Abstract. In this study dimensionally consistent governing equations of continuity and motion for transient soil water flow and soil water flux in fractional time and in fractional multiple space dimensions in anisotropic media are developed. Due to the anisotropy in the hydraulic conductivities of natural soils, the soil medium within which the soil water flow occurs is essentially anisotropic. Accordingly, in this study the fractional dimensions in two horizontal and one vertical directions are considered to be different, resulting in multi-fractional multi-dimensional soil space within which the flow takes place. Toward the development of the fractional governing equations, first a dimensionally consistent continuity equation for soil water flow in multi-dimensional fractional soil space and fractional time is developed. It is shown that the fractional soil water flow continuity equation approaches the conventional integer form of the continuity equation as the fractional derivative powers approach integer values. For the motion equation of soil water flow, or the equation of water flux within the soil matrix in multi-dimensional fractional soil space and fractional time, a dimensionally consistent equation is also developed. Again, it is shown that this fractional water flux equation approaches the conventional Darcy equation as the fractional derivative powers approach integer values. From the combination of the fractional continuity and motion equations, the governing equation of transient soil water flow in multi-dimensional fractional soil space and fractional time is obtained. It is shown that this equation approaches the conventional Richards equation as the fractional derivative powers approach integer values. Then by the introduction of the Brooks–Corey constitutive relationships for soil water into the fractional transient soil water flow equation, an explicit form of the equation is obtained in multi-dimensional fractional soil space and fractional time. The governing fractional equation is then specialized to the case of only vertical soil water flow and of only horizontal soil water flow in fractional time–space. It is shown that the developed governing equations, in their fractional time but integer space forms, show behavior consistent with the previous experimental observations concerning the diffusive behavior of soil water flow.

Abstract. Using fractional calculus, a dimensionally consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a dimensionally consistent continuity equation for transient saturated groundwater flow in fractional time and in a multi-fractional, multidimensional confined aquifer is developed. For the equation of water flux within a multi-fractional multidimensional confined aquifer, a dimensionally consistent equation is also developed. The governing equation of transient saturated groundwater flow in a multi-fractional, multidimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations. To illustrate the capability of the proposed governing equation of groundwater flow in a confined aquifer, a numerical application of the fractional governing equation to a confined aquifer groundwater flow problem was also performed.

Abstract. In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy-Buckingham-theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Abstract. A dimensionally-consistent governing equation of transient, saturated groundwater flow in fractional time in a multi-fractional confined aquifer is developed. First, a continuity equation for transient groundwater flow in fractional time and in a multi-fractional, multi-dimensional confined aquifer is developed. An equation of water flux is also developed. The governing equation of transient groundwater flow in a multi-fractional, multi-dimensional confined aquifer in fractional time is then obtained by combining the fractional continuity and water flux equations.

Abstract. In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Abstract:Complex flows in heterogeneous confined and unconfined aquifers is a phenomenon that continues to present difficulties in flow mapping and modelling in the field, laboratory, and through numerical simulations. It is often the case with complicated phenomena that transformative scaling and reduction of the problem through symmetry is of great efficacy in the formation of predictive models in both the laboratory and computational settings. A detailed a study of the application of a broad class of Lie scaling transformations on a set of equations representing the groundwater flows in heterogeneous confined and unconfined aquifers has produced a set of scaling relationships between the spatial variables, hydrologic variables, and parameters. The set of scaling transformations preserve the structure of the equations in the sense that the scaling transformations leave the initialboundary value system representing the invariant groundwater flows. This theoretical approach elucidates not only the scaling relationships but also the properties that hydrologic variables and parameters must satisfy in order for calling to be possible. Validation of the theory developed is carried out through a series of four numerical simulations using the USGS MODFLOW-2005 software package. The results of these experiments demonstrate that the derived scaling transformations can effectively form predictive models of large-scale phenomena at small scales with negligible error in many cases. Comments on the limitations of the approach and directions for future research are made in the closing sections.

Abstract. In the past two decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics including but not limited to groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non dimensionalization, and the Buckingham Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is done with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

Abstract. Observations have been made in studies that in watersheds with moist soils and lush vegetation, water does not pond on the soil surface, even during significant rainfall events. In these cases, all the water infiltrates into the soil. Furthermore, the water content in the pore space is below saturation.The tools for modeling such situations are limited. A universal method for estimating groundwater in the unsaturated zone is the Richards' equation, a non linear system of 1D or 3D equations based on physical flow principles. While effective, the 5 difficulties in solving Richards' equation can pose a significant problem in a complex system, which may require extensive amounts of calibration and numerical effort in procuring a solution. A second method is based on the idea of approximating the movement of moisture through the soil as a rectangular profile, e.g. assuming that the water moves through a soil column like a piston. Rectangular profile methods have been developed in the past for cases in which ponding occurs at the surface, and the water content inside the column is at saturation. More general rectangular profile methods allow the water in the column 10 to move at sub-saturation. Both of these existing methods involve solving a non linear algebraic equation for the hydrologic system, involving constant flux at the boundary, due to a wetting event such as rainfall. The method proposed in this article may be used for situations in which the soil does not become saturated, and the soil and rainfall properties, specifically the rates of hydraulic conductance and rate of rainfall respectively, are allowed to vary in space or time. The method proposed is based on a simple principle in infiltration hydrology, that the rate at which water will infiltrate through the soil equilibrates with the 15 rate of rainfall, when the rate of rainfall is smaller than the infiltration capacity of the soil. In application, this method does not require the solution of a non linear algebraic (or differential) system of equations, thus affording modelers computational economy.Furthermore, this new method can be made to interact with the saturated profile methods in the event that the rainfall overwhelms the ability of soil to absorb moisture. If this happens, a portion of the field will come to saturation while other 20 parts will be below saturation. The sub-saturation models can be made to interact with the redistribution and evapotranspiration processes by providing the initial and boundary condition for those events.