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The medium through which the groundwater moves varies in time and space. The Hantush equation describes the movement of groundwater through a leaky aquifer. To include explicitly the deformation of the leaky aquifer into the mathematical formulation, we modify the equation by replacing the partial derivative with respect to time by the time-fractional variable order derivative. The modified equation is solved numerically via the Crank-Nicolson scheme. The stability and the convergence in this case are presented in details.

An aquifer is an underground layer of water-bearing permeable rock or unconsolidated materials (gravel, sand, or silt) from which groundwater can be extracted using a water well. The study of water flow in aquifers and the characterization of aquifers is called hydrogeology. Related terms include aquitard, which is a bed of low permeability along an aquifer, see [

Leaky aquifer.

Hantush was the first to derive a partial differential equation describing such phenomena. However, due to the deformation of some aquifers, the Hantush equation is not able to account for the effect of the changes in the mathematical formulation. The purpose of this work is therefore devoted to the discussion underpinning the description of the groundwater flow through the deformable leaky aquifer, on one hand. On the other hand, we present the derivation of the approximate solution of the modified equation via the Crank-Nicolson scheme. We will start with the definition of the variational order derivative and the problem modification.

For the readers that are not acquainted with the concept of the variational order derivative, we start this section. We present the basic definition of this derivative.

Let

The above derivative is called the Caputo variational order differential operator; additionally the derivative of the constant is zero.

Groundwater models describe the groundwater flow and transport processes using mathematical equations based on certain simplifying assumptions. These assumptions typically involve the direction of flow, geometry of the aquifer, and the heterogeneity or anisotropy of sediments or bedrock within the aquifer. This geological formation through which the groundwater flows changes in time and space.

The simplest generalization of groundwater flow equation, which incidentally is also in accord with true physics of the phenomenon, is to assume that water level is not in a steady but transient state. In 1935, Theis [

The above equation is classified under parabolic equations. However, very few geological formations are completely impermeable to fluids. Leakage of the water could thus occur, should a confined aquifer be over- or underlain by another aquifer. The behaviour of such an aquifer, often referred to as a leaky or semiconfined aquifer, needs thus not be the same as that of a confined aquifer. Although the nature of a semiaquifer differs from that of a true aquifer, it is still possible to use the basic principles of confined flow to arrive at the governing equation for such aquifer. This is in particular true in those situations where the confining layer between the two aquifers is not too thick and the flow is mainly in the vertical direction.

According to Hantush and Jacob [

However, when we consider the diffusion process in the porous medium, if the medium structure or external field changes with time, in this situation, the ordinary integer-order and constant-order fractional diffusion equation model cannot be used to well characterize such phenomenon [

Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (

Solving difficult equations with numerical scheme has been passionate exercise for many scholars [

Before performing the numerical methods, we assume that (

The Crank-Nicolson scheme for the VO time-fractional diffusion model can be stated as follows:

Now replacing (

For simplicity, let us put

Equation (

In this section, we will analyze the stability conditions of the Crank-Nicolson scheme for the Hantush equation for a deformable aquifer.

Let

Then, the function

It was established by [

Observe that for all

It is customary in groundwater investigations to choose a point on the centreline of the pumped borehole as a reference for the observations, and therefore, neither the drawdown nor its derivatives will vanish at the origin, as required [

If we assume that

Equation (

Our next concern here is to show that for all

To achieve this, we make use of the recurrence technique on the natural number

For

Assuming that for

Making use of the triangular inequality, we obtain

Using the recurrence hypothesis, we have

If we assume that

Here,

From (

From the above, we have that

The following inequality

This can be achieved via the recurrence technique on the natural number

When

Now suppose that

The Crank-Nicolson scheme is convergent, and there exists a positive constant

An interesting and detailed research can be found on the solvability of the Crank-Nicolson scheme in the work [

In this section, we present the numerical simulation of the solution of the modified Hantush equation obtained via the Crank-Nicolson scheme. Here let us consider the following equation:

Numerical comparison of the observed data and solution of the modified equation for different values of time.

Numerical simulation of the water flowing in the leaky aquifer.

The main purpose of this paper was to consider the deformation of the leaky aquifer in the mathematical formulation. However, there are some leaky aquifers that change in time and space. Feature that the Hantush equation cannot be used to describe such situation. Recently, the variational order derivative was found very useful to describe efficiently such situation. We then modified the Hantush equation by replacing the partial derivative with respect to time by the variational order derivative. The result equation was solved numerically using the Crank-Nicolson scheme. The stability and convergence were presented in details. We compared the numerical simulation together with the observed drawdown from field observation. The comparison shows that the modified equation predicts more accurately the real field observation.