Analysis of General Groundwater Flow Equation within a Confined Aquifer Using Caputo Fractional Derivative and Caputo–Fabrizio Fractional Derivative

Abstract

The main aim of this research was to analyze the exact groundwater flow equation within a confined aquifer with nonlocal operators. There were several derivatives we looked at, but for the sole purpose of this chapter we are going to present analysis for Caputo fractional and Caputo–Fabrizio fractional derivatives. Recently, researchers have introduced a more general groundwater flow model which considers the forgotten terms by Theis; however, their model uses the concept of classical differential operators and therefore could not capture complexities of the media through which the sub-surface water flows. It is in this regard that their model was extended to the concept of nonlocal operators. New numerical schemes were developed using the classical Adams–Bashforth method and the newly developed Newton polynomial method. Stability analysis of the numerical schemes is presented. Numerical simulations were developed and showed evidence of three different types of flow according to fractional orders. Slow flow was observed in fractional order from 0.99 to 0.51, depicting that the geological media is not well connected and thus retards flow. When the fractional order was below 0.5, fast flow was observed and when the fractional order was 1, normal flow was recovered, and the geology was assumed to be homogenous.