##### Abstract

This work consists of a study of particular aspects of nonlinear flow equations and a non-traditional mass conservation equation. First, methods of constructing approximate polynomial solutions to nonlinear groundwater flow equations are presented. In addition to this work, another researcher's solution of the Boussinesq equation is discussed. Second, a derivation of a non-traditional conservation of mass equation, involving left and right Caputo fractional derivatives, is presented. The resulting equation can be used to derive fractional governing equations for fluid flow and contaminant transport. In Chapter 1, the Introduction, the two nonlinear unconfined groundwater flow equations considered in this study are described. Next, a brief review is given of other works in which fractional differential equations are used for flow and contaminant transport modeling. The later chapters consist of three papers that were previously published and one unpublished chapter. In Chapter 2, published in Water Resources Research 49(5), polynomial approximate solutions are constructed for infiltration into an initially dry, horizontal unconfined aquifer. The flows are described by a porous medium equation with a power-law head condition at the inlet of the aquifer. These approximate solutions can be used to validate numerical solutions to nonlinear equations when exact solutions do not exist. The approximate solutions are compared to numerical solutions computed using a modification of a method of Shampine. The polynomial approximate solutions reproduce known exact solutions and closely match the numerical solutions. Chapter 3, published in Water Resources Research 50(9), is a comment on another researcher's solution of the Boussinesq equation. This researcher used a traveling wave transformation and a perturbation series to generate a solution of the Boussinesq equation. He assumed a constant wave speed in the derivation, but later noted that the derivation implies that the wave speed is time dependent. The comment on this work outlines how his derivation will change if the wave speed is assumed to be time dependent from the beginning of the derivation. In Chapter 4, published in Advances in Water Resources 91, May 2016, a two-sided fractional conservation of mass equation is derived. The equation uses left and right Caputo fractional derivatives and extends the work of other researchers who derived a one-sided fractional conservation of mass equation. The mass conservation equation is based on fractional mean-value theorems. A case is also presented in which a fractional Taylor series is used to derive the mass conservation equation.In Chapter 5, the techniques described in Chapter 2 are applied to a different class of unconfined groundwater flow equation. This equation is derived using a form of the Forchheimer equation in place of Darcy's equation. The resulting groundwater flow equation can be used to model turbulent flow in coarse and fractured porous media. The techniques described in Chapter 2 are used to construct polynomial approximate solutions to the governing equation for power-law head, exponential head, power-law flux and exponential flux conditions at the inlet of the aquifer. The constructed approximate solutions closely match numerical solutions generated using a modification of the method of Shampine.Chapter 6 concludes this work with a summary, conclusions and recommendations for further research.