in the st~'eam-aqt,~%r system has attained its steady condition. After storage of water, the river stage begins m rise with a u n i f o r m rate o f 0.2 m da3? -~ . The relative g r o t m d w a t e r level changes, s2/Vot, in an observation well at a distance o f 200 m from the river are as s h o w n in Fig. 9, a n d the seepage rate from the river at t = 5 days is 1.692 m 2d a y - ~. We wish to establish the hydrogeological p a r a m e t e r s o f the a q u i f e r - a q u i t a r d systems. M a t c h i n g the early time field d a t a to a type A curve, we obtain 5 = 1.0 and c o r r e s p o n d i n g abscissas at a point t = 5 and t = 1.0 day. i~datching the late field d a t a to a type B curve, we have ,~ = 1.0 and c o r r e s p o n d i n g abscissas o f a n o t h e r m a t c h point t = 2 and t = 8.0day. F r o m early time data, ~2 = 5/ ~ 1 5 d a y - ~ , S: ~ x/B ~ 1, B ~ 200m, a 2 = ~2 B2 ~ .,00 0 0 0 m 2 da? i - 1 . F r o m late time d a t a 0q = ~t = 0.25, *7 - 1 = 5/0.25 = 20, r/ = 21. F r o m (40)
Bv0D(G h )
W h e n t~ = 5 day, ~ = ~2 h = 1.25, using (25), a table for D can be prepared, f r o m which D = 1.001. Substituting these values into (5), finally, we obtain I ,qo~
~i P2 T =
200 x .0.2 x 1,001 ~! ,7-
a:#2
-
I
=
0.04225 20 200f:)0
=
0.04225
-
0.00211
: 0.00211
=
422m-oa3.
w.z. ZHANG
256
i +
i ,
+ -+-V
i
i
I !
•
i
i
1 +
+-- i ! !
--
7 6
"i
5
--i
"52
,
~
z
i
.....
+I~
"--+ + i i+ ' i +~: ~ -
--
--7"
T1
1
I
I :!
t
+
OJ
+i!
I
i
t
I
i ,
[
+
i-
8
i__
+
7
+ i
6
l+ _ _
5
i-
it
i
+.
t
, ~.+0! _ _ _ _ i _ _ . o.t Z
i
}++ 3
~i]
+ ~ ii t
e
5
+
i
i 6 78~+.0
?.
a
e+
5~
8 S IO
+, (da6) Fig. 9. Relative groundwater level changes in an observation welt at a distance of 200 m from the river.
kl
T
422
m~
B-"
40 000
-
v..+l, o d a y
CON CLUSION
In this paper, solutions cor sidering the sterativity of an aquifer have beea derived for t~e calculation of groundwater level, seepage rate and the total amount of seepage water in response to changes of river stage. Approximate formulae for early and late time and graphs for special functions involved in them are also given. Computational fornmlae show that at early time the elastic storati'~+ity plays an important ro!e in groundwater fl+ow, in the course of continuous upward recharge from an aquifer, the role of gravity storage in the aquitard graduai!y increases and the rate of groundwater level rise reduces. After considerable time the water levels in the aquitard and the piezemetric head in the aquifer layer gradually equalize; at this time the equanons for a homogeneous unconfined aquifer can be used approximately
T R A N S I E N T G R O U N D W A T E R F L O W IN AN AQI.JIFER-AQU1TARD SYSTEM
257
to estimate water level changes in the aquifer-aquitard system. To determine the groundwater level during periods of rapid change of river stage, the equations concerning the storativity of the aquifer should be ~ased, whereas in periods of stow change of river stage that last for a long time, the use of the solutions Ibr an unconfined aquifer will not introduce notable error. REFERENCES Averianov, S.F., 1957. Influence of irrigation system on regime of ground°water, ~rt: A.N. Kostiakov, N.N. Favo~in and S.F. Averianov (Editors), The Influence of Irrigation on Regime of Groundwater, Collection of Papers, Vot. :. Press of Academy of '~eic~,,es, Moscow (in Russian). Bear, J.. 1972. Dynamics of Fluids in Porous Media. Elsevier, New York. Boulton, N.S., 1954. The drawdown of water table under nonsteady conditions near a pumped wel~ in an uncov,fined formation. Proc. Inst. Civil Eng.. 3(3): 564-579. Ferris, ~.G,, Knov !,-~;, D.B., Brown, R.H. and Statlman, R.W., 1962, Theory of aquifer tests. U.S. Geol, Surv, Water Supply Pap., 1536-E. She~tako',, V.M, i963. Unsteady flow in two-!ayered media. Proceedings of Academy of Sciences USSR, Division of Engineering Sciences, Mechanics a,nd Mechanical Engineering, No. 6, pp. 93-96 (in Russian). Stehfest, H,, 1970. Numerical inversion of Laplace transform, Cemman. A C M , 13 (1): Vefigie, N.N. and Shestakov, V.M., 1954. Methods for Computing the Grol~:ndwater Movement in Two,,layercd Media. Press of all Union Research Institute of Wat(:~r Supp~i and Hydrogeology. Moscow (in Russian), Zhang, W., 1955, The influence of Irrigation on Regime of Groundwater, Ph.D. Thesis, The Research ":~ ion of Problems of Water Economy, Academy of Sciences, Moscow qn Russian). Zhang, W., 1959, Unsteady regime of groundwater under the influence of irrigation. In Inflvence of Irrigation on Regime of Groundwater, Collection .~f Papers. 'CoL 2, Press of Academy of Sciences, Moscow (in Russian). Zhang. W., 1983. Transient Groundwater Hydraulics and Evaluation ~f Groundwater Resource,,. Academic Prc-;,~,, Beijing (in Chinese).
233
Elsevier Science Publishers B.V., Amsterdam [3]
Transient groundwater flow in an aquifer-aquitard system in response to water level changes in rivers or canals W.Z. Zhang Wuhan University of Hydraulic and Electric Engineering, Department of Irrigation and Drainage, Wuhan, Hubei, People's Republic of China (Received 30 January 1990; revision accepted 24 August 1991)
ABSTRACT Zhang, W.Z., 1992. Tiansient groundwater flow in an aquifer-aquitard system in response to water level changes in rivers or canals. J. Hydrol., 133:233-257 Previous authors have studied unsteady flow in aquifer-aquitard systems with negligible storativity in the confined aquifer layer. Formulae have lyze~ developed for groundwater level in the aquifer (or aquitard) and no adequate tables and graphs have been provided. In this paper, solutions for transient flow in an aquifer-aquitard system that consider storativity in a :onfined layer in response to both abrupt and uniform water level changes, as well as steady seepage rate frc,m rivers (or canals), are develo0e~i, Analytical formulae for groundwater levels both in the aquifer a:~d .quitard, and for seepage rate and total amount of water percolated from rivers, have been derived. To facilitate computation, graphs for the special functions involved in the proposed equations have been prepared.
INTRODUCTION
In many plain areas, rivers flow through alluvial deposits comprising shallow water-bearing strata that consist mainly of two layers: an upper s?lty loam or clay aquitard, and a lower sand or gravel aquifer. The study of groundwater flow in a stream-aquifer system is of theoretical and practical interest: (1) to evaluate the influence of,'iver stages on the groundwater regime in nearby areas, (2) to predict the change of groundwater level and the range of waterlogging induced by water storage in reservoirs in plain areas, and (3) to assess the recharge of rivers to aquifers. Formulae, tables and graphs for unsteady groundwater flow in a homogeneous aquifer have been given by Zhang (1955, 1959, 1983), Averianov (1957) and Ferris et al. (1962). Verigin and Shestakov (1954) and Shestakov (1963) have investigated unsteady flow in the case of negfigible storativity in a lower confined aquifer. However, fcrmulae were given for an upper (or lower) layer only, and the tables for
0022-1694[92[$05.00
© 1992 - - Elsevier Science Publishers B.V. All rights reserved
234
W.Z.
NOTATION
B
D gl, g2 G1E, G I L G2E, G2L ho hz h2 /-/m, H,,, , /40 kl, k2 Dl In l tli 2
MI, M2 MIE, M I L M2E, M2L P P q~ q0
Q $1 s, SIE, SIL S2E, S2L I Ira, l,): 1 l, t T2 Vm"
Vm -- I
vo V
leakage factor, B = ~/k2m2m~/k~ [L] function in eqr~. (25) gl = # 2 ( r / - 1)//fl(q- 1 ) + 1. g2 = / t 2 ( q - l)/B"(r/- 1 ) functions in eqns. (53) and (55) functions in eqns. (54) and (56) initial water table height [L] water table he;~ght in aquitard [L] piezometric head in aquifer [L] change of water level in river at t = t,, and t = t,,,_, [L] change of water level in river at time t = 0 [L] hydraulic conductivity of aquitard and aquifer [LT -1] ordinal number of occurrence of water level change in river saturated thickness of aquitard [L] thickness of aquifer [L] functions in eqns. (30) and (31) functions in eqns. (36) and (38) functions in eqns. (37) and (39) variable of Laplace transform for early time solutions variable of Laplace transform for late time solutions seepage from canal [L-'T-'] seepage rate at x = 0 [L-'T ~] function in eqn. (24) water table change in aquitard [L] piezometric head change in aquifer [L] functions in eqns. (17') and (18') functions in eqns. (21) and (22) time [T] time of occurrence of ruth and m - lth water level change [T] dimensionless time. 7 = ~2t. t ' = ~lt transmissivity of aquifer [L2T -~] rate of change of water level at t = t,., and t = t,,,_ ~ [LT- ~] rate of uniform change of water level at t = 0 [LT ~] total volume of percolated water [L: L ~]
Greek letters ~, = k l l m j l ~ j ~2 ~;* = k)Imj
q ttl It2
specific yield of aquitard storativity of aquifer
ZHANG
235
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
special functions presented in their papers are not sufficient ¢or practical use. In the present work, solutions for unsteady flow in an aquifer-aquitard system that consider storativity in a confined layer are developed: formulae and tables for early and late time are also given. THE FUNDAMENTAL EQUATIONS FOR TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
The aquifer-aquitard system studied is shown in Fig. 1(a). The thickness of the aquitard below the groundwater table is ml, its hydraulic conductivity is kl and specific yield/zl. The thickness of the lower aquifer is m2, its hydraulic conductivity is k2 and elastic storativity P2. The aquifer is underlain by a horizontal impervious layer. As the hydraulic conductivity in the aquitard k~ is far less than that in the aquifer (kl '~ k2), in practice, we can assume that the flow direction is vertical in the upper layer and horizontal in the lower layer (Bear, 1972). When the changes of groundwater level are far less than the initial saturated thickness ofaquitard, for simplicity rnl can be taken as the average thickness of the upper layer without introducing notable error. Owing to the pressure difference in aquifer and aquitard, the intensity of leakage e from the former to the latter can be expressed as kl e
-
(h2
--
hI)
mi
where h2, h I denote the elevations of the piezometric head and water table in aquifer and aquitard respectively. Since the intensity of recharge should be equal to the change of storage in aquitard per unit timex we have the following flow equation for the upper layer: t~hl
kl (h 2
Pl ~t
= ~5 i
jfl//
(1)
,
~Z
.."
/.t~ • / ' ,"
112 "'-//2
..../
I.
ha
The aquifcr-aquitard system.
" t ,~f.tt /i. i~, i . . . i~ "-l~-~"/~i'~'~.,,~l ~t ~//'t/ ;t"
O"
///////J/i//"J/1/////////.(a),/////,/i..t/iiiil Fig.
ts
." ." / ,
" / /
v/
I
h I)
/ / /'.///,.'"
i '-
--
m-"~
l ll
i
ll 'J~Z
l~.O
//l/~lillf lll/lli///I/il/i//l/iTllllllii/Jll ] l fb)
236
w.z. ZHANG
In the confined layer, the change of flow rate per unit distance in the x direction Oq
Cq(k2m2 -~)
~X
~X
82h, =
- - k 2 m 2 63x 2
the intensity of leakage loss to the aquitard kl (h2 - hi) ml
and the change of storage in aquifer per unit time 8h2 #2 c3t
should keep balance. Thus, we have t;3h2
#2 8t
=
632h2 k2m2 ¢3x2
k,
ml
(h 2 -- hi )
(2)
Let kl
=
k2m 2 =
8*,
ml
T2
where 7"2 is the transmissivity of the aquifer, and sl
=
hi - ho,
s2 =
h2 - ho
where s~ and s2 denote the changes of water table and piezometric head in aquitard and aquifer respectively. Substituting ~*, T2, s~ and s2 into (1) and (2), we obtain ~s 1
#1 c~t = ~*(s2 - s~)
(1')
#2 -~-Os2 =
(2')
T2 O2s--2-~2Ox - ~*(s2 - s,)
TRANSIENT G R O U N D W A T E R FLOW IN THE CASE OF A B R U P T WATER LEVEL CHANGES
Derivation o f formulae f o r groundwater levels
When river stage fluctuates accerding to a hydrograph consisting of stepwise abrupt changes, as shown in Fig. 2(a), the water level in the river at any
237
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
S~J s2co,t~
I
sz~o.t,
Hm
T
H~
?t
,--t =
trn
(a)
-
(b)
Fig. 2. Step-wise changes on hydrograph of river stage.
time t > t,. can be expressed as o/
S2(0, t) = E (Hm -- H.,_I)
t > tm,
(3)
0
W h e n 0 < t < fi, only a single abrupt water level change exists, so that s2(0, t) =
H0
(3')
If the initial groundwater levels in both aquifer and aquitard are equal to ho before river stage changes, then the initial conditions will be
s2(x, O) =
0
(4)
s,(x,O)
0
(5)
=
In the case o f a semi-infinite aquifer, we have s 2 ( ~ , t)
=
0
(6)
s l ( ~ , t)
=
0
(7)
Introducing dimensionless variables/" = ~2t and £ = x/~ into (1') and (2'), we obtain (r/ -
1) Osl &t =
63S2
C~2S2
3t': =
022
s2-
(8)
sl
(s2 - sl)
(9)
where kl o¢, "
=
, m i/h
.ut -P2
/~l + =
~2 --
/~2
1
=
q--
1,
238
w.z. Z H A n G
k m 2m
P =
,i-
Applying Laplace transformation to (8) and (9), we obtain
(rl
-
1)pgl
pS2
=
g2
d292 dx2
=
-
(lO)
gi
(li)
(32 -- $1)
From (10), we obtain $2
g~
(to')
( q - l)p + 1
Substituting (10') into (11), we have
d2s2 dg 2
(0 - 1)P2 + tlP = 0 (r/-
(ll')
l)p + 1
The general solution for (11') is
'~2
=
CI
exp
2p + ~ q
+ c, e x p [ - 2 " L
1 1
~ q p + ~ 1 q-I
1
]
(12)
Determining the constants ct, c2 from the boundary conditions (6) and (3'), we have q
r/l p + ~ I/-1
Ho
g2(5c, p) =
- - exp P
s~(.~, p)
p[(q-
(13)
No
1)p + 11 exp
_~
~/
..... f-P+n-1
(14)
239
TRANSIENT GROUNDWATER FLOW IN J~N AQU1FER-AQUITARD SYSTEM
Similarly, by introducing the dimensionless variables F = ~lt and ~l =
k t/mlp I, another group of Laplace transform solutions can be obtained: g,( 2, P) "
gl(2,
Hoexp[_2
-
P
P(P+ q)
/,
,_,,,q-
I)(P + l ) j
P) - p(p + 1) exp - 2
-]
(13')
(14')
~- l)(P + 1)
With the aid of the inverse theorem of Laplace transformation, we obtain the solutions for water levels:
s,(x, t) = H° - ~1{
2 [ f es xi np ( 2- gu l\ F/ q) - g l ) d U o u
t/v',/.z~
(15)
7,,_ I
U
•,/nln
sin(2u~/rl--g')du no (1 - 2 { f exp ( - g , [ ) u[l + u2(q - 1)1
s2(x, t)
1~
0
1t,7--i
-
~
exp
(-g2"i')
sin (2ux/rg~2- q ) } ) u[u2(r / _ 1) -- 1] du
(16)
~;~/q- I
where
u2(q- 1) gl
u2(q- 1)
1 + u2(q- 1)'
g2 = uZ(q_ 1 ) - 1
Approximate solutionsfor groundwaler levels Approximate solutions ]'or aquitard From the theory of Laplace transformation, at an early time 7 < corresponds to p > 20 (in this case p + 1 ~ p), so that (14) becomes gt = ( q _
H°
l)p2
exp
(-Scxfp + 1)
After inverse transformation, we obtain sl(2,[) -
(q_
1) 2
I + 5- e x , ( 2 ) erfc
+(1-~-)exp(-.~)erfc(2 ~_
+
vG-)]
(17)
240
w.z. ZHANG
At a late time, when f > 20/q, P < r//20, ~I + P '~ q, then (14') becomes H° gm =
exp ~ - .~
p ( p + 1)
+
1
After inverse transformation, we obtain
sl(x, t) = Ito 1 - yrr
~exp u
1 +
u:
du
(18)
In (I7), 7 = k~t/m~#2; in (18), T = k~t/m~tq. Equatior~s (17) and (18) can be rewritten as
s~(x, t) . . . . . . if_o_. S ! E r/-I
(i7')
s,,(x, t) = 11o" S I L
(18')
where S!E
= ~
S1L
=
1 + ~_ exp(2) erfc
1 --2
~z
i~sin~Uexp(~ u
--~_+ 2x/t
u2
1 + u2
0
du
o
On the basis of (18), graphs for S1L have been prepared and are shown in Fig. 3.
Approximate solutions for aqu~l'er Similarly, from (13) and (14) we obtain approximate solutions for water levels in an aquifer: r <
, s2(x, t) = Ho 2 exp (2) erfc
F > -q- , s2(x, t) = Ho 1 - - rt
~--~/~_+ -x/t
u(i + u 2) exp
1 + u2
du
(20)
TRANSIENT GROUNDWATER
Z
3 4 5678~t~I
F L O W IN A N
~
AQUIFER-AQUITARD S Y S T E M
3 4 56?~9~
2
3
4 5678~10
2
3
241
a 5 6 789JDz
2
3
,~ 5 5 7 8 9 0
~D 8 6 5 4
3
.,J f-O
ool 8 6 5 4
o0Ol
0(
d
~
4
D O I 0~*~
"t Fig. 3. Type curves for function SIL.
or
7< n~-~,s'(x, t) = Ho" S2E
(21)
/=> ~,20 s,(x,~ q
(22)
t) =
Ho" S 2 L
where SZE -- l[exp (2)erfc (2w//. + / ~ _ ) + exp (-2)erfc S2L = 1 - ~2 i u(~sin£u + 7,2)exp (
2x/?
1 +u2u" t~ d ,
Type curves for S2E (type A curve) and S2L (type B curve) are given in Fig. 4.
\
t
'%~,,
\
I
I
!
\ \
G" " ~
i....
~ ~
i~~ "---.. i ¸~,
~:iii
~L
TRANSIENT GROUNDWATER FLOW IN AN AQUIFER-AQUITARD SYSTEM
243
The seepage rate and the total amount of percolation water from river
From Darcy's law, at x = 0 qo =
c~s2 - 7" ~x
(23)
Os2/Ox can be obtained from Os2/CY, b'" i-~werse transformation. Substituting Cs2/&x into (23), we have qo =
T2no
T
Q
(24)
where O(r/,7)
=
~2 i 1 +x / u' ~2 -( r- g/ -,
2 I/~i-'~ ~4~
1) exp ( - g t 7) du
~
-
-i u 2 ( r / -
r/
1)-
1
exp (--g27) du
The total amount of seepage water from the river is
(25)
V = IqHo'B'D
where O(r/,i-)
=
~
o u2(r/ -- 1)
[1 -- exp (--g~ T)] du
f u 2 ( r / - 1) ~,~/,7-I
_
TRANSIENT GROUNDWATER FLOW IN THE CASE OF UNIFORM WATER LEVEL CHANGES
Derivation of formulae for groundwater levels
When the water level in a river or canal rises or falls with a uniform rate, its hydrograph is as shown in Fig. 2. The water level at any time t > tm can be expressed by m
s2(0, t) = ~ (v,, - vm)(t - tin)
(26)
0
whenO< s2(O, t) =
t < h, rot
(2 L~
254
W.Z. ZHANG
curve. The value of 2 0 l he selected type A curve and the abscissas o f any m a t c h point t, i are n,,ted. From t" ~
0~21
we have
c¢2
=
lit
k l /tlrl '
~
I ]2 2
from 2
=
x/8,
e
=
we obtain
a2 =
T/~2
=
0~2 B2
The field.data curve is then m o v e d horizontally to the right until it is m a t c h e d to a type B curve of the same 2 value as that o b t a i n e d in m a t c h i n g the early time data. In the second position the c o r r e s p o n d i n g abscissas o f any m a t c h point t, Zare noted. Since t = ~ t, we obtain e~ = ~t. From ~2 -
kl
and ~
mi l1~.
=
kl
m: i~!
we have q - 1
-
" eL
-
- 1 ~2
W h e n seepage rate q0 from the river at any given time t~ is available, we can estimate #1 f r o m
qo(fi)
=
f~,BvoD(tl, fi)
or
]d I
-~-
qo(~) BvoDOl, tl )
From q
/-ll
--
+
we have ]-'/1 t~2
-
t~2
TRANS'IEL~F G R O U N D W A T E R F L O W IN A N A Q U I F E R - A t d u I T A R D SYSTEM
255
F r o m T = a2p2, T can be determined. F r o m 9 2
Tm 1
=
kl
we have k, ml
T2 B2,
T ~ m
or kl
T h u s all the parameters
k,/m~(or k~ ), #~,
T ( o r k?.m2), k2) and p_, are obtained.
Example A two-layered succession consisting o f an upper silty loam aquitard 13.13 m thick (below initial water table) a n d a lower sand aquifer of 30.0 m thick is used as an example. A river fully penetrates the a q u i t a r d and extends to the aq~dfer layer. Before storage of water; in the river channel the flow in the st~'eam-aqt,~%r system has attained its steady condition. After storage of water, the river stage begins m rise with a u n i f o r m rate o f 0.2 m da3? -~ . The relative g r o t m d w a t e r level changes, s2/Vot, in an observation well at a distance o f 200 m from the river are as s h o w n in Fig. 9, a n d the seepage rate from the river at t = 5 days is 1.692 m 2d a y - ~. We wish to establish the hydrogeological p a r a m e t e r s o f the a q u i f e r - a q u i t a r d systems. M a t c h i n g the early time field d a t a to a type A curve, we obtain 5 = 1.0 and c o r r e s p o n d i n g abscissas at a point t = 5 and t = 1.0 day. i~datching the late field d a t a to a type B curve, we have ,~ = 1.0 and c o r r e s p o n d i n g abscissas o f a n o t h e r m a t c h point t = 2 and t = 8.0day. F r o m early time data, ~2 = 5/ ~ 1 5 d a y - ~ , S: ~ x/B ~ 1, B ~ 200m, a 2 = ~2 B2 ~ .,00 0 0 0 m 2 da? i - 1 . F r o m late time d a t a 0q = ~t = 0.25, *7 - 1 = 5/0.25 = 20, r/ = 21. F r o m (40)
Bv0D(G h )
W h e n t~ = 5 day, ~ = ~2 h = 1.25, using (25), a table for D can be prepared, f r o m which D = 1.001. Substituting these values into (5), finally, we obtain I ,qo~
~i P2 T =
200 x .0.2 x 1,001 ~! ,7-
a:#2
-
I
=
0.04225 20 200f:)0
=
0.04225
-
0.00211
: 0.00211
=
422m-oa3.
w.z. ZHANG
256
i +
i ,
+ -+-V
i
i
I !
•
i
i
1 +
+-- i ! !
--
7 6
"i
5
--i
"52
,
~
z
i
.....
+I~
"--+ + i i+ ' i +~: ~ -
--
--7"
T1
1
I
I :!
t
+
OJ
+i!
I
i
t
I
i ,
[
+
i-
8
i__
+
7
+ i
6
l+ _ _
5
i-
it
i
+.
t
, ~.+0! _ _ _ _ i _ _ . o.t Z
i
}++ 3
~i]
+ ~ ii t
e
5
+
i
i 6 78~+.0
?.
a
e+
5~
8 S IO
+, (da6) Fig. 9. Relative groundwater level changes in an observation welt at a distance of 200 m from the river.
kl
T
422
m~
B-"
40 000
-
v..+l, o d a y
CON CLUSION
In this paper, solutions cor sidering the sterativity of an aquifer have beea derived for t~e calculation of groundwater level, seepage rate and the total amount of seepage water in response to changes of river stage. Approximate formulae for early and late time and graphs for special functions involved in them are also given. Computational fornmlae show that at early time the elastic storati'~+ity plays an important ro!e in groundwater fl+ow, in the course of continuous upward recharge from an aquifer, the role of gravity storage in the aquitard graduai!y increases and the rate of groundwater level rise reduces. After considerable time the water levels in the aquitard and the piezemetric head in the aquifer layer gradually equalize; at this time the equanons for a homogeneous unconfined aquifer can be used approximately
T R A N S I E N T G R O U N D W A T E R F L O W IN AN AQI.JIFER-AQU1TARD SYSTEM
257
to estimate water level changes in the aquifer-aquitard system. To determine the groundwater level during periods of rapid change of river stage, the equations concerning the storativity of the aquifer should be ~ased, whereas in periods of stow change of river stage that last for a long time, the use of the solutions Ibr an unconfined aquifer will not introduce notable error. REFERENCES Averianov, S.F., 1957. Influence of irrigation system on regime of ground°water, ~rt: A.N. Kostiakov, N.N. Favo~in and S.F. Averianov (Editors), The Influence of Irrigation on Regime of Groundwater, Collection of Papers, Vot. :. Press of Academy of '~eic~,,es, Moscow (in Russian). Bear, J.. 1972. Dynamics of Fluids in Porous Media. Elsevier, New York. Boulton, N.S., 1954. The drawdown of water table under nonsteady conditions near a pumped wel~ in an uncov,fined formation. Proc. Inst. Civil Eng.. 3(3): 564-579. Ferris, ~.G,, Knov !,-~;, D.B., Brown, R.H. and Statlman, R.W., 1962, Theory of aquifer tests. U.S. Geol, Surv, Water Supply Pap., 1536-E. She~tako',, V.M, i963. Unsteady flow in two-!ayered media. Proceedings of Academy of Sciences USSR, Division of Engineering Sciences, Mechanics a,nd Mechanical Engineering, No. 6, pp. 93-96 (in Russian). Stehfest, H,, 1970. Numerical inversion of Laplace transform, Cemman. A C M , 13 (1): Vefigie, N.N. and Shestakov, V.M., 1954. Methods for Computing the Grol~:ndwater Movement in Two,,layercd Media. Press of all Union Research Institute of Wat(:~r Supp~i and Hydrogeology. Moscow (in Russian), Zhang, W., 1955, The influence of Irrigation on Regime of Groundwater, Ph.D. Thesis, The Research ":~ ion of Problems of Water Economy, Academy of Sciences, Moscow qn Russian). Zhang, W., 1959, Unsteady regime of groundwater under the influence of irrigation. In Inflvence of Irrigation on Regime of Groundwater, Collection .~f Papers. 'CoL 2, Press of Academy of Sciences, Moscow (in Russian). Zhang. W., 1983. Transient Groundwater Hydraulics and Evaluation ~f Groundwater Resource,,. Academic Prc-;,~,, Beijing (in Chinese).