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

- Abdon Atangana
^{1}Email author and - Joseph Francois Botha
^{1}

**2013**:53

https://doi.org/10.1186/1687-2770-2013-53

© Atangana and Botha; licensee Springer 2013

**Received: **30 January 2013

**Accepted: **23 February 2013

**Published: **14 March 2013

## Abstract

In this paper, the groundwater flow equation is generalized using the concept of the variational order derivative. We present a numerical solution of the modified groundwater flow equation with the variational order derivative. We solve the generalized equation with the Crank-Nicholson technique. Numerical methods typically yield approximate solutions to the governing equation through the discretization of space and time and can relax the rigid idealized conditions of analytical models or lumped-parameter models. They can therefore be more realistic and flexible for simulating field conditions. Within the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. We perform the stability and convergence analysis of the Crank-Nicholson method and complete the paper with some illustrative computational examples and their simulations.

### Keywords

groundwater flow equation variable order derivative Crank-Nicholson scheme stability convergence## 1 Introduction

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.

## 2 Modified groundwater flow equation

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.

### 2.1 Variational order differential operator

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

### 2.2 Problem formulation

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, the heterogeneity or anisotropy of sediments or bedrock within the aquifer. This geological formation, through which the groundwater flows, changes in time and space.

## 3 Numerical solution

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]

## 4 Stability analysis of the Crank-Nicholson scheme

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

*et al.*[19] that

*S*, and transmissivity

*T*are positive constants. Then the following properties of the coefficients ${T}_{l}^{k+1}$, ${G}_{l}^{k+1}$, ${\lambda}_{j}^{l,k+1}$, and ${b}_{l}^{k+1}$ can be established:

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

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

which this completes the proof.

## 5 Convergence analysis of the Crank-Nicholson scheme

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].

**Lemma 1**${\parallel {\mathrm{\Omega}}^{k+1}\parallel}_{\mathrm{\infty}}\le K({\tau}^{1+{\alpha}_{l}^{k+1}}+{h}^{2}{\tau}^{{\alpha}_{l}^{k}}){({\mathrm{\Omega}}_{j}^{l,k+1})}^{-1}$

*is true for*($k=0,1,2,\dots ,N-1$),

*where*${\parallel {w}^{k}\parallel}_{\mathrm{\infty}}={max}_{1\le l\le M-1}({\mathrm{\Omega}}^{k})$,

*K*

*is a constant*.

*In addition*,

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

which completes the proof.

**Theorem 1**

*The Crank*-

*Nicholson scheme is convergent*,

*and there exists a positive constant*

*V*

*such that*

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.

## 6 Numerical results

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 $t\in [0,10]$.

**Example**

## 7 Conclusion

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.

## Declarations

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors would like to thank the referee for some valuable comments and helpful suggestions. This study was supported by the National Research Fund of South Africa.

## Authors’ Affiliations

## References

- Botha, JF, Buys, J, Verwey, JP, Tredoux, G, Moodie, JW, Hodgkiss, M: Modelling groundwater contamination in the Atlantis aquifer. WRC Report No 175/1/90. Water Research Commission, P.O. Box 824, Pretoria 0001 (2004)Google Scholar
- Van Tonder GJ, Botha JF, Chiang WH, Kunstmann H, Xu Y: Estimation of sustainable yields of boreholes in fractured rock formations.
*J. Hydrol.*2001, 241: 70-90. 10.1016/S0022-1694(00)00369-3View ArticleGoogle Scholar - Barker JA: A generalised radial flow model for hydraulic tests in fractured rock.
*Water Resour. Res.*1988, 24(10):1796-1804. 10.1029/WR024i010p01796MathSciNetView ArticleGoogle Scholar - Van Der Voort, I: Risk-based decision tools for managing and protecting groundwater resources. PhD dissertation, University of the Free State, P.O. Box 339, Bloemfontein (2001)Google Scholar
- Atangana A: Numerical solution of space-time fractional order derivative of groundwater flow equation.
*International Conference of Algebra and Applied Analysis*2012, 20.Google Scholar - Botha JF, Cloot AH: A generalized groundwater flow equation using the concept of non-integer order.
*Water SA*2006, 32(1):1-7.Google Scholar - Solomon TH, Weeks ER, Swinney HL: Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow.
*Phys. Rev. Lett.*1993, 71: 3975-3978. 10.1103/PhysRevLett.71.3975View ArticleGoogle Scholar - Magin RL, Abdullah O, Baleanu D, Zhou XJ: Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation.
*J. Magn. Reson.*2008, 190: 255-270. 10.1016/j.jmr.2007.11.007View ArticleGoogle Scholar - Sun HG, Chen W, Chen YQ: Variable order fractional differential operators in anomalous diffusion modeling.
*Phys. A*2009, 388: 4586-4592. 10.1016/j.physa.2009.07.024View ArticleGoogle Scholar - Umarov S, Steinberg S: Variable order differential equations and diffusion with changing modes.
*Z. Anal. Anwend.*2009, 28: 431-450.MathSciNetView ArticleGoogle Scholar - Ross B, Samko SG:Fractional integration operator of variable order in the Hölder space $H\lambda (x)$.
*Int. J. Math. Math. Sci.*1995, 18: 777-788. 10.1155/S0161171295001001MathSciNetView ArticleGoogle Scholar - Chen YQ, Moore KL: Discretization schemes for fractional-order differentiators and integrators.
*IEEE Trans. Circ. Syst. I: Fund. Theoret. Appl.*2002, 49: 363-367. 10.1109/81.989172MathSciNetView ArticleGoogle Scholar - Theis CV: The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using groundwater storage.
*Trans. Am. Geophys. Union*1935, 16: 519-524. 10.1029/TR016i002p00519View ArticleGoogle Scholar - Zhang Y: A finite difference method for fractional partial differential equation.
*Appl. Math. Comput.*2009, 215: 524-529. 10.1016/j.amc.2009.05.018MathSciNetView ArticleGoogle Scholar - Tadjeran C, Meerschaert MM, Scheffler HP: A second order accurate numerical approximation for the fractional diffusion equation.
*J. Comput. Phys.*2006, 213: 205-213. 10.1016/j.jcp.2005.08.008MathSciNetView ArticleGoogle Scholar - Meerschaert MM, Tadjeran C: Finite difference approximations for fractional advection dispersion equations.
*J. Comput. Appl. Math.*2004, 172: 65-77. 10.1016/j.cam.2004.01.033MathSciNetView ArticleGoogle Scholar - Deng WH: Numerical algorithm for the time fractional Fokker-Planck equation.
*J. Comput. Phys.*2007, 227: 1510-1522. 10.1016/j.jcp.2007.09.015MathSciNetView ArticleGoogle Scholar - Li CP, Chen A, Ye JJ: Numerical approaches to fractional calculus and fractional ordinary differential equation.
*J. Comput. Phys.*2011, 230: 3352-3368. 10.1016/j.jcp.2011.01.030MathSciNetView ArticleGoogle Scholar - Chen CM, Liu F, Turner I, Anh V: A Fourier method for the fractional diffusion equation describing sub-diffusion.
*J. Comput. Phys.*2007, 227: 886-897. 10.1016/j.jcp.2007.05.012MathSciNetView ArticleGoogle Scholar - Yuste SB, Acedo L: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations.
*SIAM J. Numer. Anal.*2005, 42: 1862-1874. 10.1137/030602666MathSciNetView ArticleGoogle Scholar - Podlubny I, Chechkin A, Skovranek T, Chen YQ, Vinagre Jara BM: Matrix approach to discrete fractional calculus II: partial fractional differential equations.
*J. Comput. Phys.*2009, 228: 3137-3153. 10.1016/j.jcp.2009.01.014MathSciNetView ArticleGoogle Scholar - Hanert E: On the numerical solution of space time fractional diffusion models.
*Comput. Fluids*2011, 46: 33-39. 10.1016/j.compfluid.2010.08.010MathSciNetView ArticleGoogle Scholar - Lin R, Liu F, Anh V, Turner I: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation.
*Appl. Math. Comput.*2009, 212: 435-445. 10.1016/j.amc.2009.02.047MathSciNetView ArticleGoogle Scholar - Crank J, Nicolson P: A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type.
*Proc. Camb. Philol. Soc.*1947, 43(1):50-67. 10.1017/S0305004100023197MathSciNetView ArticleGoogle Scholar - Diethelm K, Ford NJ, Freed AD: Detailed error analysis for a fractional Adams method.
*Numer. Algorithms*2004, 36: 31-52.MathSciNetView ArticleGoogle Scholar - Li CP, Tao CX: On the fractional Adams method.
*Comput. Math. Appl.*2009, 58: 1573-1588. 10.1016/j.camwa.2009.07.050MathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.