Mass Conservation

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

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

where

$\begin{array}{l}t\end{array}$ : time [T]
$\begin{array}{l}\boldsymbol{V} = (u,v)^T\end{array}$ : velocity vector [L/T]
$\begin{array}{l}u\end{array}$ and $\begin{array}{l}v\end{array}$ : velocity components in the x- and y- direction respectively [L/T]
$\begin{array}{l}\phi(z_s)\end{array}$ : porosity [-]
$\begin{array}{l}\bar \phi\end{array}$ : depth-averaged porosity [-]
$\begin{array}{l}q\end{array}$ : is a source/sink flux term.

Following the standard HEC-RAS sign conventions, sinks are negative and sources are positive. The velocity $\begin{array}{l}\boldsymbol{V}\end{array}$ represents the average velocity over the fluid volume. If porosity is less than 1, then the velocity is known as the pore velocity, intrinsic velocity, or interstitial velocity. The velocity over the volume including both the solid and fluid is given by the Dupuit–Forchheimer relationship $\begin{array}{l}\boldsymbol{v} = \phi \boldsymbol{V}\end{array}$ . The velocity $\begin{array}{l}\boldsymbol{v}\end{array}$ is known as the filtration velocity, seepage velocity, superficial velocity, Darcy velocity, macroscopic velocity, and volumetric flux velocity. The interstitial velocity is also known as the intrinsic, pore velocity, or simply flow velocity.
In vector form, the continuity equation takes the form:

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:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial}{\partial t} \iint_{A} \phi h dA + \iint_S \bar \phi h ( \boldsymbol{V} \cdot \boldsymbol{n} ) dS = Q\end{array}$

where

$\begin{array}{l}A\end{array}$ : horizontal area [L²]
$\begin{array}{l}\boldsymbol{n}\end{array}$ : unit vector normal to boundaries $\begin{array}{l}S\end{array}$ [-]
$\begin{array}{l}Q\end{array}$ : sum of flows at the boundaries such precipitation, infiltration, evaporation, pumps, etc. [L³/T]

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.