1. Introduction
Pumping from a well, group of wells or any kind of water reserve has been the subject of a deep study by specialized geologists in the last century. One of the most important subjects of investigation in this area is the prediction of the water level in the aquifer due to pumping mechanisms and pumping techniques that have evolved over time due to continuous technological improvements. It would be an impossible mission to cite all the important contributions to this topic, most of which were published in specialized journals of geology. Without the aim of being exhaustive, a few of them that give an idea about the state of the art in the problem of predicting water levels in aquifers due to different pumping mechanisms are mentioned: [
1,
2,
3,
4,
5,
6,
7,
8,
9]. In order to obtain a complete picture of the problem, a review of these references and references therein is suggested.
Until 1971, the mathematical models that describe the water level changes were based on Theis’ theory [
8] and the superimposition of the effects of each pumping well [
7,
9] for both leaky and nonleaky aquifers. The first precise mathematical model was proposed in [
4], which is used to predict water level changes at a certain point of an aquifer when a variable pumping is acting from one or more wells. Their main contribution is the derivation of an integral representation of the drawdown of water in the aquifer in the form of a convolution integral, which is valid when the pumping is constant. Their theory is improved and generalized by [
6] for an arbitrary pumping. The key formula that gives the response of the aquifer due to an arbitrary pumping is the convolution integral
In this formula,
represents the drawdown as a function of the distance
r to the pumping point and of the time
t elapsed since the beginning of the pumping, all of which are measured in natural unities. The parameter
S is the (dimensionless) storage coefficient;
T is the transmissivity coefficient (measured in units
);
P is the vertical permeability of the confining layer (measured in speed units
);
m is the thickness of the confining layer (measured in length units
); and
is the pumping as a function of time (measured in units
). The three parameters
m,
S and
T are positive, and
P is non-negative. Hantush and Jacob’s formula [
1,
2] is the particular case of Moench’s Formula (
1) when the pumping is constant,
. On the other hand, as a difference with Theis’ theory, Formula (
1) assumes a negligible storage capacity of the confining layer and that the leakage rate through this layer is proportional to the drawdown in the aquifer. When
, the aquifer is said leaky, and it is nonleaky when
.
Formula (
1) gives a quite accurate description of the level changes in the aquifer. However, it has some limitations, since there are some physical situations, more or less relevant in practice, that are not taken into account in Moench’s theory. They are described by Moench himself in [
6]. One of them is, for example, the presence of boundaries in the aquifer that are not included in the theoretical analysis, although it could be handled by using image theory [
6]. For a more detailed description of the limitations of Moench’s theory, see [
6].
A partially penetrating pumping well has an effect on the vertical components of the water flow. Thus, in the presence of partially penetration effects in a leaky confined aquifer, some terms must be added on the right-hand side of (
1) [
10,
11]: terms that are a combination of elementary functions as well as Bessel functions. Other variations and alternative representations in terms of Bessel functions may be found in [
5]. In the case of the presence of parallel unsteady-state flow, Vanderberg showed that the denominator in Formula (
1) must be slightly modified
or, in other words, that a constant pumping translates into
[
12,
13].
Therefore, from a mathematical point of view, the following integral transform of a function
g for which the integral exists is of interest,
with
or
(in the case
, only the first integral above is considered). The integral transform considered in Chapter 27 of [
14] is the particular case
of the above formula. Chapter 27 in [
14] contains several asymptotic expansions in terms of Bessel functions as well as several illustrative examples. In addition, the particular case
and
is a Krätzel integral, which was introduced in [
15]. Apart from a constant factor, integral (
1) is integral (
2) with the identification
and
. Therefore, integral (
2) is from now on called
Moench’s transform of the function
g.
Even for elementary pumping functions
, the right-hand side of (
2) cannot be evaluated in terms of elementary functions. Many numerical techniques have been considered in the literature for the evaluation of the right-hand side of (
2) and eventually implemented in computer codes and compared with experimental data obtained from several aquifer systems (see for example [
2,
6,
9]). In this paper, analytic approximations of (
2) are of interest. Several analytic approximations, convergent or asymptotic, have been proposed in the literature in several limits of the parameters
x,
y and
t. According to the experiments described in [
1,
2,
3,
4,
5,
6,
7,
8,
9] and from a practical point of view, it is of interest to evaluate
for a domain of the time variable
t that is as large as possible: small and large values of the parameter
x and moderate (positive) values of the parameter
y. In addition, the most interesting pumping functions
are power functions of the form
,
.
Several asymptotic or convergent expansions, accurate for small values of
x and
y, have been derived in [
9,
16,
17] for
. A convergent expansion for large
x and small
y was derived in [
18,
19] for
. Asymptotic expansions for large
t, unbounded values of
y and
, are given in [
20]. A detailed summary of the known analytic approximations is given in [
21]. To our knowledge, analytic approximations of
for large values of
x, moderate values of
y and valid for any value of the time variable
t are not given in the literature. This region of the parameters is important in practical situations (see the experiment described in
Section 4), and then, the objective of the present paper is the analytic approximation of
in this region. For the sake of generality, in principle, pumping functions
are considered in a large functional space with special attention to the more interesting family
. In the next section, the set of functions
that we consider in this paper is specified.
In
Section 2 an integral representation of
appropriate for the asymptotic analysis when
is large is derived. Observe that when
x is large and
y bounded from below, the product
is a large variable. In
Section 3, the family of pumping functions
,
is analyzed in more detail as they are more interesting from a practical point of view. An asymptotic expansion of
for large values of the product
(large
x with
y bounded from below) and arbitrary values of
is obtained with explicit and recurrent expressions for the coefficients of the expansion. In
Section 4, the accuracy of the approximations is analyzed by means of several numerical experiments based on an experiment carried out by the Layne-Western Company in Illinois in 1953. Some conclusions are pointed out in
Section 5. A general pumping function
is discussed in a separate appendix, deriving an asymptotic expansion of
for large values of
and arbitrary values of
. In this more general case, obtaining an analytic expression for the coefficients of the expansion is not possible in general, and the coefficients must be computed numerically or by using an algebraic manipulator.
2. Preliminaries
For an appropriate analysis of
, when
is large, a suitable integral representation of the Moench transform (
2) must be found. With this aim, a sequence of changes of the integration variable is introduced below. After this sequence of transformations, the integral (
2) is written in the form of a combination of Laplace transforms, which is much more appropriate for an asymptotic analysis.
Either of the two integrals in (
7) has the form of a Laplace transform, which is suitable for an asymptotic analysis when
is large by using Watson’s lemma ([
14], Chap. 2) for any value of
, whenever the asymptotic expansion of
at
is uniformly valid for large
x and bounded values of
y. The two integrals in (
7) shall be studied separately. However, for the sake of clarity, and because it is more common in practice [
9], in the next section, the particular case of a pumping function
of power type is considered:
, and the study of a general pumping function
is relegated to the
Appendix B. As it will be seen below, in the case of a pumping function of power type, the coefficients of the asymptotic expansion of
may be computed explicitly and do not depend on the parameters
x,
y (and then, it is trivially uniform in these variables), whereas in the case of a general pumping function, the asymptotic expansion of
must be written in terms of arbitrary coefficients, and their dependence on the parameters
x,
y must be analyzed.
3. Asymptotic Approximation of (7) for a Pumping Function of Power Type
Setting
in (
7), the Moench transform of this pumping function reads
where
with
given in (
5). From [
14], Watson’s lemma can be applied to the two integrals above whenever the two functions
are infinitely differentiable at
, and their derivatives are uniformly bounded by a positive constant times an exponential function
, with
. From the definition (
5) of the functions
, it is clear that both functions
are infinitely differentiable at
, and their derivatives are power functions bounded by a positive constant times an exponential
for any
. In order to apply Watson’s lemma, the asymptotic expansion of
needs to be considered at
([
14], Chap. 2),
The coefficients
may be computed by using a symbolic manipulator. They may also be computed by means of a recurrence relation. In order to derive this recurrence, the above expansion must be introduced into the following differential equation satisfied by
,
After some simplifications, and equating the coefficients of equal powers of
s, it can be checked that the coefficients
satisfy the recurrence relation
, with
The first few coefficients of the expansion are
From Watson’s lemma ([
14], Chap. 2), an asymptotic expansion of (
8) for large
follows by introducing the expansion (
10) into the two integrals on the right-hand side of (
8) and interchanging the sum and integral. For the first integral, the asymptotic expansion is given by
and for the second one,
with
The integrals in the asymptotic expansion (
11) may be computed exactly in terms of elementary functions,
For the values of
z required in the above formula, we have that
where
denotes the double factorial of the integer number
([
22], Equation 5.4.2) with
.
On the other hand, the computation of the integrals (
13) in the asymptotic expansion (
12) is more elaborated. Integrating by parts in the integral (
13), it is straightforward to obtain the recurrence relation
with
where
is the well-known error function [
23] that models fast but continuous transitions between two constant values. The solution to the recurrence relation (
15) is, for
,
where empty sums must be understood as zero.
Therefore, from (
8), (
11), (
12), (
14) and (
16), the following asymptotic expansion of the Moench transform for pumping functions of power type is finally derived,
where for
, the sequence of functions
is defined in the form
where empty sums (for
) must be understood as zero.
Observe from (
18) that the functions
depend on the variables
through the combination
. Moreover, the right-hand side of (
18) is a sum of constants, an error function (whose absolute value is bounded by 1) and terms of the form
, all of them bounded for any value of
. Therefore, the numerators
in terms of the expansion (
17) are bounded functions of
, which means that the terms of the expansion (
17) are of the order
when
uniformly in the time variable
.
In particular, the first-order approximation (obtained by considering only the first term
of the sum (
17)) is given by the following formula:
The accuracy of Formula (
19) is illustrated in
Figure 2.
The argument of the error function on the right-hand side of (
19) (and also in all the error functions in the successive terms of the expansion (
17)) is
, where
is defined in (
6). At the critical value
, we have that
, and then this error function, multiplied by the sign function, presents a fast transition from the value
to the value 1 as
t crosses this critical value
. Then, this function encodes the fast (but smooth) transition that Moench’s transform experiments at the critical time
, from a value close to the zero value (as
) to a value close to
It is shown in
Figure 2.
4. Numerical Experiments
In
Figure 2, the accuracy of Formula (
19) (particular case of Formula (
17)) for certain selected values of the parameters
x,
y,
is shown. In this section, one step forward is given, and the accuracy of the approximation (
17) in a more realistic situation is shown. Consider the experimental data obtained from the experiment described in Part 3 of [
9] that was conducted by Layne-Western Company on 2 July 1953 on a village well at Gridley, McLean County, Illinois (for more details, see Part 3 in [
9]). For this experiment, the parameter values are
10,100 gpd/ft
feet
min,
,
feet,
gpd/feet
min
, and a pumping function
gpm
feet
min. These parameter values translate into the values
used in Moench’s transform
.
In
Table 1,
Table 2,
Table 3 and
Table 4, the absolute value of the relative error obtained by using the approximation given by (
17) is computed when
(to the right of the critical time), when
(to the left of the critical time) and when
(near the critical time), for several values of the degree
n of the approximation and the above values of
x and
y. In every table, four different values of the time
t and a pumping function of the form
are considered with a different value of
in every table. Four different values for
are considered:
to exactly reproduce Moench’s theory, and three other values that represent certain Vanderberg-type modifications, with
and
. All the computations in the tables and figures have been carried out by using the symbolic manipulation software
Wolfram Mathematica 12.2. In particular, the “exact” value of the integral
defined in (
2) was computed by means of numerical integration with the command
NIntegrate.
5. Conclusions
A complete asymptotic expansion of Moench’s transform (
2) for large values of the parameter
x with
y bounded from below (or large values of
) that is uniformly valid in the time variable
is derived. It is given in (
17) for the particular case (and more interesting in practice) of a power-type pumping function. For a more general pumping function, a complete asymptotic expansion of the Moench transform is given in
Appendix B in Formula (
A6). In any case, the expansion is given in terms of elementary functions and an error function with fixed argument
. For a power-type pumping function, Moench’s transform, which measures the water level in the aquifer, experiences a fast (but continuous) transition between a low level and a top limit level: more precisely, from the zero value attained at
(
) to the limit value attained at
,
In the case of a more general pumping function
, the transition is similar but with the factor
replaced by
. This transition occurs around the critical time
, where the Moench transform takes half of its maximum value,
. This fast but smooth transition is encoded in the error function that appears in the expansions (
17) or (
A6) at any order
n of the approximation, as all the terms
on the right-hand side of (
17) or (
A6) contain the error function. The argument of this error function is (see (
18) and (
6))
that vanishes at
, and precisely the error function experiences the above-mentioned transition when its argument vanishes. Moreover, the larger
x is, the faster the transition is, as small variations of the time
t around the critical value
are amplified by the factor
x.
The approximations (
17) or (
A6) have been tested by means of numerical examples that consider realistic parameter values, corresponding to actual physical experiments (see
Section 4). As it can be observed in
Figure 2 and
Figure A1 and
Table 1,
Table 2,
Table 3 and
Table 4, the accuracy of the approximations is remarkable.
We have analyzed Moench’s transform for large values of the parameter x with y bounded from below and . It would be interesting to find other approximations of Moench’s transform valid in new regions of the parameters, such as small values of x and/or y. Moreover, it would be interesting to find approximations uniformly valid for x and/or y in . This is the subject of future research.