We described a groundwater model with prolate spheroid coordinates, and introduced a new parameter, namely τ the silhouette influence of the geometric under which the water flows. At first, we supposed that the silhouette influence approaches zero; under this assumption, the modified equation collapsed to the ordinary groundwater flow equation. We proposed an analytical solution to the standard version of groundwater as a function of time, space and uncertainty factor α. Our proposed solution was in good agreement with experimental data. We presented a good approximation to the exponential integral. We obtained an asymptotic special solution to the modified equation by means of the Adomian decomposition and variational iteration methods.

## INTRODUCTION

The mathematical equation underpinning the description of the conservative and flooded underground water for density-independent flow is provided in the following formula:
1
where is the specific storativity, K the hydraulic conductivity tensor of the aquifer or the transmissivity in the case of Theis, the piezometric head, the strength of any sources or sinks, with x and t the usual spatial and time coordinates, the gradient operator and ∂t the time derivative (Bear 1972; Barker 1988; van Tonder et al. 2001; Cloot & Botha 2006). Equation (1) is subject to the following initial and boundary conditions:

A problem that arises naturally in groundwater investigations is the need to choose an appropriate geometry for the geological system in which the flow occurs (Atangana 2014). We think that describing the groundwater flow with one equation for the whole aquifer is unrealistic, because properties, including geology and geometry, change from one point of the aquifer to another, thereby affecting the flow. In this paper, we describe the groundwater flow equation with prolate spheroid coordinates. The rest of this paper is structured as follows: we start with the coordinate transformation, followed by proposing a solution to the standard version of the groundwater flow equation, the proposed solution is compared to Theis and Cooper–Jacob solutions for several sets of experimental data, we propose a good approximation to the exponential integral and we end by solving the new groundwater flow equations using many methods including Adomian decomposition and variational iteration methods.

## COORDINATE TRANSFORMATION

The prolate spheroidal coordinates can be interconnected to the Cartesian coordinates as follows (Atangana 2014):
2
Here, . It is worth noting that, by rotating an ellipse about the main z-axis, we get a prolate spheroid; however, by rotating about the minor y-axis, we get an oblate spheroid. For small values of , prolate spheroids are rod-shaped and so can be looked upon as an approximation to a cable antenna (Acho 1992; Atangana 2014). In the case of oblate spheroids, the surface is a disc . From the coordinate system (2) we obtain the Laplacian as
3
where are metric coefficients given by
so that
4
The spherical polar coordinates are given as
5
Here, . The first aspect of this section is to obtain the prolate spheroid in spherical coordinates. The first guess here is that a Laplacian would be the sum of a sphere and a small term (Acho 1992; Atangana 2014). To achieve this we first find a relation between spheroidal coordinates to the polar spherical coordinates. The equation of an ellipse with its centre at the origin will then be (Acho 1992; Atangana 2014)
6
It is notable that if , then the ellipse falls down to a circle with ratio a. We can therefore conclude that if , the prolate spheroid falls down to a sphere, since a sphere is generated by a circle (Acho 1992; Atangana 2014). The shape factors in this paper will be , where l is one half the interfocal distances and a, which is for physical purposes finite, is the radius of the approximating sphere. Establishing a comparison between Equation (2) and Equation (5) yields

Implying . The next concern is to establish the relationship between for small τ such that
7
For , looking for the dominant term and making use of some asymptotic expansion technique, we get the following relation (Acho 1992; Atangana 2014):
8

Thus, from the above equation, the following relation may be established:
for Thus, putting everything together, we obtained the Laplacian for the prolate spheroidal in the following form (Equation (9)):
9
Here, we simplify Equation (1) by considering as a scalar, and we suppose that there is no sink or source in the system so that Equation (1) becomes
10
Employing the Laplacian for the spheroidal in Equation (10) produces the new version of the groundwater equation flow (Equation (11)).
11

## ANALYSIS AND SOLUTIONS

The above Equation (11) is very difficult to solve analytically. However, some assumptions are made to achieve the reduced version of this equation; the first assumption we make here regards the shape factor introduced in this paper, i.e., the shape factor, is zero. If the shape factor tends to zero, and we assume that the flow is in the direction, the new groundwater equation flow collapses to the standard version of the groundwater flow equation given below
12
Several solutions to the above partial differential equation have been recently proposed, see for example Theis (Theis 1935) and Cooper & Jacob (1946). The Theis solution was an exact solution to the partial differential equation and this solution is given below as
13
For practical purposes this solution is very difficult to implement. However, Cooper and Jacob proposed an approximate solution of the partial differential equation for a later time and this solution is given below as (Jacob & Lohman 1952)

14
The Cooper–Jacob method is the one most used in groundwater studies because it is very easy to handle; at the time the solution was proposed, it was much easier to manipulate using log paper than using the Theis solution (the Theis solution involves the exponential integral), but nowadays many versions of computational software are available to handle Theis without log paper (Atangana 2014). Nevertheless, the Cooper–Jacob solution has limitations, because it is a large time approximation of the Theis non-equilibrium method (Atangana 2014). The approximation involves truncations of an infinite series expansion for the Theis well function that is valid when the variable: is small enough, which is not always the case in a groundwater study. In this paper, we propose an approximated solution of the groundwater flow equation in form of
15
Then, putting Equation (15) into Equation (12) produces
By applying the boundary condition on the solution, we approximate the solution of the groundwater flow equation as
16

where is a small positive constant such that .

The above proposed solution takes into account the events that take place for later and earlier times during the pumping test. The proposed solution, however, contains a new parameter alpha, which can be viewed as a new physical parameter that may characterise the epistemic uncertainty; that is, the uncertainty associates to the model, more precisely this new parameter will be considered as an uncertainty factor in the geological formations associated with the mathematical formulation of the model in this paper. To test the validity of the above proposed solution, we compare it to the Theis and Cooper–Jacob solutions. In order to compare these solutions we choose a set of aquifer parameters, the discharge rate Q, the transmissivity T and the storativity S0. The following figures illustrate the comparison of these solutions in space and time (full colour versions of these figures are available online at http://www.iwaponline.com/ws/toc.htm). The red line is the graphical representation of the solution proposed by Cooper and Jacob, the green line is the graphical representation of the solution proposed by the author and the blue line is the graphical representation of the solution proposed by Theis (Figure 1).

Figure 1

Comparison of Theis, proposed and Cooper–Jacob solutions for T = 2.6, Q = 200, S = 0.65 and r = 10, .

Figure 1

Comparison of Theis, proposed and Cooper–Jacob solutions for T = 2.6, Q = 200, S = 0.65 and r = 10, .

## COMPARISON OF SOLUTIONS

Figure 2 shows that the solution proposed by the author is in perfect agreement with the Theis solution for an earlier time, in this case . From this analysis, we can conclude that for a given set of aquifer parameters and an appropriate uncertainty factor defined earlier, the solution proposed by the author gives a better approximation of the Theis equation than the Cooper–Jacob solution. The following graphs show the graphical representation of the solution for a fixed time and a function of distance; as stated earlier, the red line presents Cooper–Jacob, the green line represents the author and the blue line represents Theis.

Figure 2

Comparison of Theis, proposed and Cooper–Jacob solutions for T = 2.6, Q = 200, S = 0.65 and t = 10, .

Figure 2

Comparison of Theis, proposed and Cooper–Jacob solutions for T = 2.6, Q = 200, S = 0.65 and t = 10, .

Figures 13 show that, for any fixed time, the solution proposed by the author gives a better approximation than the Cooper–Jacob solution.

Figure 3

Comparison of Theis, proposed and Cooper Jacob solutions for T = 2.6, Q = 200, S = 0.65 and t = 10,000, .

Figure 3

Comparison of Theis, proposed and Cooper Jacob solutions for T = 2.6, Q = 200, S = 0.65 and t = 10,000, .

## EXPONENTIAL INTEGRAL APPROXIMATION

It important to single out from the comparison between the solution proposed by the author and the Theis solution that the exponential integral can be equated as
The following graphs show the comparison between the exponential integral and the proposed approximation (Figure 4).
Figure 4

Comparison of exponential integral and the proposed approximation .

Figure 4

Comparison of exponential integral and the proposed approximation .

## COMPARISON WITH EXPERIMENTAL DATA

In order to examine the validation of this solution, the above asymptotic solution is compared with four sets of experimental data (Figure 5).

Figure 5

Comparison of experimental data from ‘Oude Korendijk (Cem et al. 2011; Wit 1963)’ and Cooper–Jacob, Theis and proposed solutions.

Figure 5

Comparison of experimental data from ‘Oude Korendijk (Cem et al. 2011; Wit 1963)’ and Cooper–Jacob, Theis and proposed solutions.

Figure 5 shows the comparison between experimental data from a pumping test conducted in the polder ‘Oude Korendijk’, south of Rotterdam with Cooper–Jacob, Theis and the proposed solutions. Here, the transmissivity was determined as T = 360 m3/day, the storativity and the uncertainty factor for a constant discharge rate of Q = 9.12 l/s at a distance of r = 90 m (Figure 6).

Figure 6

Comparison of experimental data from ‘Oude Korendijk’ and Cooper–Jacob, Theis and proposed solutions.

Figure 6

Comparison of experimental data from ‘Oude Korendijk’ and Cooper–Jacob, Theis and proposed solutions.

Figure 6 shows the comparison between experimental data from a pumping test conducted in the polder ‘Oude Korendijk’, south of Rotterdam with Cooper–Jacob, Theis and the proposed solutions. Here the transmissivity was determined as T = 360 m3/day, the storativity and the uncertainty factor for a constant discharge rate of Q = 9.12 l/s at a distance of r = 215 m.

## NUMERICAL SOLUTION VIA ADOMIAN DECOMPOSITION AND VARIATIONAL ITERATION METHODS

We next assume that the flow is radial, which implies that all derivatives with respect to and reduce to zero and the reduced equation is given below as
The above equation can be rewritten in the form of
17
The above equation describes the groundwater flow governed by the elliptic flow. Let and then we have Equation (17), which can be reduced to
18
The above Equation (18) can be very difficult to handle analytically; therefore, in this paper we make use of variational iterationed and Adomian decomposition methods to find an asymptotic analytical solution for this equation.

The method used here is based on applying the inverse operator of on both sides of Equation (18) to obtain
19
The Adomian decomposition method (Adomian 1988, 1994) suggests the series solution of Equation (19) in the form of
20
Substituting Equation (20) into Equation (19) and following the decomposition technique, we introduce the recursive formula:
21

Since the zeroth component is defined, the remaining components n ≥ 1, can be completely determined such that each term is determined by using the previous terms, and the series solutions are thus entirely determined. Finally, the solution is approximated by the truncated series (Atangana & Kılıçman 2013)
Thus, from the recursive formula Equation (4) the following components are determined:

In this way the components of the decomposition series were obtained, of which ϕ(r, t) was evaluated to the following asymptotic expansion:
22

### Variational iteration method

The values of the variational iteration method and its applications for a range of categories of differential equations can be viewed in (He 2001, 2003, 2004). To solve Equation (18) by means of the variational iteration method, we put Equation (18) in the form

23
The correction functional for Equation (6) can be approximately expressed as follows:
where is a general Lagrange multiplier (Inokuti et al. 1978), which can be recognised optimally by means of variation assumption, here and are considered as constrained variations. Making the above functional stationary,
capitulates the next Lagrange multipliers
In this way, we investigate a possible solution for the case of m = 1. Therefore for m = 1 we obtain the following recursive formula:
Hence, we commence with

In this way three components of the decomposition series were obtained, of which ϕ(r, t) was evaluated to the following asymptotic expansion (Atangana & Kılıçman 2013):
24

It is important to highlight that self-cancelling noise terms appear between various components of the two asymptotic solutions. Then we cancelled the noise terms in the decomposition solution (22) and the variational iteration solution (24).

## DISCUSSION AND CONCLUSION

In this work, the groundwater flow equation was modified using prolate spheroid coordinates. A new parameter was introduced, which was called the ‘silhouette influence of the geometry’. The new version of the groundwater flow equation collapsed to the standard version as the shape parameter tends towards zero. We proposed an analytical solution to the standard version of the groundwater flow equation as a function of time, space and uncertainty factor. This analytical solution was compared to the existing solutions including, Theis and Cooper–Jacob solutions using several sets of parameters. The comparison revealed that the new analytical solution proposed in this paper was for a given alpha predicting exactly what can be predicted by the Theis solution rather than the Cooper–Jacob solution. However, the new solution takes into account the event that could take place at the vicinity of the borehole for earlier and later time; it also takes into account the event that could take place at a long distance from the borehole for earlier and later times.

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