Modelling the vertical flow of groundwater in a saturated-unsaturated medium

Abstract

The classical equation developed to replicate the vertical groundwater flow through the saturatedunsaturated zone is based on the concept of rate of change. While a lot of success has been achieved using this concept, researchers discovered that only classical mechanical homogenous problems with no memory could be depicted using these mathematical tools. Therefore, a need to formulate new mathematical tools that can be used to depict non-conventional behaviour was urgent as some physical problems were not understandable. For this reason, we introduced to the saturatedunsaturated groundwater flow equation differential operators that are based on the power-law kernel, exponential decay kernel and Mittag-Leffler function to develop new models. These operators are the Caputo, Riemann-Liouville, Caputo-Fabrizio and the Atangana-Baleanu. Some of these operators are local and unable to model problems that display fractal behaviours. To account for this, we used the recently introduced fractal-fractional approach to develop more models. However, the above differential operators are based on sets of constant parameters/coefficients and largely ignore random fluctuations that are likely to occur in nature. To account for randomness, a stochastic approach is used. However, this approach is unable to capture fading memory and long-range behaviour. To help us capture problems with randomness, fading memory and long-range behavior we used the recently established stochastic-fractional approach to generate more models that give us a better understanding of the complex flow. The numerical solutions and stability analysis for all the models are detailed. We also suggested a new approach for modelling groundwater flow based on the Sumudu and Laplace transform. The new models presented in this thesis will be relevant to the majority if not all problems associated with unsaturated-saturated groundwater flow. For this reason, we have not focused on any specific problem/study area, however, theoretical parameters and assumptions are presented to illustrate the effectiveness of the new models, in which readers can make possible modifications to parameters and assumptions to match their specific groundwater problem/study area.