The Meyer-Peter and Müller (MPM) equation (1948) was one of the earliest equations developed and is still one of the most widely used. It is a simple excess shear relationship. Parker (2006)4 casts MPM in its simplest, dimensionless, volumetric form as.

q_b^* = 8 \left( \tau^* - \tau^*_c \right) ^{3/2} \space , \space \tau^*_c = 0.047

Where qb* and τ* are dimensionless transport and mobility parameters respectively, where:

\tau^* = \frac{\tau}{\left(\gamma_s - \gamma \right) d_m} \space and \space q_b^* = \frac{q_b}{\sqrt{Rgd_m}d_m}

MPM is strictly a bedload equation developed from flume experiments of sand and gravel under plane bed conditions. The MPM experiments mostly examined uniform gravel, making the transport function MPM most applicable in gravel systems. MPM tends to under predict transport of finer materials.

HEC-RAS uses the version of MPM from Vanoni (1975), ASCE Manual 54, the version used in HEC 6.

\left( \frac{k_r}{k^'_r} \right) ^{\frac32} \gamma RS = 0.047 \left( \gamma_s - \gamma \right) d_m + \left( \frac {\gamma}{g} \right) ^{\frac13} \left( \frac {\gamma_s - \gamma}{\gamma_s} \right) ^{\frac23} g_s^{\frac23}

where:
gs = Unit sediment transport rate in weight/time/unit width
kr = A roughness coefficient
kr' = A roughness coefficient based on the grains
γ = Unit weight of water
γs= Unit weight of sediment
g = Acceleration of gravity
dm = Median particle diameter
R = Hydraulic radius
S = Energy gradient

Solving for transport, this equation starts to take the familiar dimensionless, form:

with an additional term in the excess shear equation. The full derivation of Parker's simple dimensionless, volumetric form from the Vanoni version used in HEC-RAS is included in the transport appendix. This version includes a form drag correction, (kr/kr')1.5, based on the roughness element ratio computed from the Darcy –Weisbach bed fiction factor. This form drag partitions bed shear stress, isolating grain shear. By imbedding the form shear correction, this version of the equation computes 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.
Therefore, HEC-RAS offers the Wong Parker correction to MPM based on their 2006 paper. The Wong Parker correction changes MPM in two ways. First, it sets the form drag correction to unity (kr/kr'=1), effectively removing it from the equation. Second, it sets the MPM coefficients to those Wong and Parker (2006) computed using the plane-bed data sets from the original MPM analysis recasting:

q_b^* = 8 \left(\tau^* - \tau^*_c \right) ^{3/2} \space , \space \tau^*_c = 0.047 \vspace{1} \space \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space As \vspace{1} q_b^* = 3.97 \left(\tau^* - \tau^*_c \right) ^{3/2} \space , \space \tau^*_c = 0.0495

Where: q*b is the Einstein bedload number (correlated with bedload), τ* is the Shield's stress which is compared to, τ*c which is the 'critical' Shields stress.

The effects of these changes can push transport higher or lower than MPM based on the magnitude of the form drag correction. Removing the form drag correction can increase transport (if it was computing a partition) and changing the coefficients decreases transport.

Wong and Parker (2006) based their work on the plane-bed data sets MPM analyzed, those without appreciable bed forms. Therefore, their correction is directly applicable only to lower-regime plane-bed conditions.

Modeling Note:

Dr. Gary Parker's ebook is an excellent place to start for anyone new to sediment transport theory.

4Parker (2006) 1D Sediment Transport Morhodynamics with applications to River and Turbidity Currents, e-book.