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

*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

*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 where are metric coefficients given by so that

*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

*τ*such that For , looking for the dominant term and making use of some asymptotic expansion technique, we get the following relation (Acho 1992; Atangana 2014): 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)): 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 Employing the Laplacian for the spheroidal in Equation (10) produces the new version of the groundwater equation flow (Equation (11)).

## ANALYSIS AND SOLUTIONS

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 *S*_{0.} 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).

## 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.

Figures 1–3 show that, for any fixed time, the solution proposed by the author gives a better approximation than the Cooper–Jacob solution.

## EXPONENTIAL INTEGRAL 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 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 m^{3}/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 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 m^{3}/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

### Adomian decomposition method

*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:

### Variational iteration method

*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

*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):

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.