Modelling groundwater flow in a confined aquifer with dual layers
Chaka, Disebo Venoliah
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Groundwater flow occurring in a confined aquifer has been modelled in the past using deterministic mathematical models such as the Theis (1935) equation providing fundamental solutions. However, for this model to be applicable, certain assumptions must be taken into consideration which sometimes leads to the misinterpretation of the real-life scenarios. Because of the limitations arising from this model, flow in a confined aquifer layer cannot be accurately modelled by this equation as it is too simplified. Therefore, the main aim of this study is to model flow in a confined aquifer with a dual layers using non-conventional differential operators and integral operators. Due to the complexity of the geological formation within which the flow is taking place, classical calculus operators are not suitable mathematical operators to be used. Further, research was done on appropriate mathematical operators in order to find the one to be used in this study and the application of the combination of fractal-fractional operators were adopted. Fractal-fractional derivatives are used to solve a complex physical problem and were found to be effective in modelling anomalous diffusion. In order to include into mathematical formulation some complexities of the geological formation, the concept of fractal-fractional differential and integral operators were used. These new differential operators are able to depict scenarios that combine behaviours following the power-law together with self-similarities, or fading memory with self-similarities or crossover behaviours with self-similarities that are observed when the geological formation is equipped with fractures with a self-similar feature. In this study, the Theis groundwater flow model was extended, where the classical differentiation was replaced by three different types of fractal-fractional operators. The modified models were solved numerically using the newly introduced numerical scheme. For each case, a detailed analysis of stability and convergence was presented. The obtained numerical solutions were used to depict numerical simulations showing different values of fractional order and fractal dimension. Obtained figures present a new class of flow different from the normal flow. The fractal dimension has brought new flow trends that can be observed in fractured rock aquifers. The application of these differential operators will open a new way to capture heterogeneity associated with the geological formation.