Skip to content

Sediment transport formulas

The following load formulas are available.

In MIKE+, not all the transport formulas are available.

Total load formulas determine the total load which is then divided between bed and/or suspended load. The user determines how fractions for bed and suspended load. The total load chosen as suspended load gives the equilibrium concentration:

\[ c_{e} = \frac{q_{susp}\ \rho s}{q_{water}}, \quad q_{water} > 0 \]

Equation: Equilibrium concentration derived from the suspended part of the total load.

If the discharge is zero, the equilibrium concentration is also zero. The internally adopted unit is [g/m3], i.e. the water density is \(\rho = 10^{6}\) g/m3.

The sediment transport formulas are documented in the following sections in which we use the notation explained below.

Notation

The Shields parameter is defined for each fraction by:

\[ \theta_{i}' = \frac{\tau'}{\left( s_{i} - 1 \right)gd_{i}} \]

Equation: Skin friction Shields parameter for a sediment fraction.

Here shown for the skin friction Shields parameter \(\theta'\), and the Einstein decomposition applies:

\[ \theta = \theta' + \theta'' \]

Equation: Einstein decomposition of the Shields parameter.

We use the standard notation for the Shields parameters:

  • \(\theta\) — total Shields parameter
  • \(\theta'\) — skin friction Shields parameter
  • \(\theta''\) — form friction Shields parameter

The calculation of these is described in Einstein decomposition. The notation also applies to shear stresses, friction velocities, water depths and resistance numbers, e.g. \(C'\) means the skin friction Chezy number used in the van Rijn formula.

Calculations involving the suspended sediment concentration profile and turbulence usually are based on the total friction rather than skin friction, while entrainment rates are generally calculated from skin friction. There are no universal laws for how this is to be done, but we follow the principle that everything related to turbulence levels are best derived from total friction, while actions on the bed are best derived from the skin friction, unless the adopted sediment transport model uses the total friction.

Most sediment transport formulas use the non-dimensional sediment transport:

\[ q_{b} = \phi_{b}\ \sqrt{\left( s_{i} - 1 \right)g{d_{i}}^{3}} \]

Equation: Relation between volumetric and non-dimensional sediment transport.

Where:

  • \(\phi\) — non-dimensional sediment transport
  • \(q\) — volumetric sediment transport
  • \(g\) — acceleration of gravity
  • \(i\) — fraction
  • \(s\) — sediment fraction relative density
  • \(d\) — sediment fraction grain diameter

The subscript on the sediment transport formula depends on whether it is bed-load (b), suspended load (s) or total load (t).

For graded sediment transport the sediment transport capacity is multiplied by the \(F_{i}\) weight function, unless explicitly stated that it is not the case.

Some formulas deviate from this, and in those cases it is pointed out in the documentation.