A Unified Framework for Governing Equations of Hydrologic Flows

Abstract

Laws of conservation of mass, momentum, and energy lead, respectively, to the equations of continuity, momentum, and energy, which are used to mathematically represent hydrologic flow systems as well as analogous systems (physical or nonphysical). For solving a range of problems, the momentum and energy equations are often either simplified or replaced by what are called flux or constitutive laws (linear or nonlinear). When a flux law is coupled with the continuity equation, the resulting equation can be called a governing equation. Depending on the type of flux law and the problem at hand, numerous governing equations exist, but have not been brought under a single framework yet. In this paper, we (1) illustrate a unified framework from which 26 governing equations are derived, each of which is a differential equation common in physics, such as Euler, diffusion, Laplace, Poisson, Boussinesq, Riccati, or others, encompassing partial differential equations (PDEs) of all three types, namely parabolic, hyperbolic, and elliptic; (2) derive 12 hydrologic problems from our unified framework, namely overland flow, surface runoff, snowmelt runoff, glacial movement, flow routing, infiltration, unsaturated flow, subsurface flow, groundwater flow, groundwater recharge, pollutant transport, and sediment transport; (3) show how this framework also applies to two nonhydrologic analogous problems describing a physical system (traffic flow on long highways) and a nonphysical one (flood frequency analysis in statistical hydrology); and (4) conclude with a strategy for analytical treatment of the error history in continuous time or space in an approximate model. Taken together, the unified framework helps establish a connection between numerous seemingly disparate flow problems that can aid in engineering education, research, and design.