Deterministic and stochastic analysis of groundwater in unconfined aquifer model

Abstract

In several developed countries, groundwater has been recognised as one of the most important natural source of fresh water. It can now be understood why many researchers from all corners from applied science have devoted their attention in developing new methods, models that could be used to monitor and understand the movement of this water within subsurface. The literature nowadays revealed different type of geological formations, including confined, leaky and unconfined aquifers. The movement of water within these three aquifers cannot be captured using the same mathematical models. The Theis model was introduced to capture flow within a confined aquifer, while the Hantush model was suggested to predict the flow within leaky aquifer, these two partial differential equations cannot account for the flow within an unconfined aquifer, and more precisely they are linear equations. To capture flow within an unconfined aquifer, a new mathematical equation was suggested and happens to be integro-differential type. The study of this model is not popular maybe due to the complexity of the mathematical setting. In this dissertation, we considered the model of groundwater flowing within an unconfined aquifer. We derived the conditions under which the exact solution can be obtained. We suggested numerical solutions using different schemes including forward Euler, Crank-Nicholson and Atangana-Batogna schemes. For each of them, we presented detailed study underpinning the stability of the used scheme. To conclude, we suggested a new numerical scheme that combines the fundamental theorem of calculus, Adams-Bashforth and the trapezoidal rule. The method is a new door for investigation in the field of modelling as it is highly accurate and efficient. In addition to this, we argued that differential equations with constant coefficient cannot capture complexities with statistical setting, to solve this problem in case; we converted all parameters included in our equation into distribution functions. The new model was also solved numerically. Lastly, we present numerical simulations from a software package called MATLAB, using normal and statistical data.