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# 5 Fundamental Hydrologic Equations

Part 1. Theory, Organization and Scale

1. Roger Beckie

Published Online: 15 APR 2006

DOI: 10.1002/0470848944.hsa004

## Encyclopedia of Hydrological Sciences

#### How to Cite

Beckie, R. 2006. Fundamental Hydrologic Equations. Encyclopedia of Hydrological Sciences. 1:5.

#### Author Information

1. University of British Columbia, Department of Earth and Ocean Sciences, Vancouver, BC, Canada

#### Publication History

1. Published Online: 15 APR 2006

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### Abstract

In this article our goal is to present an overview of the fundamental principles that are the basis of most models used in hydrology. We develop the fundamental principles of mass, momentum, and energy conservation and express them in mathematical form. We first outline the general approach that can be used to develop a mathematical statement of a conservation law, using a so-called Eulerian framework, where we consider volumes fixed in time and space through which material may flow. We then derive the general conservation equations for mass, momentum, and energy for the case of flowing fluids. We next provide examples from hydrology that illustrate the application of the general conservation principles. We begin with relatively straightforward applications of the conservation equations and progress to more complex and less direct applications. Our first and simplest example is the advection–dispersion equation, which is a relatively transparent application of the conservation of mass principle, augmented with a so-called gradient-flux model, Fick's law, which describes the dispersion and diffusion of solute mass within the bulk flowing fluid. Next we present the Navier–Stokes equations, which are the conservation of momentum equations for a Newtonian fluid. The next suite of examples involves flow in porous media, which is described by more than one conservation principle applied simultaneously. Our last example is from engineering hydraulics, the Saint Venant equations, which are gross but practical simplifications of the general conservation statements.

### Keywords:

• conservation laws;
• conservation of mass;
• conservation of momentum;
• conservation of energy;