Application of the New Numerical Method with Caputo–Fabrizio Fractal-Fractional Derivative on the Self-Similar Leaky Aquifer Equations

Abstract

One can ask a question “Can groundwater sustain the future bulk water supply for upcoming generations?” to answer that, yes it can, however it depends on human activities and impacts. Both quantity and quality issues are of concern for groundwater conservation. Groundwater occurrence in fractured-rock aquifers (90% of aquifers in South Africa occur in these domains) was not studied fully during previous years. The article dwells more on fractured aquifers and their properties. Fractures represent a certain type of replication known as self-similarity. The replicated aquifers can have similar characteristics and descriptions. These layers will have artificial fractals, which commonly reproduce self-similar patterns at increasingly small scales and constitute self-affinity. The research was conducted in 2018 and the article published covering self-similar leaky aquifers, and two equations were presented for two different scenarios. The first scenario was the flow of water into and out of self-similar leaky aquifers. The second scenario was the flow within a self-similar leaky aquifer and outflow on a normal leaky aquifer. This chapter intends to cover the concept of self-similarity and the newly introduced operator known as the fractal-fractional operator where the fractional operator considered is Caputo–Fabrizio. Caputo–Fabrizio derivative represents the exponential decay law. Exponential decay law appears in many scenarios, it was reapplied without proper definition and use. As opposed to the power law, it can capture the initial or beginning condition and it is complete.