A method for the estimation of dual transmissivities from slug tests
A method for the estimation of dual transmissivities from slug tests
Filip Wolny
Marek Marciniak
Mariusz Kaczmarek
Aquifer homogeneity is usually assumed when interpreting the results of pumping and slug tests, although aquifers are essentially heterogeneous. The aim of this study is to present a method of determining the transmissivities of dual-permeability water-bearing formations based on slug tests such as the pressure-induced permeability test. A bi-exponential rate-of-rise curve is typically observed during many of these tests conducted in heterogeneous formations. The work involved analyzing curves deviating from the exponential rise recorded at the Belchatow Lignite Mine in central Poland, where a significant number of permeability tests have been conducted. In most cases, bi-exponential movement was observed in piezometers with a screen installed in layered sediments, each with a different hydraulic conductivity, or in fissured rock. The possibility to identify the flow properties of these geological formations was analyzed. For each piezometer installed in such formations, a set of two transmissivity values was calculated piecewise based on the interpretation algorithm of the pressure-induced permeability test-one value for the first (steeper) part of the obtained rate-of-rise curve, and a second value for the latter part of the curve. The results of transmissivity estimation for each piezometer are shown. The discussion
Dual-permeability; Groundwater hydraulics; Heterogeneity; Preferential flow; Poland
-
1 Institute of Physical Geography and Environmental Planning, Adam
Mickiewicz University, Krygowskiego Street 10,
61-680 Poznan, Poland
2 Institute of Mechanics and Applied Computer Science, Kazimierz
Wielki University, Chodkiewicza Street 30,
85-064 Bydgoszcz, Poland
presents the limitations of the interpretational method and
suggests future modeling plans.
Introduction
The accurate determination of hydraulic conductivity (k)
and transmissivity (T) of an aquifer is a crucial part of
many hydrogeological studies, especially those related
to groundwater control in mining areas, vulnerability
mapping and the delineation of protection zones for
water intakes
(Rafini et al. 2017; Renard 2005)
. Pumping
tests performed in wells can provide the average
hydraulic conductivity within the cone of depression. The
calculated transmissivity, i.e. the product of hydraulic
conductivity and aquifer thickness, will also be an
average value.
More local values of hydraulic conductivity and
transmissivity can be obtained via slug tests, which require
initiating and measuring water-level fluctuations in a
piezometer. In order to achieve such fluctuations, water is
typically added to or removed from the piezometer.
Alternatively, objects heavy enough to sink into water
can be inserted. Fluctuations can also be achieved by
sealing off the piezometer and inducing air pressure
changes inside
(Marciniak 1999, 2012)
. Slug tests are of
great significance, since they can even be conducted in
cases of aquifer contamination, when water pumping is
strongly inadvisable.
The most common analytic models used to interpret slug
tests were developed by
Bouwer and Rice (1976)
,
Bredehoeft
and Papadopulos (1980)
,
Cooper et al. (1965
, 1967) and
Hvorslev (1951)
. These models, however, usually assume
homogeneity of the analyzed water-bearing formations, when
many of them are, in fact, characterized by the presence of
different systems of void space which may lead to preferential
flow
(Beckie and Harvey 2002; Copty et al. 2011; Pechstein
et al. 2016)
. In such cases, qualitative deviations in the return
of the water level to its initial state can be observed.
The research described here is motivated by the results
of many pressure-induced permeability tests which show
deviations from the exponential rise (or fall) of the water
level that can be approximated by bi-exponential curves
(Fig. 1). It was assumed that these curves characterize
heterogeneous, dual-permeability aquifers and indicate
the presence of two distinct pathways for water flow. In
general, a dual-permeability aquifer can be envisioned as
a special type of dual-porosity reservoirs and should be
defined as a medium composed of two subdomains, in
which both are responsible for water flow.
Fissuredporous rock—if water flow occurs through both (the
fissures and the pores)—are classified as dual-permeability
reservoirs, as are layered formations with layers of highly
permeable sediments
(Balogun et al. 2009; Leij et al.
2012)
. For example, two different sand layers or a layer
of sand and a layer of fissured limestone or even one
consolidated but fissured porous material like sandstone
can fall under the definition of a dual-permeability
medium.
The paper aims to present the procedure of a piecewise
estimation of two transmissivity values, whenever a
biexponential curve is observed in dual-permeability formations
during a pressure-induced permeability test. However, this
procedure can easily be extrapolated to any other type of slug
test, since all result in recording water level movements inside
a piezometer.
Fig. 1 Bi-exponential water level
rise recorded during a
pressureinduced permeability test (bold
red line) and attempts to fit
theoretical exponential curves
(thin black lines) to field data
Materials and methods
The pressure-induced permeability test
The pressure-induced permeability (slug) test conducted
in field conditions determines the in situ hydraulic
conductivity of an aquifer in the area around a piezometer
(Kaczmarek et al. 2016; Marciniak 2001; Marciniak
et al. 2013)
. The test is performed in the following
manner. First, the upper part of the piezometer is sealed off
using a sealing device. Simultaneously, a probe is placed
inside the piezometer in the area of expected water level
fluctuations. Next, air is pumped into the sealed
piezometer to decrease the water level. After pumping is stopped
and the water level has stabilized, the air valve of the
sealing device is opened and the air inside the piezometer
decompresses, leading to the free rise of the water level to
its initial state. An alternate way of conducting the test is
to apply suction and successively increase the water level
in the piezometer. In such case, a decrease of the water
level is observed after the air valve is opened. The rate at
which the water returns to its initial level determines the
transmissivity of the investigated aquifer.
The mathematical model of water level changes within the
piezometer characterizes two possible cases: damped
oscillations or aperiodic movement
(Krauss 1974, 1977; Marciniak
2001, 2012)
. By using the model in the identification
algorithms of the pressure-induced permeability test
(PARAMEXtype), it is possible to calculate the transmissivity of an aquifer
based on the free rise or fall of the water level that is recorded
within a piezometer (Marciniak 2001). Moreover, hydraulic
conductivity of the investigated aquifer can be easily
determined by dividing the obtained transmissivity value by the
thickness of the aquifer.
In a vast majority of cases, an aperiodic movement of the
water level can be observed. The water level’s rate-of-rise or
rate-of-fall as a function of time, h(t), can be described by the
following exponential function:
ð1Þ
hðtÞ ¼ h0½1−expðλtÞ
where h0 is the initial water level (cm) and λ < 0 (1/s) is an
exponent dependent on the transmissivity.
Unusual bi-exponential water-level movement has been
observed during ca. 12% of these tests. The possible causes
of such movement have been analyzed thoroughly. Initially,
piezometer malfunctions, i.e. leaky isolating clay seals or
faulty sand packs around the piezometers, were taken into
consideration. Such types of damage have been observed
during many pressure-induced permeability tests but, in such
cases, the initial stage of water level rise is rapid and short
and the next stage is exponential.
In order to find the hydraulic connections that result in the
bi-exponential behavior of the water level, several laboratory
models of water flow have been investigated by
Marciniak
et al. (2013)
. Only one model called the BW-tube^ allowed
such behavior to be observed. This laboratory system of water
flow indicates that dual-permeability conditions are required
to record curves deviating from the exponential rise. Such
conditions are observed in aquifers characterized by the
presence of two subdomains (e.g. two layers) with different
transmissivities and hydraulic conductivities, as well as a weak
hydraulic connection between them.
When conducting any type of slug test in a piezometer
installed in a dual-permeability aquifer, and if the screen
section covers both subdomains, faster water flow should be
present in the subdomain with a higher transmissivity
compared to the subdomain with a lower transmissivity. Generally,
higher contrast between transmissivity and hydraulic
conductivity values will result in a more apparent bi-exponential
water level movement.
The laboratory W-tube model of a permeability test
conducted in heterogeneous aquifers
The BW-tube^ is a laboratory model that was thoroughly
discussed in two earlier publications
(Kaczmarek et al. 2016;
Marciniak et al. 2013)
. The model is considered to reenact the
hydraulic conditions of a field pressure-induced permeability
test conducted in heterogeneous, dual-permeability aquifers. It
consists of three hydraulically connected columns. The two
outer columns are equipped with cylindrical chambers where
samples of different sands are placed. The central column,
which represents a piezometer, contains a measuring probe
that records the water level in this column as a function of
time (Fig. 2).
The preparation of a test requires filling all three columns
with water to the same level. Afterwards, by pumping air into
the central column using an air compressor, the water level in
the central column is displaced down and the water level in the
outer columns is displaced up. Over time, the water level
stabilizes, then a test is initiated when the air valve at the top
of the central column is opened, causing water to return to the
initial level due to its filtration through the sand samples in the
outer columns.
The mathematical model of the W-tube test was formulated
using a balance of forces for water flowing through the sand
samples. It was assumed that the inner cross-sections of all
three columns, denoted as S, are equal. S1 and S2 are the
crosssections of the sand samples, L1 and L2 denote the heights of
these samples, and their hydraulic conductivities are k1 and k2.
Initially, the water level in the central column is lower than
that in the adjacent columns and serves as the reference level
for the measurements of the water level hr in the central
column and the levels h1 and h2 in the adjacent columns. The
mathematical model neglects all inertial forces related to the
motion of water in the columns and in the pore space of the
sand samples. The interaction forces between water and the
porous material are approximated according to the linear
Darcy’s law. Taking into account the hydrostatic forces caused
by the water columns as well as the gravitational forces and
the interaction forces between water and the porous material,
After some basic transformations presented by
Marciniak
et al. (2013)
, the analytical form of the solution, height hr, is
obtained:
hr ¼ H30 ½2 þ ðB−1Þexpðλ1tÞ−ðB þ 1Þexpðλ2tÞ
ð5Þ
where the constant B is derived from the following formula:
A2 þ A1
B ¼ 2pffiAffiffi1ffiffi2ffiffiffiffiffiffiffiffiAffiffiffi2ffiffi2ffiffi−ffiffiffiAffiffi1ffiffiAffiffiffi2ffiffi
þ
and λ1,2 equals:
λ1;2 ¼ −A1−A2
qffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A1 A2 −A1A2
Constants A1 and A2 are related to the geometric parameters
of the analyzed sand samples: S1, S2, L1, L2 and to their
hydraulic conductivities k1 and k2:
k1S1 k2S2
A1 ¼ L1S ; A2 ¼ L2S
After making use of Eq. (8), constants B and λ1,2 can be
written in the following form:
k1L2S1 þ k2L1S2
B ¼ 2pffikffiffi1ffiffi2ffiffiLffiffi2ffiffi2ffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffikffiffi2ffiffi2ffiffiLffiffi1ffiffi2ffiffiffiffiffiffiffi2ffiffi−ffiffikffiffiffi1ffiffikffiffi2ffiffiLffiffiffi1ffiffiLffiffi2ffiffiSffiffiffi1ffiffiSffiffi2ffiffi
S1 þ S2
the following balance equations for pore water can be written
(Kaczmarek 2009; Rehbinder 1992)
:
ρg dh1
8>< ρgðh1−hrÞS1 þ k1 L1 dt S ¼ 0
ρg dh2
>: ρgðh2−hrÞS2 þ k2 L2 dt S ¼ 0
where ρ is the density of water, g is the acceleration due to
gravity, and the dh1/dt and dh2/dt derivatives denote the rate at
which the water height changes. This rate is equal to the
filtration velocities in the appropriate samples.
Due to the conservation of mass in the system, the changes
of water heights in the columns for any time increment dt are
given by the following relation:
h ¼
h1 ¼ h01 ½1−expðλ1tÞ01
for t0 ≤ t < tbh1 ¼ h02½1−expðλ2tÞ
for tb ≤ t < t f
ð2Þ
ð3Þ
ð4Þ
ð6Þ
ð7Þ
ð8Þ
ð9Þ
λ1;2 ¼ − k1S1 − k2S2
L1S L2S
sffikffiffi1ffiffi2ffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffi2ffiffiffiffiffiffiffi2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L12SS12 þ kL222SS22 − kL1S1 L1k2S2S22 ð10Þ
Hence, with the solution of Eq. (5), it is possible to determine
the water height in the central column as a function of time.
The results of laboratory tests performed on the W-tube
were also presented in the authors’ previous publications. In
Marciniak et al. (2013)
, the so-called direct problem is solved,
i.e. bi-exponential rate-of-rise curves experimentally obtained
in the W-tube for various combinations of sand samples are
compared with the predictions of the mathematical model
(generated in MATLAB). A root-mean-square error (RMSE)
was calculated for every rate-of-rise curve and the
corresponding prediction, good convergence was obtained.
In
Kaczmarek et al. (2016)
the inverse problem is solved—a
numerical optimization procedure is applied in order to jointly
estimate a pair of hydraulic conductivity values based on a
biexponential rate-of-rise curve recorded in the W-tube. In this
case, the predictions of the mathematical model varied in
accuracy. Relatively small percent errors between the
numerically obtained hydraulic conductivity values and the values
derived from constant head measurements were usually obtained
for highly permeable sands. Higher errors were obtained for
sands with a low permeability or in cases when the contrast in
permeability between the two analyzed sand samples in a pair
was small.
A piecewise exponential model for dual transmissivities
in field conditions
When interpreting field pressure-induced permeability tests
that result in recording bi-exponential curves, it should be
assumed that the piezometer screen is in contact with two
subdomains, both characterized by a distinct system of water
flow, even though dual permeability is not always apparent
when analyzing hydrogeological data (Fig. 3). Thin layers of
sediment or zones of weathered material are frequently
omitted in the lithological description of boreholes. Since it is
impossible to calculate a pair of hydraulic conductivity values
for the two screened subdomains if one of them is unnoticed
or if their thickness is unknown, the authors propose a
simplified analytic method of estimating dual transmissivity values
piecewise (in two intervals).
The bi-exponential movement of the water level is
approximated using two theoretical exponential curves:
Fig. 3 a Piezometer screening
two subdomains with different
conductivities and b the
biexponential rise of the water level
observed during permeability
tests
and a pair of transmissivity values, Tb1 and Tb2, is calculated
based on the interpretation algorithm of the slug test. The
transmissivity of the subdomain with a higher conductivity
(Tb1) is calculated from the exponential curve that
approximates the first (steeper) part of the bi-exponential rise of the
water level – from the initial time t0 to Bbreakthrough^ time tb.
Transmissivity of the subdomain with lower conductivity
(Tb2) is calculated from the exponential curve that
approximates the second part of the bi-exponential movement
observed during the test—from time tb to time tf, i.e. the time
when water reaches its initial level). Slightly shifting time tb
(altering the intervals) does not cause any significant change
in the estimated T values.
Furthermore, a goodness-of-fit analysis can be performed
to estimate how well the observed water level corresponds to
the predictions of the model. The authors usually calculate
root-mean-square errors (RMSE) and normalized
root-meansquare errors (NRMSE) to assess the differences between
observed and predicted data:
1 sffinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
RMSE ¼ n ∑ hpred−hobs 2
1
RMSE
NRMSE ¼ hmax−hmin
where hobs is the water level observed during the slug test in a
piezometer, hpred is the water level estimated using the analytic
method, n is the number of measurements, hmax and hmin are
the maximum and minimum water levels achieved during the
test, respectively. When estimating the goodness-of-fit, the
two theoretical exponential curves are analyzed jointly to
obtain one RMSE and one NRMSE value for the entire duration
of a slug test (time t0 to tf).
ð12Þ
ð13Þ
of Poland. Usually, single tests are performed in a small number
of piezometers, e.g. in water-intake-protection zones. The only
region with a significant number of performed tests is the
Belchatow Lignite Mine. In this mine, a crew of hydrogeologists
routinely performs PARAMEX tests to determine the technical
condition of the piezometer network
(Marciniak 2001)
. So far,
over 200 piezometers have been analyzed at the mine. Most of
the them (154) are located within the Belchatow mining field,
one of the two fields where opencast mining takes place;
therefore, this field was chosen as the study area (Fig. 4).
The mine is located in the Kleszczow Graben (central
Poland). This arc-shaped graben, developed on the Epi–
Variscan platform, is 40–50 km long, 4–5 km wide, and
stretches in a E–W direction
(Gruszka and van Loon 2007)
. Its
depth reaches 350 m, which makes it the deepest neotectonic
graben in the Polish Lowland. The Kleszczow Graben was
formed due to the movement of faults in the substratum which
were reactivated during the Alpine orogeny
(Gruszka 2001)
.
The basement is represented by Jurassic and Cretaceous
sediments, mainly sandstones, limestones, shales and marls.
Permian rocks are solely present in a small area of the
BDebina^ salt dome
(Gotowala and Haluszczak 2002;
Haluszczak 2007)
. The graben is filled with Neogene (early
Miocene–Pliocene) and Quaternary deposits, predominantly
lignite, sands and clays. Their displacements are caused by tectonic
deformations of the basement, especially near the fault zones
and the salt dome. Moreover, the exploitation of lignite involves
relocating vast amounts of gangue to surrounding areas, which
evokes tectonic activities in the region such as earthquakes, rock
displacement and rock fracturing (Widera 2016).
Results
Study area
Pressure-induced permeability tests (PARAMEX-type) have
been performed in over 1,400 piezometers in various regions
As already mentioned, pressure-induced permeability tests
were conducted in 154 piezometers located in the Belchatow
mining field; however, water movement could not be achieved
in all of them. In some piezometers, the water level was
located at the height of the screen, preventing air compression
and decompression, whereas in several other cases, the
analyzed piezometers were damaged. Ultimately, rate-of-rise
curves were successfully recorded in 92 piezometers.
Biexponential water level movement was observed in 11 of these
piezometers—6 of them screened Mesozoic formations
(Upper Jurassic or Cretaceous), 4 others screened Paleogene
sediments underlying the lignite deposit, and 1 piezometer
was installed in Quaternary material (Table 1).
In all piezometers where a bi-exponential curve was
recorded (Fig. 5), the screen was 5 m long. Furthermore, in most
cases the lithology of the water-bearing formations
surrounding the screen suggests the presence of at least two distinct
pathways for water flow (Table 2). Most piezometers either
screen weathered/fissured sedimentary rock layers (Mesozoic)
or layers of two different unconsolidated sediments
(Paleogene or Quaternary). Only in two cases—piezometers
PW-424-2 and PW-354—the lithology suggests the presence
of one single water-bearing formation around the screen.
However, it should be noted that these piezometers and most
of the other analyzed piezometers in the Belchatow field are
over 200 m deep. It cannot be ruled out that thin layers and the
presence of weathered or fissured zones were omitted in the
lithological description of the boreholes; moreover, the Rock
Quality Designation (RQD) index was not always established.
Such information is available for the screen section of
piezometers KT-89, KT-117-BIS and PW-420. RQD values
calculated for these piezometers, ranging from 10 to 50%, indicate the
presence of a weathered or completely weathered rock mass
around the screen
(Deere 1964; Deere et al. 1967)
.
The results of transmissivity estimation for each piezometer
are presented in Table 3. Calculations of dual-transmissivities
(Tb1 and Tb2) were performed in accordance with the
methodology presented in section ‘A piecewise exponential model for dual
transmissivities in field conditions’. Table 3 also includes
Bstandard^ transmissivity (Ts) estimates from the mathematical
model of the pressure-induced permeability test, i.e. from a single
exponential function that treats the geological medium to be
homogeneous. A comparison of transmissivity values Tb1 and
Tb2 is also presented in Fig. 6. The distance from the diagonal
line illustrates the heterogeneity of the analyzed
dualpermeability formation (high Tb1/Tb2 ratio). Moreover, graphs
showing the bi-exponential rise of the water level in four
piezometers are depicted in Fig. 7, along with the best-fit exponential
curves for dual and single transmissivity estimation.
Discussion
A goodness-of-fit analysis was performed for all investigated
slug tests which resulted in observing bi-exponential
waterlevel movement. Root-mean-square errors (RMSE) and
normalized root-mean-square errors (NRMSE) were calculated to
measure the difference between bi-exponential rate-of-rise
curves recorded in the field and the adjusted exponential
curves. Error values are presented in Table 4.
In all cases, relatively small RMSE and NRMSE errors
are obtained for the dual-transmissivity approach, i.e.
when the bi-exponential rate-of-rise curve is
approximated piecewise by two exponential curves. When using one
single exponential function the errors are roughly 3–8
times larger.
Table 2 List of piezometers
characterized by a bi-exponential
movement of the water level
during permeability tests
Age of screened formations
Lithology of geological formations around the screen
Quaternary
Paleogene
Mesozoic
Piezometer
symbol
PW-424-2BIS
KT-93-1
KT-103-1
PW-424-2
PW-425-1
KT-89
KT-117
KT-117-BIS
It should also be noticed that single transmissivity (Ts)
estimates fall between the two dual transmissivity values (Tb1
and Tb2). This is understandable, considering the fact that the
Ts curve is merely an Baveraged^ approximation of the
biexponential movement.
Since the area of contact between each of the two
subdomains and the piezometer screen is unknown, the
authors could only determine dual transmissivities accurately,
without determining the exact hydraulic conductivities of both
subdomains near the piezometer screen. However, knowing
the transmissivity values, it is possible to calculate several
potential pairs of hydraulic conductivities assuming different
subdomain thickness ratios (e.g. assuming that the thickness
of one subdomain is in contact with 10% of the screen,
whereas the other covers the remaining 90%). In such case, the k
values should be calculated for each subdomain by dividing
the obtained transmissivity values by the appropriate
thickness (i.e. the length of the screen covered by the subdomain).
Fig. 6 A comparison of the
transmissivity values Tb1 and Tb2
for each of the analyzed
piezometers
Transmissivity Tb1
[m2/s]
Transmissivity Tb2
[m2/s]
Transmissivity Ts
[m2/s]
Eventually, the authors plan to expand the W-tube model
and make it applicable in field conditions where the thickness
of both subdomains is known. It would therefore be possible
to calculate a set of hydraulic conductivities using
optimization techniques, whenever a slug test results in observing
biexponential movement of the water table.
Conclusions
During some pressure-induced permeability tests performed
in the Belchatow Lignite Mine, an unusual bi-exponential
type of water-level movement was observed in several
piezometers. The movement clearly differs from the exponential
rise usually recorded in such experiments. The location of
these piezometers and the lithology of the screened formations
suggest the possible presence of a dual-permeability system of
water flow that may be responsible for such deviations.
Some analogies are apparent between slug tests and other
methods such as the pumping test. Hydrogeological
parameters of heterogeneous formations are determined during a slug
test based on the shape of the experimentally recorded
rate-ofrise or rate-of-fall curve, whereas during a pumping test this
shape indicates a possible leakage of water from adjacent
layers. When analyzing dual-permeability formations, slug
tests (e.g. pressure-induced permeability tests) can be used
to determine a pair of transmissivity values, even if the
thickness of both subdomains responsible for water flow is
unknown.
Determining hydraulic conductivities of a water-bearing
formations composed of two layers of porous material is
possible if the exact thickness of these layers around the
piezometer screen, or the ratio of two flow paths, are known.
Determining hydraulic conductivities in fissured or
fissuredporous rock is more difficult, since the ratio of different flow
paths in such rock is rarely examined.
The proposed piecewise method of determining
transmissivity values of heterogeneous water-bearing formations is an
approximate one. The limitations of the method may serve as
a motivation to elaborate new mathematical models of water
flow and interpretation algorithms based on slug tests. These
algorithms could be implemented in field conditions,
whenever a bi-exponential water level rise is observed. Such
algorithms would increase the reliability of slug tests conducted in
heterogeneous formations.
Acknowledgements This work was partially supported by the Polish
Ministry of Science and Higher Education
(NCN Grant No. 0697/B/P01/
2010/39)
.
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Balogun A , Kazemi H , Ozkan E , Al-Kobaisi E , Ramirez B ( 2009 ) Verification and proper use of water-oil transfer function for dualporosity and dual-permeability reservoirs . SPE Res Eval Eng 12 ( 2 ): 189 - 199 . https://doi.org/10.2118/104580-PA
Beckie R , Harvey CF ( 2002 ) What does a slug test measure: an investigation of instrument response and the effects of heterogeneity . Water Resour Res 38 ( 12 ): 1290 . https://doi.org/10.1029/2001WR001072
Bouwer H , Rice RC ( 1976 ) A slug test for determining hydraulic conductivity of unconfined aquifers with completely or partially penetrating wells . Water Resour Res 12 ( 3 ): 423 - 428 . https://doi.org/10. 1029/WR012i003p00423
Bredehoeft JD , Papadopulos SS ( 1980 ) A method for determining the hydraulic properties of tight formations . Water Resour Res 16 ( 1 ): 233 - 238 . https://doi.org/10.1029/WR016i001p00233
Cooper HH Jr, Bredehoeft JD , Papadopulos IS , Bennett RR ( 1965 ) The response of well-aquifer systems to seismic waves . J Geophys Res 70 ( 16 ): 3915 - 3926 . https://doi.org/10.1029/JZ070i016p03915
Cooper HH Jr, Bredehoeft JD , Papadopulos IS ( 1967 ) Response of a finite diameter well to an instantaneous charge of water . Water Resour Res 3 ( 1 ): 263 - 269 . https://doi.org/10.1029/WR003i001p00263
Copty NK , Trinchero P , Sanchez-Vila X ( 2011 ) Inferring spatial distribution of the radially integrated transmissivity from pumping tests in heterogeneous confined aquifers . Water Resour Res 47 : W05526 . https://doi.org/10.1029/2010WR009877
Deere DU ( 1964 ) Technical description of rock cores for engineering purposes . Rock Mech Eng Geol 1 ( 1 ): 16 - 22
Deere DU , Hendron AJ , Patton FD , Cording EJ ( 1967 ) Design of surface and near surface constructions in rock . In: Fairhurst C (ed) Proceedings of the 8th US Symposium on Rock Mechanics. AIME , New York, pp 237 - 302
Gotowala R , Haluszczak A ( 2002 ) The late alpine structural development of the Kleszczów graben (central Poland) as a result of a reactivation of the pre-existing regional dislocations . EGS Stephan Mueller Special Publication Series 1 , European Geosciences Union , Munich, Germany, pp 137 - 150
Gruszka B ( 2001 ) Climatic versus tectonic factors in the formation of the glaciolacustrine succession (Bełchatow outcrop, central Poland) . Glob Planet Chang 28 : 53 - 71 . https://doi.org/10.1016/S0921- 8181 ( 00 ) 00064 - 3
Gruszka B , van Loon AJ(T) ( 2007 ) Pleistocene glaciolacustrine breccias of seismic origin in an active graben (central Poland) . Sediment Geol 193 ( 1-4 ): 93 - 104 . https://doi.org/10.1016/j.sedgeo. 2006 . 01 . 009
Haluszczak A ( 2007 ) Dike-filled extensional structures in Cenozoic deposits of the Kleszczów graben (central Poland) . Sediment Geol 193 : 81 - 92 . https://doi.org/10.1016/j.sedgeo. 2005 . 05 .013
Hvorslev MJ ( 1951 ) Time-lag and soil permeability in ground-water observations . Bull no. 36 , Waterways Experimental Station , US Army Corps of Eng., Vicksburg, MS, 50 pp
Kaczmarek M ( 2009 ) Role of inertia in falling head permeability test . Int J Numer Anal Meth Geomech 33 : 1963 - 1970 . https://doi.org/10. 1002/nag.818
Kaczmarek M , Wolny F , Marciniak M ( 2016 ) Joint estimation of hydraulic conductivities of two sand samples in a W-tube system with a biexponential response . Hydrol Res 47 ( 2 ): 344 - 355 . https://doi.org/ 10.2166/nh. 2015 .320
Krauss I ( 1974 ) Die Bestimmung der Transmissivitat von Grundwasserleitern aus dem Einschwingverhalten des BrunnenGrundwasserleitersystems [The determination of the transmissivity of aquifers from the transient behaviour of the well-aquifer system] . J Geophys 40 : 381 - 400
Krauss I ( 1977 ) Das Einschwingverfaren: Transmissivitatsbestimmung ohne Pumpversuch [The transient process: transmissivity determination without a pumping test] . GWF-Wasser/Abwasser 118 : 407 - 410
Leij FJ , Toride N , Field MS , Sciortino A ( 2012 ) Solute transport in dualpermeability porous media . Water Resour Res 48 : W04523 . https:// doi.org/10.1029/2011WR011502
Marciniak M ( 1999 ) Identyfikacja parametrów hydrogeologicznych na podstawie skokowej zmiany potencjału hydraulicznego . Metoda PARAMEX) [ In situ identification of the hydrogeological parameters of the aquifers and evaluation of the technical state of piezometers using sudden change in the hydraulic potential (the PARAMEX method)] . Wyd Nauk UAM , Poznan, Poland
Marciniak M ( 2001 ) Identification of hydraulic conductivity coefficient of aquifers using the PARAMEX test . In: Seiler KP , Wohnlich S (eds) New approaches characterizing groundwater flow, vol 2 . XXXI Congress IAH , Munich, Germany
Marciniak M ( 2012 ) The PARAMEX test: a method of determining hydrogeological properties of groundwater recharge zone . In: P r o c e e d i n g s o f t h e 3 9 t h I n t er n a t i o n a l A s s o c i a t i o n o f Hydrogeologists Congress BConfronting Global Change^ . Niagara Falls , Canada, September 2012
Marciniak M , Kaczmarek M , Wolny F ( 2013 ) W-tube system with biexponential response: a model for permeability tests in heterogeneous aquifers . J Hydrol 501 : 175 - 182 . https://doi.org/10.1016/j. jhydrol. 2013 . 08 .012
Pechstein A , Attinger S , Krieg R , Copty NK ( 2016 ) Estimating transmissivity from single-well pumping tests in heterogeneous aquifers . Water Resour Res 52 : 495 - 510 . https://doi.org/10.1002/ 2015WR017845
Rafini S , Chesnaux R , Ferroud A ( 2017 ) A numerical investigation of pumping-test responses from contiguous aquifers . Hydrogeol J. https://doi.org/10.1007/s10040-017-1560-x
Rehbinder G ( 1992 ) Measurement of the relaxation time in the Darcy flow . Transport Porous Med 8 ( 3 ): 263 - 275 . https://doi.org/10.1007/ BF00618545
Renard P ( 2005 ) The future of hydraulic tests . Hydrogeol J 13 : 259 - 262 . https://doi.org/10.1007/s10040-004-0406-5
Widera M ( 2016 ) Characteristics and origin of deformation structures within lignite seams: a case study from polish opencast mines . Geol Q 60 ( 1 ): 179 - 189