Water Table Fluctuations Due to Harmonie Variations in the Source
JAMES N. LUTHIN
1. Introduction
The problem of water table fluctuations near a tidal river is of importance in all the delta regions of the world. There are many hundreds of thousands of acres of land lying near sea level and having a high water table in deltic areas. Permeable aquifers transmit water from the river to adjacent agricultural areas and the water table fluctuations result from tidal variations in the river. In the delta region of the Sacramento-San J oaquin Rivers these water table fluctuations have been studied in the field by ScoTT and LUTHIN (1959).
This paper presents an analysis of the problems associated with the water table fluctuations induced by periodic changes in the river level. !he basic equation that must be solved is that due to Boussinesq. Extensive use is inade of solutions pub
lished by CARSLAW and JAEGER (1959) as well as the developments contained in
CHURCHILL (1944).
2. Assumptions
The basic equation to · be solved is the Boussinesq equation as used by LuTHIN and HOLMES (1960) and ScHMID and LUTHIN (1964) and others.
E ö 4' (x , t )
km cl t (1)
where c is the storage coefficient of the soil above the aquifer, k is the hydraulic conductivity of the aquifer, m is the thickness of the aquifer, <l> is the hydraulic head, t is the time and x is the horizontal distance from the source.
lt is assumed that the tidal fluctuations can be represented by a sine wave of constant amplitude. In actual fact tidal fluctuat:ions do not have a constant ampli
tude but vary in an irregular fashion.
The storage coefficient, c, is defined as the amount of water that must be added to or subtracted from the soil to induce a unit change in the water table height.
lt is here assumed to be a constant value but will in fact depend on whether the table is rising or falling.
The thickness of the aquifer, m, is known and it is assumed that this thickness does not vary. In actuality, as the water table rises and falls there will be a change in the thickness of the conducting region. However, this is unimportant where the 147
thickness of the aquifer is large compared to the soil above it or where the perme
ability of the aquifer is large compared to the permeability of the soil.
However, all of these assumptions must be considered in applying the results of the analysis presented here.
3. The problem
Equation (1) is to be solved subject to the following boundary conditions I :
I I :
Q (o , t) 0
· q>(l , t) A cos wt
I I I : � (x , O) 0
X = + Ü , t > Ü X = l , t > Ü Ü < X < l
The first boundary condition indicates that the hydraulic head is maintained at zero at the end of the aquife.r away from the river. This might be accomplished by a drainage ditch or some other facility.
In this first analysis the river is treated as a line source.
By introducing new variables t' = kmt/ et and w' = wdkm and applying the Laplace transform, equation (1) becomes
p<Ji Cx , p )
I . I I .
d-f, (x,p) ? dx2
4 '(x, p)
<jJ (l ,p )
0 X = + Ü p > Ü
X = l p > Ü
The solution of the above differential equation is
t (x , p ) (p + 2 A w ' - ) ? sinh sinh x l /p
/p
(2)
(3) Writing the hyperbolic functions in exponential form and applying transform
82 p. 299 CHURCHILL (1944) we have the inverse transform given by
L -1
{ r
s inh x s i nh Z /p/p} -
. 2 ../JI t '� 1 .., n=oi
[ (2n + l) l _ x] e- [ (2n+l) l-x] /4 t ' 2
- [ (2n + l) l + x] e - [ (2n+i) l+x ] 2/ 4t ' 148
With the aid of the convolution theorem the inverse of equation (3) can be written
< (x, t) = 2
rn
n=o li {J
Ot /43
[ cos w ( t-,)J [ (2n + l) Z - x] e- [ (2n+l) Z-x] 2/4,d,J
0 lr 0 [ cos w C t - T)...,j
[ (2n + l ) l + x] e- [ (2n + l) Z+x]2/4,d, In order to evaluate the above integral it is necessary to ignore the transient part of the solution by choosing t very large. If this is done and proper notice is taken of the substitution made initially for t and w, the result is<P(x, t)
ro
� [ - [ (2n+l) Z-x] I �� 2,
A n=o
L
e Km cos wt - [ ({
2n + l ) Z - x] / 2km;;;;}
_ e- [ (2n+l) Z+xJ; � cos { wt - [ (2n + l) L + x] � }
rru
This is the steady periodic solution for the finite region considered here. If the aquifer is considered to be semi-infinite in extent the second term in the series becomes zero and the solution becomes similar to the one given by CARSLA w and JAEGER (1959) page 64 for the analogous problem with heat. For the semi-infinite case the amplitude of the water table fluctuations will diminish like
- [ (2n + l ) Z - x] rz-;;;
e
l fu
Literature Cited
CARSLAW, H. S., and JAEGER, J. C., 1959: Conduction of heat in solids. 2nd Ed., 510 pp., Oxford, Clarendon Press.
CHURCHILL, R. V., 1944: Modem operational mathematics in engineering. 306 pp., New York, McGraw-Hill Book Co.
LunnN, J. N ., and HOLMES, J. W., 1960: An analysis of the flow of water in a shallow, linear aquifer, and of the approach to a new equilibrium after intake. J. geophys. Res. 65:
1573-1576.
SCHMID, P., and LUTHIN, J., 1964: The drainage of sloping lands. J. geophys. Res. 69: 1525-1529.
Scorr, V. H., and LUTHIN, J. N., 1959: Investigation of an artesian well adjacent to a river.
ASCE J. Irrigation Drainage 85: 45-62.
149
Zusammenfassung
Grundwasserschwankung infolge harmonischer Schwingung des Flusses In Mündungsgebieten schwanken die Wasserstände der Flüsse mit den Gezeiten.
Diese Schwankungen teilen sich dem Grundwasserstrom mit. Ein mathematisches Modell wird hergeleitet, das die Bewegung des Grundwassers als Folge der zeit
lichen Wasserstandveränderung des Flusses beschreibt.
Es wird angenommen, daß der Wasserstand des Flusses harmonisch mit kon
stanter Amplitude schwingt und daß die Schwankung des Grundwasserspiegels in einer endlichen Entfernung vom Fluß vernachlässigbar klein wird, was zum Bei
spiel durch eine Drainage erreicht wird. Ferner wird der Fluß als Linienquelle für die Speisung des Grundwassers betrachtet.
Grundlage für das Rechenmodell ist die Gleichung von Boussinesq, wie sie von SCHMID und LUTHIN (1964) benutzt wurde. Durch eine Laplace-Transformation kann die Gleichung für den stationären Fall gelöst werden. Die Rücktransforma
tion erfolgt mit dem Faltungssatz aus der Theorie der Differentialgleichungen.
Übersetzung P. Germann
150
