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We present a model of groundwater dynamics under stationary flow and, governed by Darcy’s law of water motion through porous media, we apply it to study a 2D aquifer with water table of constant slope comprised of a homogeneous and isotropic media; the more realistic case of an homogeneous anisotropic soil is also considered. Taking into account some geophysical parameters we develop a computational routine, in the Finite Difference Method, which solves the resulting elliptic partial equation, both in a homogeneous isotropic and in a homogeneous anisotropic media. After calibration of the numerical model, this routine is used to begin a study of the Ayamonte-Huelva aquifer in Spain, a modest analysis of the system is given, and we compute the average discharge vector as well as its root mean square as a first predictive approximation of the flux in this system, providing us a signal of the location of best exploitation; long term goal is to develop a complete computational tool for the analysis of groundwater dynamics.

One of the most powerful tools to advance theoretical and practical knowledge in the characterization of water flow through ground porous media are computational numerical models, which must always be compared and calibrated with experimental measurements [

Although actual aquifers are inhomogeneous and anisotropic, in order to construct a model, they are usually decomposed in a great number of elements since it is feasible to solve certain problems assuming that each element behaves in its neighborhood as a small ideal aquifer [

This paper is organized as follows: in Section

The Finite Difference Method essentially transforms a differential equation into a system of algebraic equations by means of a spatial discretization of a physical problem’s domain. This is performed considering a finite set of points on a rectangle, called grid, as shown in Figure

Index distribution of a point on the grid with step size of

Let

In order to show how this approximation is done let us focus on a single variable function

Interval centered at

Similarly, one can sum the Taylor expansions above and solve for the second derivative at

Let us say for the moment that we are interested in solving the Laplace equation in 2-dimensions

Along with appropriate boundary values one can compute quantities in the inner part of a grid as exemplified in Figure

Finite difference grid for the domain of certain problem.

In practice one makes a very large partition of the domain, since it is advisable that the partition size

There are various iterative methods to solve a system of linear equations; we will refer here to three of them: Jacobi iteration, Gauss-Seidel iteration, and Successive Overelaxation, although we present results only for the second one. These methods in general are the most efficient to solve equations containing a great number of unknowns since they get rid of the need to store data [

We can simply describe the Jacobi procedure as follows: to each interior point in Figure

Unlike the method described above Gauss-Seidel iteration is performed in an orderly fashion; we compute across the grid from left to right and downwards line per line so that on each node we can use the newest values to compute the next one during the same iteration; in this way we accelerate the convergence of the computation. The iterative expression one uses is

To implement the SOR method we must define the residue,

We say then that one works in the over relaxation scheme since we add more residue to the value at hand; if on the contrary we have

In order to obtain the mathematical model of groundwater dynamics, we assume that water, or any other fluid under study, obeys Darcy’s law of flow through porous media; that is to say that the fluid is incompressible and that its Reynolds number

Assuming that our coordinate axes coincide with the principal axes of the conductivity tensor then for an isotropic homogeneous medium

The equation governing groundwater flow can be obtained by the Control Volume approach [

In the following we restrict ourselves to the case of a 2-dimensional confined aquifer and to the geometry proposed by Toth [

Sketch of the mathematical model and boundary conditions for the regional groundwater system studied by Toth.

Profile of Toth’s analytical solution for the aquifer describe above and the domain configuration depicted in Figure

It is in order to say that the study of analytical solutions for steady and transient flow, though maybe not totally realistic, can give us a great deal of insight of an aquifer and groundwater flow systems. For instance, the stationary solution can be used to determine the response time of a system under transient flow, one of the methods to compute this time consists in solving both equations, for steady and transient flow, with which the response time will be the amount of time required for the transient solution to approach to the stationary one within a predefined tolerance [

Moreover problems susceptible of being analytically solved are still the battleground to probe new predictive techniques, as the mean action integral recently proposed in [

Making use of the results of Section

To compute the solution of this system we use the Gauss-Seidel iteration method since it will converge faster than the simple Jacobi iteration, also given the typical values of

We implement the numerical solution using Python 3.6, recalling we use a second order approximation let us say that we take

For the aquifer setup described at the end of the previous section we have

Profile of the numerical solution to the isotropic aquifer problem sketch by Toth domain configuration depicted in Figure

Overlapping of analytical and numerical solution profiles, here we have taken

In Figure

Convergence curves of the Jacobi and Gauss-Seidel numerical solution methods for the case of flow in an isotropic homogeneous aquifer, we have taken

These results make us confident to apply our computational code to a 2D-cross section of an homogeneous anisotropic aquifer within the same approximations as the problem described above. We take for instance the Ayamonte-Huelva aquifer, localized in the Spanish province of Huelva in Andalucía County, whose hydrogeological characteristics suggest the presence of materials such as sand, sandstone, clay, slate, gravel, lime and chalk [

Profile of the numerical solution to the homogeneous aquifer problem sketch by Toth domain configuration depicted in Figure

Overlapping of the numerical solutions for the isotropic and homogeneous anisotropic cases, here we have taken

As we can see from Figures

Convergence curves of the Jacobi and Gauss-Seidel numerical solution methods for the case of flow in an homogeneous anisotropic aquifer, we have taken

From Figure

Directional field (red) of the water flow inside the aquifer according to (

We should state that while groundwater flow in aquifers can be modeled via specialized software, based on finite differences and directly oriented to the problem, as Visual MODFLOW and MODFLOW-SURFACT in our opinion they present certain limitations, for instance MODFLOW requires additional subroutines, not included as part of the suit, in order to discretize the data one feeds to the software (via text archives) in the form it needs. On the other hand Visual MODFLOW’s major disadvantage resides in the numerical formulation, and it is impossible to “interpolate” between cell values when there is no measured or fed value available, this software dries out the cell assigning to it a zero head value, therefore producing a computational error; another problem is that its numerical resolution allows only work with regular aquifers, generally rectangular or prismatic; therefore to model irregular, anisotropic, or heterogeneous aquifers via this software is very difficult. All this situations can be overcome with careful, though laborious, coding, and detailed calibration of the computer model, as well as a sufficiently accurate and precise finite difference scheme [

Also we must emphasize that the role computer groundwater models have gained a wide variety of applications from critic environmental problems to direct industrial applications, considering, for instance, a contaminated aquifer by heavy metals for which having a 3-dimensional transport model in the case of heterogeneous flow is of vital importance in order to implement soil remediation strategies. These models are generally described in terms of the water balance equation [

We have studied the numerical solution of the steady groundwater flow equation obtained from application of Darcy’s law for flux in porous media; the precise computation of the discrete version of such equation was given considering the general case of an inhomogeneous anisotropic confined aquifer with steady flow. We tested our second-order approximation scheme by confronting numerical results with a known analytical solution. Application of this methodology in the case of the Ayamonte-Huelva aquifer showed that its actual hydraulic parameters, within the second-order finite difference approximation of the differential operators and with the tolerance we work with, this aquifer can be regarded as comprised of an homogenous and isotropic medium since consideration of anisotropy produces negligible changes in the numerical values of the hydraulic potential (with respect to the isotropic case) and thus in the generic behaviour of water flow. We should emphasize that this is valid for the vertical and horizontal hydraulic conductivity values, computed as the weighted mean of the conductivities reported in the literature for the components of the aquifer’s soil. With this setup we were able to determine a first approximation of the optimal location for exploitation of the aquifer as well as an estimation of the flux. Further work is required; we are preparing a 3-dimensional fourth-order code to model Ayamonte-Huelva aquifer which incorporates actual water table data from several locations and soil properties according to depth and including extraction wells and recharge sources also, thus requiring matching conditions depending on the depth; setting the way for the end purpose of developing and alternative computational toolkit for the analysis of groundwater flow, this will allow us to predict the behaviour of the system and to give policies for the sustainable exploitation and management of the aquifer.

The hydrogeological data supporting this study are from previously reported studies and datasets, which have been cited.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors thank VIEP-BUAP and DGDI-BUAP for their financial support during the realization of this work and also PRODEP-SEP for supporting publication expenses. They also thank Javier M. Hernandez-Lopez for his useful insight and comments in the subject of this paper.