Deterministic and Stochastic Analysis of Groundwater in Unconfined Aquifer Model

Abstract

In several developed countries, groundwater has been recognized as one of the most important natural source of fresh water. It can now be understood why many researchers from all corners of applied science have devoted their attention to developing new methods, models that could be used to monitor and understand the movement of this water within the sub-surface. The literature nowadays reveals different types of geological formations, including confined, leaky and unconfined aquifers. The movement of water within these three cannot be captured using only 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 a leaky aquifer, but 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 an integro-differential type. The study of this model is not popular, perhaps because of the complexity of the mathematical setting. In this chapter, we considered the model of groundwater flowing within an unconfined aquifer. We derived the conditions under which an exact solution can be obtained. We suggested numerical solutions using different schemes, including forward Euler, Crank–Nicolson and Atangana–Batogna schemes. For each of them, we presented a detailed study underpinning the stability of the scheme used. 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 opening for investigation in the field of modeling as it is highly accurate and efficient. In addition to this, we argued that differential equations with constant coefficients are unable to capture complexities with stochastic behavior. To solve this problem, we converted all parameters included in our equation into distribution functions. The new model was also solved numerically. Finally, we presented numerical simulations from a software package MATLAB, using both normal and statistical data.