When the bed material is composed of multiple grain sizes, larger grains have a greater probability of being exposed to the flow while smaller particles have a greater probability of being hidden from the flow. The figure below shows an example of a sediment grain d_j being hidden by d_k.

Figure 2 23. Schematic of the exposure height of bed sediment grains.

Depending on the form of the sediment transport formula, the hiding and exposure effect may be included in different ways. In many cases, the correction is done by adjusting the incipient motion in which case the hiding and exposure correction is defined as

1)	$\begin{array}{l}\displaystyle \xi _{k}=\frac{\theta _{crk}}{\theta _{cr}}=\frac{\tau _{crk}}{\tau _{cr}}=\frac{U_{crk}^{2}}{U_{cr}^{2}}\end{array}$

in which

$\begin{array}{l}τ_{cr}\end{array}$ = uncorrected critical shear stress [M/L/T²]

$\begin{array}{l}τ_{crk}\end{array}$ = corrected critical shear stress [M/L/T²]

$\begin{array}{l}\theta _{cr}=\frac{\tau _{cr}}{(\rho _{sk}-\rho )gd_{k}}\end{array}$ = uncorrected critical Shields parameter [-]

$\begin{array}{l}\theta _{crk}=\frac{\tau _{crk}}{(\rho _{sk}-\rho )gd_{k}}\end{array}$ = corrected critical Shields parameter [-]

$\begin{array}{l}U_{cr}\end{array}$ = uncorrected critical depth-averaged current velocity [L/T]

$\begin{array}{l}U_{crk}\end{array}$ = corrected critical depth-averaged current velocity [L/T]

Examples of hiding and exposure correction factors for incipient motion are Egiazaroff (1965), Ashida and Michiue (1971), and Wu et al. (2000). Some transport formulas such as (e.g. Ackers-White and Engelund-Hansen) do not have a threshold for transport. In these cases, a different form of a hiding and exposure correction is utilized as

2)	$\begin{array}{l}\displaystyle \eta _{k}=\frac{q_{k}^{}}{q^{}}\end{array}$

where

$\begin{array}{l}q_{k}^*\end{array}$ = corrected sediment transport potential for hiding and exposure

$\begin{array}{l}q^*\end{array}$ = uncorrected or original sediment transport potential

Examples of the above formulation are Day (1980) and Proffitt and Sutherland (1983).

It is possible to utilize hiding and exposure correction factors for both the incipient motion and transport potential utilizing the following relationship

3)	$\begin{array}{l}\displaystyle \eta _{k}\approx \frac{1}{\xi _{k}^{a}}\end{array}$

where a is an empirical coefficient (close to one) which depends on the hiding and exposure function. For example, Wu and Lin (2011) extended the Lund-CIRP (Camenen and Larson 2007) sediment transport formula to multiple grain sizes in this way and found a ≈ 0.6 when applying the Wu et al. (2000) hiding and exposure correction formula developed for the Shields parameter.

Ashida and Michiue

The Ashida and Michiue (1971) formula is given by

4)	$\begin{array}{l}\xi _{k}=\begin{cases} \left[\frac{\log _{10}(19)}{\log _{10}(19d_{k}/d_{m})}\right]^{2}\,\,\,\mathrm{for}\,\,d_{k}/d_{m}\geq 0.4\\ d_{m}/d_{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,d_{k}/d_{m}< 0.4 \end{cases}\right.\end{array}$

where d_m is mean particle diameter. The Ashida and Michiue (1971) formula is basically a slightly modified version of the Egiazaroff (1965) formula described below for ratios d_k/d_m < 0.4.

Day

The hiding and exposure correction factor of Day (1980) is given by

5)	$\begin{array}{l}\displaystyle \eta _{k}=\frac{1}{0.4(d_{k}/d_{A})^{-0.5}+0.6}\end{array}$

where d_A is a reference diameter determined by

6)	$\begin{array}{l}\displaystyle d_{A}=1.6d_{50}\left(\frac{d_{84}}{d_{16}}\right)^{-0.28}\end{array}$

in which d₁₆ and d₈₄ are the 16^th and 84^th percentile diameters. The Day (1980) formula was developed specifically for the Ackers and White (1973) transport potential formula.

Egiazaroff

The Egiazaroff (1965) formula is given by

7)	$\begin{array}{l}\displaystyle \xi _{k}=\left[\frac{\log _{10}(19)}{\log _{10}(19d_{k}/d_{m})}\right]^{2}\end{array}$

where d_m is the arithmetic mean particle diameter. Egiazaroff (1965) also assumed a critical Shields parameter of θ_cr = 0.06, which is relatively high for most sediments.

Hayashi et al.

The Hayashi et al. (1980) formula is given by

8)	$\begin{array}{l}\xi _{k}=\begin{cases} \left[\frac{\log _{10}(8)}{\log _{10}(8d_{k}/d_{m})}\right]^{2}\,\,\,\mathrm{for}\,\,d_{k}/d_{m}\geq 1\\ d_{m}/d_{k}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,d_{k}/d_{m}< 1 \end{cases}\right.\end{array}$

where d_m is the arithmetic mean particle diameter. The Hayashi et al. (1980) formula is a recalibrated version of the Ashida and Michiue (1971) which itself is based on the Egiazaroff (1965) formula.

Parker et al.

The hiding and exposure formula by Parker et al. (1982) and others has the form

9)	$\begin{array}{l}\displaystyle \xi _{k}=\left(\frac{d_{k}}{d_{50}}\right)^{-m}\end{array}$

where m is an empirical coefficient between 0.5 to 1.0.

Proffitt and Sutherland

The hiding and exposure correction factor of Proffitt and Sutherland (1983) is given by

10)

$\begin{array}{l}\eta _{k}=\begin{cases}0.4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,d_{k}/d_{u}\leq 0.075\\ 0.53\log _{10}(d_{k}/d_{u})+1\,\,\,\mathrm{for}\,\,0.075< d_{k}/d_{u}\leq 3.7\\ 1.3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,d_{k}/d_{u}>3.7 \end{cases}\right.\end{array}$

where the reference diameter d_u is given in graphical form by Proffitt and Sutherland and may be approximated by the curves (HEC 2008)

11)

$\begin{array}{l}\frac{d_{u}}{d_{k}}=\begin{cases}1.1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\theta \leq 0.04\\ 2.3-30\theta \,\,\,\mathrm{for}\,\,0.04< \theta \leq 0.045\\ 1.4-10\theta \,\,\,\mathrm{for}\,\,0.045< \theta \leq 0.095\\ 0.45\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \end{cases}\right.\end{array}$

The Proffitt and Sutherland (1983) formula was developed specifically for the Ackers and White (1973) transport potential formula.

Wilcock and Crowe

Wilcock (2001) and Wilcock and Crowe (2003) developed a surface-based bed-load transport equation for graded beds with sand and gravel. In their formulation the hiding and exposure correction is calculated as

12)	$\begin{array}{l}\displaystyle \xi _{k}=\frac{\tau _{r,k}}{\tau _{r,sm}}=\left(\frac{d_{k}}{d_{sm}}\right)^{{b_{k}}}\end{array}$

where

$\begin{array}{l}τ_{r, k}\end{array}$ = reference shear stress corresponding to d_k

$\begin{array}{l}τ_{r,m}\end{array}$ = reference shear stress corresponding to d_m

$\begin{array}{l}d_k\end{array}$ = grain size diameter [L]

$\begin{array}{l}d_{sm}\end{array}$ = surface mean grain size diameter [L]

The empirical exponent b_k is given by

13)	$\begin{array}{l}\displaystyle b_{k}=\frac{0.67}{1+\exp \left(1.5+\frac{d_{k}}{d_{sm}}\right)}\end{array}$

Wu et al.

The hiding and exposure correction for each sediment size class is based on Wu et al. (2000):

14)	$\begin{array}{l}\displaystyle \xi _{k}=\left(\frac{P_{ek}}{P_{hk}}\right)^{-m}\end{array}$

where

$\begin{array}{l}m\end{array}$ = empirical coefficient that varies for each transport formula (approximately between 0.6-1.0) [-]

$\begin{array}{l}P_{hk}\end{array}$ = hiding probability [-]

$\begin{array}{l}P_{ek}\end{array}$ = exposure probability [-]

The total hiding and exposure probabilities (P_hk and P_ek, respectively) are calculated as

15)	$\begin{array}{l}\displaystyle P_{hk}=\sum _{j=1}\hat{f}_{1j}\frac{d_{j}}{d_{k}+d_{j}}\end{array}$

16)	$\begin{array}{l}\displaystyle P_{ek}=\sum _{j=1}\hat{f}_{1j}\frac{d_{k}}{d_{k}+d_{j}}\end{array}$

where f̂_1j are the active layer fractions by volume. Therefore, the formulation takes into account the size composition of the bed material by using the probabilities of each grain class of being exposed or hidden by other grain classes in the bed. The Wu et al. (2000) was developed in conjuction with the bed- and suspended-load transport potential formulas developed in the same reference.

Download PDF

Hiding and Exposure Corrections

Ashida and Michiue

Day

Egiazaroff

Hayashi et al.

Parker et al.

Proffitt and Sutherland

Wilcock and Crowe

Wu et al.