A new method for modeling groundwater flow problems: fractional-stochastic modeling
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Date
2019-06
Authors
Mahantane, Mohau L.
Journal Title
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Publisher
University of the Free State
Abstract
To date, groundwater flow problems are still increasingly becoming a great environmental concern worldwide. This is among some of the reasons that many researchers from various fields of science have focused much of their attention in formulating new mathematical equations and models that could be used to capture and understand the behavior of groundwater flow with respect to space and time. The main aim of this study was to develop a new concept for modeling groundwater flow problems. The approach involved coupling of differential operators with stochastic approach. Literature proves that each of these two concepts has shown a great success in modeling complex real-world problems. But we argued that differential equations with constant coefficient are not fit to capture complexities with statistical setting. Therefore, to solve such a problem in this study, we considered a classical one-dimensional advection-dispersion equation for describing transport in porous medium and then applied stochastic approach to convert groundwater velocity (v), retardation (R) and the dispersion (D) constant coefficients into probability distribution. The next step was to employ the concept of fractional differentiation where we substituted the time derivative with the time fractional differential operator. Thereafter, we applied the Caputo, Caputo-Fabrizio and the Atangana-Baleanu fractional operators and derived conditions under which the exact solution for each derivative can be obtained. We then suggested the numerical solutions using the newly established numerical scheme of the Adams-Bashforth in the case of the aforementioned three (3) different types of differential operators. By combining the two concepts, we developed a new method to capture groundwater flow problems that could not be possible to capture using differential operators or stochastic approach alone. This new approach is believed to be a future technique for modeling complex groundwater flow problems. After solving the new model numerically, the condition for stability was also tested using the Von Neumann stability analysis method. Lastly, we presented numerical simulations using a software package called MATLAB.
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Keywords
Dissertation (M.Sc. (Geohydrology))--University of the Free State, 2019, Advection-dispersion equation, Stochastic approach, Fractional differentiation, Adams-Bashforth, Numerical analysis, Stability analysis