Garcia & Parker (1991)

Garcia & Parker¹ is a true multi-fraction suspended load model. The advection-dispersion equation form used by Garcia & Parker is different from what we normally use in DHI's models:

\[ Erosion - Deposition = w_{s}(E - c_{b}) \]

Equation: Erosion and deposition expressed through the near-bed concentration.

This is reformulated from near bed concentration to depth-integrated concentration by using:

\[ c_{b} = \gamma C \]

Equation: Relation between near-bed and depth-integrated concentration.

Where \(\gamma\) is the ratio between near bed and depth-integrated concentration, and is calculated from a Rouse profile using \(b = 0.05\), i.e. the 5% of the depth used by Garcia & Parker (1991). This results in a function \(\gamma(Z)\), where \(Z\) is the Rouse number.

Now the source and sink have the generic form:

\[ Erosion - Deposition = w_{s}(E - \gamma C) \]

Equation: Erosion and deposition expressed through the depth-integrated concentration.

Garcia & Parker (1991) provide the entrainment function for sediment mixtures:

\[ E_{i} = F_{i}\ \frac{A\ Z_{ui}^{5}}{1 + \frac{A}{0.3}Z_{ui}^{5}} \]

Equation: Garcia & Parker entrainment function for sediment mixtures.

Where:

• \(E_{i}\) — entrainment rate for the fraction (\(i\))
• \(Z_{ui}\) — parameter calculated for each fraction (see below)
• \(A\) — constant, \(A = 1.3 \times 10^{-7}\)
• \(F_{i}\) — fraction of (\(i\)) in the surface layer, i.e. surface supply modification

\(E_{i}\) is a volumetric concentration, which can be translated to the internally adopted mass concentration units [g/m³] by multiplying by \(\rho s_{i}\), where:

• \(\rho\) — water density \(\rho = 10^{6}\) g/m³
• \(s_{i}\) — relative sediment density [-] for the considered fraction

The parameter \(Z_{ui}\) is calculated as:

\[ Z_{ui} = \lambda_{m}\ \frac{u_{f}'}{w_{si}}\ Re_{pi}^{0.6}\ \left( \frac{d_{i}}{d_{50}} \right)^{0.2} \]

Equation: Parameter Zᵤᵢ in the Garcia & Parker entrainment function.

Where:

• \(u_{f}'\) — skin friction velocity [m/s]
• \(w_{si}\) — settling velocity for the considered fraction [m/s]
• \(\lambda_{m}\) — parameter derived from the geometric grain size distribution [-]
• \(Re_{pi}\) — grain Reynolds number for the fraction
• \(d_{i}\) — grain size for the considered fraction [mm]
• \(d_{50}\) — local median grain size [mm] in the bed surface

The grain Reynolds numbers for the individual fractions are given by:

\[ Re_{pi} = \frac{\sqrt{\left( s_{i} - 1 \right)gd_{i}^{3}}}{\nu} \]

Equation: Grain Reynolds number for an individual sediment fraction.

The value of \(\lambda_{m}\) is given by Garcia and Parker (1991) as:

\[ \lambda_{m} = 1 - 0.288\ \sigma_{\phi} \]

Equation: Lambda parameter from the standard deviation of the sediment mixture.

Where \(\sigma_{\phi}\) is the standard deviation of the sediment mixture:

\[ \sigma_{\phi}^{2} = \sum{\frac{m_{i}}{\sum m_{i}}}\left( \phi_{i} - \phi_{mean} \right)^{2} \]

Equation: Standard deviation of the sediment mixture on the phi scale.

Where:

• \(m_{i}\) — mass of each fraction in the bed surface [g/m²]

with the geometric grain sizes \(\phi_{i}\) defined using the phi scale (grain size in [mm]):

\[ \phi_{i} = -\log_{2} d_{i} \]

Equation: Geometric grain size on the phi scale.

and the mean geometric grain size (i.e. mean phi size) defined as:

\[ \phi_{mean} = \sum{\frac{m_{i}}{\sum m_{i}}\ \phi_{i}} \]

Equation: Mean geometric grain size on the phi scale.

Calculation of the γ-function

The derivation of \(\gamma\) is done from the Vanoni concentration profile:

\[ c(y) = c_{b}\left( \left( \frac{y}{b} \right)\left( \frac{h - b}{h - y} \right) \right)^{-Z} \]

Equation: Vanoni concentration profile.

Where:

• \(c(y)\) — concentration profile
• \(y\) — distance above the bed
• \(h\) — water depth
• \(b\) — reference level where \(b = 0.05h\)
• \(Z\) — Rouse number [-]

The Rouse number is calculated from the total friction velocity \(u_{f}\) to reflect turbulence:

\[ Z = \frac{w_{s}}{\kappa u_{f}} \]

Equation: Rouse number based on the total friction velocity.

The function \(\gamma = c_{b}/C\) is tabulated by numerically integrating the depth-integrated concentration:

\[ C = \frac{1}{h - b}\int_{b}^{h}{c(y)\ dy} \]

Equation: Depth-integrated concentration.

The Vanoni profile now gives a tabulated function:

\[ \gamma(Z) = \frac{c_{b}}{C} \]

Equation: The γ-function as the ratio of bed to depth-integrated concentration.

Figure: The γ-function.

The numerically integrated \(\gamma(Z)\) function is shown in the γ-function chart. The parameter is unity for zero Rouse number (i.e. evenly distributed sediment has the same depth-integrated and bed concentrations) and increases quite fast with increasing Rouse number.

Notes regarding Garcia & Parker

Garcia & Parker is a suspended load formula, it does not apply to bed-load.

Garcia & Parker was developed from graded sediment data, and is a true multi-fraction suspended load model.

The model was originally developed for single fraction but was extended to multi-fraction. The single fraction form is imbedded in the presented formulation originally published by the authors.

Garcia & Parker does not use the critical Shields parameter, and therefore does not react to the Ashida & Michiue hiding function, and the formula already includes the surface weight function \(F_{i}\).

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1. Garcia, M., and G. Parker (1991) "Entrainment of bed sediment into suspension", Journal of Hydraulic Engineering, 117(4): 414-435. ↩