Mass Conservation

Assuming that the flow is incompressible, the unsteady differential form of the mass conservation (continuity) equation is:

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial h}{\partial t} + \frac{\partial (hu)}{\partial x} + \frac{\partial (hv)}{\partial y} = q\end{array}$

where $\begin{array}{l}t\end{array}$ is time, $\begin{array}{l}u\end{array}$ and $\begin{array}{l}v\end{array}$ are the velocity components in the x- and y- direction respectively and q is a source/sink flux term. Following the starndard HEC-RAS sign conventions, sinks are negative and sources are positive.
In vector form, the continuity equation takes the form:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial h}{\partial t} + \nabla \cdot (h\boldsymbol{V}) = q\end{array}$

where $\begin{array}{l}\boldsymbol{V} = (u,v)^T\end{array}$ is the velocity vector and $\begin{array}{l}\nabla\end{array}$ is the gradient operator given by $\begin{array}{l}\nabla = (\partial / \partial x , \partial / \partial y )^T\end{array}$ .

Integrating over a horizontal region with boundary normal vector n and using Gauss' Divergence theorem, the integral form of the equation is obtained:

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial}{\partial t} \int \int_{\Omega} \int d \Omega + \int \int_S (\boldsymbol{V} \cdot \boldsymbol{n}) dS = Q\end{array}$

The volumetric region $\begin{array}{l}\Omega\end{array}$ represents the three-dimensional space occupied by the fluid, and $\begin{array}{l}\textbf {n}\end{array}$ is the unit vector normal to the side boundaries $\begin{array}{l}S\end{array}$ . It is assumed that $\begin{array}{l}Q\end{array}$ represents any flow that crosses the bottom surface (infiltration) or the top water surface of $\begin{array}{l}\Omega\end{array}$ (evaporation or rain). The source/sink flow term $\begin{array}{l}Q\end{array}$ is also convenient to represent other conditions that transfer mass into, within or out of the system, such as pumps.

This integral form of the continuity equation will be appropriate in order to follow a sub-grid bathymetry approach in subsequent sections. In this context, the volume Ω will represent a finite volume cell and the integrals will be computed using information about the fine underlying topography.