Some of the recent physically-based algorithms for sediment transport in shallow waters.

Table 3: Some of the recent physically-based algorithms for sediment transport in shallow waters.

Source	Algorithm	Parameters
[29]	$\frac{\partial (q c_{i})}{\partial x} + \frac{\partial (c_{i} D)}{\partial t} = e_{i} - r_{i} - d_{i}$	D = Water depth, c_i = The suspended sediment concentration of class size i in the overland ﬂow, q = The volumetric water ﬂux per unit width of slope, e_i = The rates of ejection of original soil, r_i = Rate of re-ejection of deposited material, d_i = Rate of deposition.
[117]	$\frac{\partial (q c_{i})}{\partial x} + \frac{\partial (c_{i} D)}{\partial t} = r_{i} + r_{r i} + r_{g i} + d_{i}$	q = Unit width flow, D = Depth of flow, c_i = Mass of sediment per unit volume of solution, r_i = Rate of entrainment, r_ri = Rate of re-entrainment, r_gi = Gravity process, d_i = Rate of deposition per unit area,
[144] Eulerian sediment transport model	$\frac{\partial (H C)}{\partial t} + \frac{\partial (H U C)}{\partial x} + \frac{\partial (H v C)}{\partial y} - \frac{\partial}{\partial x} (D \frac{\partial H C)}{\partial x}) - \frac{\partial}{\partial y} (D \frac{\partial H C)}{\partial y})$ $= - γ H C + (U, V) ._{γ s}$	C(x, y, t) is depth averaged concentration, γ is the deposition coefficient, ε(U,V) = (U² + V²)(m²s^-2) is a function of flow velocities and the term λ_s•ε(U,V) models erosion of sediment particles. The particle pick up function is parameterized as λ_s•ε(U,V), where, λ_s is the erosion coefficient, it can be related to sediment properties.
[84]	$\begin{array}{l} \frac{\partial Q_{s}}{\partial x} + \frac{\partial (C h)}{\partial t} = D_{r} + F_{r} \end{array}$	Q_s = The sediment load (kg m^-1s^-1), h = The flow depth (m), C = The sediment concentration in flow (kg m^-3), D_r and F_r are, respectively, the rainfall detachment rate and overland flow entrainment rate (kg m^-2s^-1), x is the distance down slope (m) and t is the time (s).
[55]	$\frac{\partial C}{\partial t} + u \frac{\partial C}{\partial x} = E C_{*} + G C_{0} \exp (- η h) - (Y + E) C$	C = The sediment transport rate or sediment discharge, C_* = Sediment transport capacity of surface runoff, u = The velocity of the flow, C₀ = The sediment discharge corresponding tso c₀, c₀ = The maximum sediment concentration generated by the raindrop impact, E and Y are coefficients, η = The damping rate of the water depth (h).
[39]	$\frac{\partial (\frac{h q_{s, s}}{q})}{\partial t} + \frac{\partial q_{s, s}}{\partial x} = D i s \frac{\partial^{2} (\frac{h q_{s, s}}{q})}{\partial x^{2}} - λ_{s} q_{s, s}$	q_s,s = The sediment loading rate of fraction s per unit width (gs^-1m^-1), Dis = Dispersion coefficient (m²s), λ_s = The trapping efficiency for fraction s per unit length (m^-1).
[120]	$q \frac{\partial c_{i}}{\partial x} + D \frac{\partial c_{i}}{\partial t} = r_{i} + r_{r i} - d_{i} = r_{i} - \frac{\partial M_{d i}}{\partial t}$	Q = Unit flux of water (m²/s), c_i = Sediment concentration of class i sediment (kg/m³), D = Water depth, r_i = Rate of entrainment of class i sediment from the soil matrix (kg/m²/s), r_ri = Rate of re-entrainment of class _i sediment from sediment in the deposited layer (kg/m²/s), d_i = Rate of deposition, M_di = Rate the mass per unit area.
[145]	$\frac{d q_{s}}{d x} = D_{r} + D_{i}$	dq_s/dx = The sediment rate per unit width of rill channel, D_r = Rate of rill and interill net detachment, D_i = Rill and interill net deposition rate.