A generalized groundwater flow equation using the concept of variable-order derivative

is used as a keystone in the derivation of Eq. (1.1). This law, proposed by Darcy early in the nineteenth century, relies on experimental results obtained from the flow of water through a one-dimensional sand column. Alternatively, Darcy’s law states that the rate of flow through a porous medium is proportional to the loss of head and inversely proportional to the length of the flow path. Note that the specific discharge $q(x,t)$ has the dimensions of velocity. Recent investigations [2] suggest that the flow is also influenced by the geometry of the bedding parallel fractures, the feature that equation (1.1) cannot account for. It is therefore possible that equation (1.1) may not be applicable to the flow in these fractured aquifers. In an attempt to circumvent this problem, Barker [3] introduced the model in which the geometry of the aquifer is regarded as a fractal. Although this model has been applied with reasonable success in the analysis of hydraulic tests from boreholes in Karoo aquifers [4], it introduces parameters for which no sound definition exists in the case of non-integer flow dimensions. Recently [5, 6], the concept of a fractional-order derivative was used to generalize the groundwater flow equation. However, it has been found that the constant-order fractional diffusion equations are not capable of characterizing some complex diffusion processes, for instance, diffusion process in aninhomogeneous or heterogeneous medium [7].

In addition, when we consider the diffusion process in a porous medium, if the medium structure or external field changes with time, in this situation, the constant-order fractional diffusion equation model cannot be used to well characterize such a phenomenon [8–19]. This is the case of the groundwater flow problem, the medium through which the flow occurs is heterogeneous and changes with time. Still in some biology diffusion processes, the concentration of particles will determine the diffusion pattern [10, 11]. To solve the above problems, the variable-order (VO) fractional diffusion equation models have been suggested for use [12]. This present work is therefore devoted to the discussion underpinning the description of the groundwater flow equation with the variable-order derivative.

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

Environmental phenomena such as groundwater flow described by variational order derivative are highly complex phenomena, which do not lend themselves readily to the analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of a numerical solution to groundwater flow equation (2.3).

Numerical methods yield approximate solutions to the governing equation through the discretization of space and time. Within the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. Deterministic, distributed-parameter, numerical models can relax the rigid idealized conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating field conditions. The finite difference schemes for constant-order time or space fractional diffusion equations have been widely studied [14–19]. For constant-order time fractional diffusion equations, Chen et al. proposed an implicit difference approximation scheme [20]. Yuste et al. introduced weighted average finite difference methods [21]. Podlubny et al. proposed the matrix approach for fractional diffusion equations [22], and Hanert proposed a flexible numerical scheme for the discretization of the space-time fractional diffusion equation [23]. Recently, Zhuang et al. considered the numerical schemes for VO space fractional advection-dispersion equation [16], Lin et al. investigated the explicit scheme for VO nonlinear space fractional diffusion equation [24]. Before applying the numerical methods, we assume Eq. (2.3) has a unique and sufficiently smooth solution. To establish the numerical schemes for the above equation, we let $x_l=lh$, $0\le l\le M$, $Mh=L$, $t_k=k\tau$, $0\le k\le N$, $N\tau =T$, h is the step and τ is the time size, M and N are grid points.

3.1 Crank-Nicholson scheme [24]

We introduce the Crank-Nicholson scheme as follows. Firstly, the discretization of first- and second-order space derivative is stated as follows:

(3.1)

(3.2)

(3.3)

The Crank-Nicholson scheme for the VO time fractional groundwater flow model can be stated as follows:

$\begin{array}{rcl}\frac{{\partial }^{{\alpha }_l^{k+1}}\mathrm{\Phi }(r_l,t_{k+1})}{\partial t^{{\alpha }_l^{k+1}}}& =& \frac{{\tau }^{-{\alpha }_l^{k+1}}}{\mathrm{\Gamma }(2-{\alpha }_l^{k+1})}(\mathrm{\Phi }(r_l,t_{k+1})-\mathrm{\Phi }(r_l,t_k)\\ +\sum _{j=1}^k[\mathrm{\Phi }(r_l,t_{k+1-j})-\mathrm{\Phi }(r_l,t_{k-j})][{(j+1)}^{1-{\alpha }_l^{k+1}}-{(j)}^{1-{\alpha }_l^{k+1}}]).\end{array}$

(3.4)

Now, replacing equations (3.1), (3.2), (3.3) and (3.4) in (2.3), we obtain the following:

$\begin{array}{c}\left[\frac{S{\tau }^{-{\alpha }_l^{k+1}}}{\mathrm{\Gamma }(2-{\alpha }_l^{k+1})}\right(\mathrm{\Phi }(r_l,t_{k+1})-\mathrm{\Phi }(r_l,t_k)\hfill \\ +\sum _{j=1}^k[\mathrm{\Phi }(r_l,t_{k+1-j})-\mathrm{\Phi }(r_l,t_{k-j})][{(j+1)}^{1-{\alpha }_l^{k+1}}-{(j)}^{1-{\alpha }_l^{k+1}}]\left)\right]\hfill \\ =T\left[\frac{1}{2}\right(\left(\frac{\mathrm{\Phi }(r_{l+1},t_{k+1})-2\mathrm{\Phi }(r_l,t_{k+1})+\mathrm{\Phi }(r_{l-1},t_{k+1})}{{(h)}^2}\right)\hfill \\ +\left(\frac{\mathrm{\Phi }(r_{l+1},t_k)-2\mathrm{\Phi }(r_l,t_k)+\mathrm{\Phi }(r_{l-1},t_k)}{{(h)}^2}\right)\left)\right]\hfill \\ +\frac{1}{r_l}\left[\frac{1}{2}(\left(\frac{\mathrm{\Phi }(r_{l+1},t_{k+1})-\mathrm{\Phi }(r_{l-1},t_{k+1})}{2(h)}\right)+\left(\frac{\mathrm{\Phi }(r_{l+1},t_k)-\mathrm{\Phi }(r_{l-1},t_k)}{2(h)}\right))\right].\hfill \end{array}$

For simplicity, let us put:

$\begin{array}{r}{b}_j^{l,k+1}={(j+1)}^{1-{\alpha }_l^{k+1}}-{(j)}^{1-{\alpha }_l^{k+1}};T_l^{k+1}=\frac{\mathrm{\Gamma }(2-{\alpha }_l^{k+1}){\tau }^{{\alpha }_l^{k+1}}}{S{h}^2}T;\\ G_l^{k+1}=\frac{\mathrm{\Gamma }(2-{\alpha }_l^{k+1}){\tau }^{{\alpha }_l^{k+1}}}{Sh}\text{and}{\lambda }_j^{l,k+1}=b_{j-1}^{l,k+1}-b_j^{l,k+1}.\end{array}$

(3.5)

Equation (3.5) becomes

$\begin{array}{rcl}{\mathrm{\Phi }}_l^{k+1}(1+2T_l^{k+1})& =& {\mathrm{\Phi }}_{l+1}^{k+1}(T_l^{k+1}+\frac{G_l^{k+1}}{r_l})+{\mathrm{\Phi }}_{l-1}^{k+1}(T_l^{k+1}-\frac{G_l^{k+1}}{r_l})\\ +{\mathrm{\Phi }}_{l+1}^k(T_l^{k+1}-\frac{G_l^{k+1}}{r_l})+{\mathrm{\Phi }}_l^{k+1}(1+2T_l^{k+1})\\ +\sum _{j=1}^k({\mathrm{\Phi }}_l^{k+1-j}-{\mathrm{\Phi }}_l^{k-j}){\lambda }_j^{l,k+1}{G}_l^{k+1}.\end{array}$

(3.6)

In this section, we will analyze the stability conditions of the Crank-Nicholson scheme for the generalized groundwater flow equation.

where ϕ is a real spatial wave number, new replacing the above equation (4.5b) into (4.5a), we obtain

(4.6)

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

Using the recurrence hypothesis, we have

(4.11)

which this completes the proof.

where $K_1$, $K_2$, and K are constants. Taking into account the Caputo-type fractional derivative, the detailed error analysis on the above schemes can refer to the work by Diethelm et al. [25] and further work by Li and Tao [26].

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

Now suppose that ${\parallel {\mathrm{\Omega }}^{i+1}\parallel }_{\mathrm{\infty }}\le K({\tau }^{1+{\alpha }_l^{i+1}}+h^2{\tau }^{{\alpha }_l^i}){({\lambda }_j^{l,i+1})}^{-1}$, $i=1,\dots ,N-2$. Then

(5.5)

which completes the proof.

An interested reader can find the solvability of the Crank-Nicholson scheme in the work done by [24]. Therefore, the details of the proof will not be presented in this paper.

An image is worth ten thousand words; therefore, we devote this section to the numerical simulations of the solution of the generalized groundwater flow equation. The parameters used in the simulation are given as $S=0.0086$, $T=400$, $r\in [0.5,35]$, and $t\in [0,10]$.

The numerical simulations of the approximate solution are presented in figures. First, Figure 2 shows the solution as a function of time and space. Figure 3 shows the solution as a function of time for a fixed distance. Figure 4 shows the solution as a function of space for fixed time.

In this paper, the groundwater flow equation was generalized using the concept of a variational order derivative. The new equation was solved numerically via the Crank-Nicholson technique. We presented in detail the stability and the convergence of this problem. We presented numerical simulations.