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THE EXTENSION OF THE THEIS EQUATION FOR NATURAL BOUNDARY CONDITIONS FOR THE DETERMINATION OF HYDRO-GEOLOGICAL PARAMETERS BY A LONG-TERM PUMPING TEST
HELMUT
SCHÖNLAUB
TIWAG-
Tiroler Wasserkraftwerke
Aktiengesellschaft
Eduard-Wallnöfer-Platz 2
A - 6020 Innsbruck
Tel.:
0043-512-506-2356
Fax:
0043-512-506-2580
e-mail:
helmut.schoenlaub@tiwag.co.at
Abstract
The solution of the partial differential equation for
nonsteady groundwater flow near a well is known as the THEIS equation. To get a
unique solution, you have to define suitable boundary and initial conditions.
Up to now a homogeneous, infinite aquifer has been required. Both the natural
gradient of the groundwater and the influence of surface waters were
disregarded.
If you consider a well in a
natural aquifer and near surface waters, the piezometric
level h follows from the
superposition of the local radially symmetric THEIS-solution with the steady
and nonsteady influence of the surface waters and the natural gradient of the
groundwater.
A new method is presented,
which incorporates these enlarged boundary conditions into an improved THEIS
equation. This formula can be solved numerically by the method of coefficient
determination. With it, the important parameters T (transmissivity) and S (storage-coefficient) can be determined for
the classification of the aquifer.
Keywords: Long-term pumping test, evaluation methods, hydrogeology, transmissivity, storage-coefficient, boundary conditions, image injection well
Introduction
For determining the hydrogeological parameters of the
aquifer in the area of the power plant Langkampfen on the river Inn (Tyrol,
Austria) a long-term pumping test
was carried out near the power plant. The depth of the well was 42 m, the
diameter of the well hole was 1000 mm. Completion of the well was done with a
600 mm steel-tube with perforated casing in the area of the confined aquifer at
a depth of 29 to
40 m under terrain. The pumping rates of the two-stage
pumping test were performed with 71 l/s resp. 116 l/s. For observation of the
groundwater lowering 12 surveying spots were drilled near the well. In addition
to these spots the water level of all other surveying spots in the influence
area of the well was recorded.
Figure 1 shows the area of investigation and the
entire hydrological monitoring network for the perpetuation of evidence
(SCHÖNLAUB, TENTSCHERT [1]). The well is situated at the northern side of the
weir next to the bank of the river Inn.
Figure
1: Area of investigation
1.
DETERMINATION
OF HYDROGEOLOCICAL PARAMETERS
Figure 2 shows the course of the hydrographs of the
river Inn (A58) and some spots for surveying the groundwater during the pumping
test.
Figure 2: Hydrographs of the
river Inn and groundwater surveying spots
Figure
3 shows the potential and flow lines for the pumping test in the confined lower
aquifer. Figure 4 shows the groundwater contour at the time of maximum pumping
rate.
Figure 3: Potential and flow lines - Section
through the central line of flow
Figure 4: Plan of groundwater contours
The hydraulic system is
characterised by a high degree of complexity, so that the procedures of
determination presented in the past cannot be applied. Therefore a modified analytical method of solution was developed. The
differential equation of the well flow is solved numerically, in compliance
with the steady and nonsteady influence of the river Inn, the natural gradient
of the groundwater level and the influence of the upper aquifer. The parameters
T (transmissivity) and S (storage coefficient) can be determined.
The
partial differential equation for nonsteady groundwater flow or fundamental
equation for horizontal plane movement of the underground water for a confined
aquifer is the solution by JACOB [2].
In the
area of a well a unique solution of the fundamental equation is the THEIS
equation [3]. The integral form is:
(1)
with
s : groundwater
lowering [m]
r : distance
of the surveying spot from the well (radial coordinates)
t : time
[sec]
Q : pumping
rate [m3/s]
T : transmissivity
[m2/s]
S : storage
coefficient [-]
There
is no unique solution of the differential equation. To obtain a unique
solution, you have to define suitable boundary and starting conditions.
Until now a homogeneous, infinite aquifer was required.
Neither the natural gradient of the groundwater nor the influence of surface
waters were taken into account.
These improved boundary
conditions now include the natural gradient of the groundwater and the influence of the fluctuation
of the water level of the river Inn.
In the area of the well the
piezometric level h follows from the superposition of the local radially
symmetric THEIS-solution with the steady and nonsteady influence of the surface
waters and the natural gradient of the groundwater.
The water
level of the river Inn affects this differential equation as a nonsteady source term, which influences the
groundwater level and the groundwater flow rate. In this model you can take
into account the influence of the groundwater within the observation area. The
fluctuation of the water level H(t) of nearby surface waters with straight line
boundaries y £ 0 can
simply be considered by solving the homogeneous equation
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(2)
with
the two boundary conditions 
and 
< 
.
x,y : horizontal
coordinates [m]
h : piezometric
level (pressure level) [m]
The
general solution of the equation is
(3)
B(w) and C(w) are arbitrary functions of w which are defined by the
necessary boundary conditions. Because of 
<
, C(w) will
disappear. Hence 
leads to an equation
for the function B(w):
(4)
Applying
the Fourier theorem leads to: 
(5)
Substituting
of equation (5) in equation (3) will give:
(6)
The
integral representation of the formula (6) is as follows[4]:
(7)
The
equation is solved numerically by the method of coefficient determination. In order to be able to calculate the
piezometric level at each location and moment, S and T have to be given. These coefficients will be found by comparison with measured and calculated variables according
to the least squares method. To achieve this the piezometric levels will be
calculated with selected starting terms for S and T at different measuring
points and with their respective time slopes. To develop the least squares the
data have to be weighted as to their reliability. After this, S and T will be
varied until the least squares reach a minimum.
2. COMPUTATIONS
The computation is made by the program "Grundwasser FKT.M" with the help of the software MATLAB. The parameters S and T are transferred to the function "Grundwasser FKT.M". After this, the expected data are calculated by this groundwater model and be compared with the real measured data. The function "Grundwasser FKT.M" yields therefore a scale for degree of correlation.
2.1
Boundary conditions
The
following limiting quantities have an effect on the results of the model:
2.1.1
The natural gradient of groundwater
For
assessing the natural gradient of the groundwater the measured groundwater
levels before the pumping test were averaged and a linear regression was
calculated in dependence on the x- and y-coordinates of the measuring points.
The result is a gradient of 0.5 ? in the direction of the river Inn and a
gradient of 1.3 ? at right angles to the Inn.
2.1.2
The unsteady influence of the river Inn
Because
of the colmation of the river bed there are different sizes of S and T in the
recharge area. To eliminate this influence in this region, the calculation was
based not on the water level of the river but on the piezometric level
immediately near the river. The measurement points D22 and D02 have a distance
of about 10m from the recharge area. The piezometric level of these two
measurement points before the pumping test varies only by a steady value, which
depends on the natural gradient. The residual variance of this difference is
only in the range of cm. In comparison with the water level of the river Inn
(measuring point A58 at the same station) the variations of the piezometric
level of the measuring point D22, over the same timespan, are in the order of
dm. The piezometric level further away from the Inn can be calculated from the
piezometric level measured near the Inn. The greatest dependence of the
reaction of groundwater distant from the Inn on the level close to the river
shows a water level t* units of time ago. The value of t* is calculated by the
model with
(8)
(d = distance from the river Inn)
(9)
The
model also shows that the water fluctuations of
the river Inn propagate in the aquifer with the following velocity v:
Wave
groups propagate with this velocity. This propagating speed is twice as high as
the phase velocity.
2.1.3
The steady influence of the river Inn
This
influence was taken into account by inserting an image injection well
mirror-inverted to the real well.
3. RESULTS AND CONCLUSIONS
The
calculation was done for a homogeneous distribution of S- and T- data.
By
optimisation at the surveying spots S2, S3 and W3 the following data were
calculated:
T = 0,02037 [m2/s]
S =
0,0475 [
- ]
By
setting the depth of the groundwater body H at 11m (drilling from depth 29 -
40m) an average hydraulic conductivity for the lower aquifer was calculated as
kf
= 1,8 . 10-3 [m/s]
In
comparison with this result, a hydraulic conductivity of kf = 1,3.10-3
m/s was determined in the drilling hole D22 by means of flowmeter
measurements at a depth of 30 - 35m
under terrain.
In figure
5 the hydrographs of the calculated and measured data are compared. The
hydrographs S2iws, S3iws and W3iws result from the optimisation of T and S by
using the improved boundary conditions.
The
hydrographs S2i, S3i and W3i are the forecasting data without discharge well
because of the unsteady water level of river Inn.
One of the measuring points (W1)
shows an extremely discontinuous reaction. A calculation with the determined
data of T and S gives a clear difference. This can be reduced to an inhomogeneity in this area in the form of an impervious
lens. Such lenses were often found in the course of the test drillings.
You may come to the conclusion
that this mathematical model gives good results valid only within a homogeneous
aquifer. Good agreement between calculated and measured data were found.
This model cannot be applied to
extremely absorbing waves measured in points within inhomogeneous areas.
Figure
5: Measured and calculated hydrographs
References
[1] H.Schönlaub, E.Tentschert: Investigation and Modelling in the
Groundwater Area of Langkampfen (Tyrol).
Mitt. Österr.Geol.Ges., 87
(1994), 29-36, (1996)
[2] C.E. Jacob: On the
Flow of Water in an Elastic Artesian Aquifer.
Trans. Am. Geophys. Union,
574-586 (1940)
[3] C.V. Theis: Relation
between the Lowering of the Piezometric Surface and the Rate and Duration of
Discharge of a Well Using Groundwater Storage.
Trans. Am. Geophys. Union,
519-524 (1935)
[4] G. Holzmüller,
H.Schönlaub: On the Determination of Hydrogeological Parameters by a Pumping
Test near Surface Waters with varying Water Level.
Unpublished Report of the
Tyrolean Water Power Company about the Analysis of the Pumping Tests for the
Power Plant Langkampfen, (1995)
Laplace - Operator