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Transport theory

Non-cohesive sediments are transported as bed load and/or suspended load and is always considered in equilibrium. Suspended load is evaluated by the AD solver, using the equilibrium concentration to derive erosion and deposition rates.

Cohesive sediments use the AD solver combined with erosion and deposition.

In MIKE+, non-cohesive sediments are transported using bed load only.

Bed-load

Bed-load is handled by moving sediment from one grid point to the next in each time step. The amount of sediment is determined from the chosen load formula but limited according to availability of sediment.

In addition to scaling the local bed-load with the weight function \(F_{i}\), bed-load is also subjected to a supply check, which essentially means that the local bed-load during the considered time-step cannot be larger than what is available in the H-point located just upwind of the Q-point in which the bed-load is calculated. This process ensures mass conservation and stability, but the user should be cautious about the morphological time-step selection, as very large morphological time-steps will not necessarily cause instabilities but will lead to inaccurate solutions. The best way to ensure that a morphological time-step is selected wisely is to run the same model with e.g. half the time-step and compare the results.

MIKE 1D does not include any gravity effects on bed-load, and therefore bed-load and suspended load will give very similar results for relative coarse sediment sizes.

The stability of bed-load is formally limited by the CFL condition, but it may not express itself clearly as instability in layered models due to the supply limitations. However, solutions that do not satisfy the CFL condition are invalid.

Suspended load (non-cohesive sediment)

The transport of suspended sediment is described by the advection-dispersion equation:

\[ \frac{\partial AC_{i}}{\partial t} + \frac{\partial QC_{i}}{\partial x} - \frac{\partial}{\partial x}\left( AD\frac{\partial C_{i}}{\partial x} \right) = W(E_{i} - D_{i}) \]

Equation: Advection-dispersion equation for suspended sediment transport.

Where \(W\) is a representative cross-section width. This form applies to both non-cohesive and cohesive sediments, with the erosion and deposition for a non-cohesive sediment fraction:

\[ E_{i} - D_{i} = \gamma_{0}w_{s}\left( C_{e,i} - C_{i} \right) \]

Equation: Net erosion and deposition for a non-cohesive sediment fraction.

where \(C\) is the depth-integrated concentration (of fraction \(i\)), \(w_{s}\) is the fall velocity, and \(C_{e}\) is the equilibrium concentration determined from the selected transport formula.

\(\gamma_{0}\) is a profile function tabulated in the code as \(\gamma_{0}(Z)\), and it is defined as the ratio between the bed concentration (\(C_{b}\)) of suspended sediment and the depth-integrated concentration (\(C\)), in other words the deposition rate is:

\[ D_{i} = \gamma_{0,i}w_{s,i}C_{i} = w_{s,i}C_{b,i} \]

Equation: Deposition rate expressed through the profile function and the bed concentration.

The actual value of \(\gamma_{0}\) depends on the distance from the bed (\(y = b \cdot h\)) for which the bed concentration applies. The \(b\)-value is default \(b = 0.02\), but in some formulas, \(b\) is specifically set to another value, e.g. Garcia & Parker (1991) uses \(b = 0.05\).

Suspended load (cohesive sediment)

Cohesive sediments are transported in the same manner as non-cohesive suspended sediment, but with different erosion and deposition functions. The erosion rate for cohesive sediment is calculated from:

\[ E_{i} = E_{0,i}\left( \frac{\tau'}{\tau_{ce,i}} - 1 \right)^{n_{i}}, \quad \tau \geq \tau_{ce,i} \]

Equation: Erosion rate for a cohesive sediment fraction.

and deposition is given by:

\[ D_{i} = w_{s,i}C_{i}\left( 1 - \frac{\tau'}{\tau_{cd,i}} \right), \quad \tau \leq \tau_{cd,i} \]

Equation: Deposition rate for a cohesive sediment fraction.

The deposition of cohesive sediment is not modified for \(\gamma_{0}\).

The following parameters need to be specified for each cohesive fraction:

  • Erosion shear stress threshold \(\tau_{ce}\)
  • Erosion coefficient \(E_{0}\)
  • Erosion exponent \(n\)
  • Deposition shear stress threshold \(\tau_{cd}\)
  • Fall velocity \(w_{s}\)

Often one will want to use the same parameters (shear stress thresholds, erosion coefficient and exponent) for all cohesive fractions to represent mass erosion, while the cohesive fractions will obviously have individual fall velocities.

Sediment transport through nodes

MIKE 1D does not simulate sedimentation in basins and manholes. Both non-cohesive and cohesive sediments continue unaffected through manholes and basins. Suspended load is handled automatically by the AD module, while the bed load entering a node is immediately transferred into outflow reach.

In case the water from the node continues into several reaches (sewer links or river branches), the sediment load is split between the reaches similarly.