Meyer-Peter and Müller

The Meyer-Peter and Müller (MPM) formula (1948) is a bed-load formula developed from flume experiments of sand and gravel under plane bed conditions. It was originally developed for uniform sediment beds. HEC-RAS uses the version of MPM from Vanoni (1975), ASCE Manual 54, the version used in HEC 6. This version includes a form drag correction (the RKR parameter, based on the roughness element ratio, (k_b/k_r)^3/2, computed from the Darcy-Weisbach bed fiction factor). The form drag correction isolates grain shear, computing transport based on the bed shear component acting only on the particles. The form drag correction should be unnecessary in plane-bed conditions, so some versions of MPM exclude it. Wong and Parker (2006) demonstrate that using MPM without the form drag correction over-predicts bed load transport.

1)	$\begin{array}{l}\displaystyle \frac{q_{bk}^{*}}{\sqrt{R_{k}gd_{k}^{3}}}=A_{M}\rho _{sk}\left(\theta '_{b}-\theta _{crk}\right)^{{E_{M}}}\end{array}$

where

$\begin{array}{l}\theta '_{b}=\frac{\tau '_{b}}{(\rho _{sk}-\rho _{w})gd_{k}}\end{array}$ = grain-related Shields parameter [-]

$\begin{array}{l}ρ_{sk}\end{array}$ = sediment density [M/L³]

$\begin{array}{l}ρ_{w}\end{array}$ = water density [M/L³]

$\begin{array}{l}θ_{crk}\end{array}$ = critical Shields parameter [-]

$\begin{array}{l}A_{M}\end{array}$ = empirical coefficient [-]

Meyer-Peter Müller (1948) estimated A_M = 8, E_M = 3/2, and θ_crk = 0.047. However, Wong and Parker (2006) recalibrated the equation and found A_M = 3.97, E_M = 1.6, and θ_crk=0.0495. The MPM formula is most applicable to uniform gravel bed and tends to under-predict transport for fine sands and silts.