The classical Darcy law is generalised by regarding the water flow as a function of a non-integer order derivative of the piezometric head. This generalised law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Numerical solutions of this equation for various fractional orders of the derivatives are compared with experimental data and the Barker generalised radial flow model for which a fractal dimension for the flow is assumed.

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.

No abstract

This book presents the finite-difference, Galerkin-based finite-element, and collocation-based finite element techniques for numerical solution of elliptic, parabolic, and hyperbolic partial differential equations. The book contains seven chapters. The first chapter reviews notation and general

Step-drawdown and multi-rate tests present convenient tools for the estimation of the long-term yield of boreholes. However, the analytical methods commonly employed for the analysis of such tests are all based on the assumption that the drawdown in a borehole is a linear function of the discharge rate. Numerous constant rate tests, of which a few are discussed in this paper, has shown that this is not necessarily the case with boreholes drilled in the Karoo formations of South Africa. The drawdowns in these boreholes are not only influenced by the peculiar geometry of the aquifers, but also the non-linear deformation of the aquifers during the pumping of a borehole. The two new non-linear models for the analysis of step-drawdown and multi-rate tests introduced here, tries to account for these factors; in particular the deformation of the aquifer, flow dimension and dewatering of discrete fractures. Although the model proposed for multi-rate tests is still based on constant time steps, the one for step-drawdown tests allows the user to use arbitrary time steps, when performing the test in the field.Non-linearities in drawdown curves should always be treated with caution, especially when used to assign sustainable yields for boreholes. However, the example of a step-drawdown test performed at the Campus Test Site of the University of the Free State, shows that non-linearities can be addressed with an appropriate model.