Improved Continuity Balance for Links¶

Urban (MIKE+)

This section applies to MIKE+ type setups only.

While solving the continuity equation:

\[ \frac{\partial q}{\partial x} + b_{s}\frac{\partial h}{\partial t} = q_{in} \]

Equation: The continuity equation, with the storage width bs introduced.

the storage width, b_s, is in the first iteration approximated at time n, and in subsequent iterations at time n+½. In situations where only one iteration is performed, as is typically the case in MIKE+ type setups, the approximation can imply a significant amount of water being generated or disappearing, giving a significant mass balance error.

A Taylor expansion of the storage width can greatly improve on the mass balance error. Expanding the storage width around time n gives:

\[ b_{s}^{n+\delta} = b_{s}^{n} + \delta\Delta t\frac{\partial b}{\partial h}\frac{\partial h}{\partial t} \]

Equation: Storage width at the new time level, from a first-order Taylor expansion.

Isolating and inserting the \(\partial h/\partial t\) term from the continuity equation:

\[ b_{s}^{n+\delta} = b_{s}^{n} + \delta\frac{\Delta t}{b_{s}}\frac{\partial b}{\partial h}\left(-\frac{\partial q}{\partial x} + q_{in}\right) \]

Equation: Storage width at the new time level, substituting the continuity equation for the time derivative of water level.

Rearrange to get the contribution as a correction term:

\[ b_{s}^{n+\delta} = b_{s}^{n} \cdot \left(1 + \delta\frac{\Delta t}{\left(b_{s}^{n}\right)^{2}}\left.\frac{\partial b}{\partial h}\right|^{n}\left(-\left.\frac{\partial q}{\partial x}\right|^{n} + q_{in}\right)\right) \]

Equation: Storage width at the new time level, rearranged into an explicit update formula.

This modification is applicable only for cross sections with relatively smooth changes of the storage width. As the width for arbitrary cross sections can vary in very unpredictable ways, the correction is only applied to the standard predefined cross sections.

It should be noted that errors in the mass balance occur when changes in storage width are significant as the water level changes. Since cross sections typically become narrow at the bottom, errors in the mass balance can sometimes be controlled by adjusting the minimum water depth (i.e. make sure that reaches do not run dry). See the user manual for information about how to control the minimum water depth.