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Mathematic Modelling of Non-Equilibrium Suspended Load Transport, Reservoir Sedimentation, and Fluvial Processes

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Advances in Water Resources Engineering

Part of the book series: Handbook of Environmental Engineering ((HEE,volume 14))

Abstract

This chapter consists of two parts: mechanism of non-equilibrium transport of non-uniform suspended load and its application to mathematical modelling. Based on a stochastic approach of sediment transport proposed by the authors, a 1D equation of non-equilibrium transport for each size group of non-uniform sediment is developed. The equations to predict the change of sediment concentration and the corresponding size distribution of suspended load and bed material are also derived. The concept that changes in size distribution are interrelated to sediment-carrying capacity is explored. These results reveal the essence of sediment transport of non-uniform sediment. In the second part, a mathematical model incorporating the mentioned equations to compute deposition and scouring in reservoirs as well as the fluvial processes of river channels has been developed. Verification of the model agrees well with field data.

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Abbreviations

\(1-{{\varepsilon }_{0\cdot l}}\) :

Probability of lth size group being stopped from motion

\({{\alpha }_{l}}\) :

Coefficient of saturation recovery

\({{\beta }_{l}}\) :

Probability of incipient suspension of lth size group

\({{\gamma }_{\text{s}}}\) :

Specific weight of sediment (kg/m3)

\({{{\gamma }'}_{\text{s}}}\) :

Specific weight of deposits or bed material (kg/m3)

ε y :

Diffusion coefficient of flow in vertical direction (m2/s)

\(\lambda \) :

Percentage of deposition (kg/m2s)

\({{\lambda }_{1\cdot 4\cdot l}}\) :

Exchange intensity of sediment from rest at bed surface to suspension of lth size group (kg/m2s)

\({{\lambda }_{4\cdot 1\cdot l}}\) :

Exchange intensity of sediment from suspension to rest at bed surface of lth size group (kg/m2s)

\({{\lambda }^{*}}\) :

Percentage of scouring of bed material

\(\omega \) :

Mean settling velocity of suspended load (m/s)

\({{\omega }_{l}}\) :

Settling velocity of lth size group (m/s)

\({{\omega }_{m}}\) :

Median value of settling velocity during deposition (m/s)

\({{\omega }^{*}}\) :

Mean settling velocity of sediment-carrying capacity (m/s)

\(\omega_{m}^{*}\) :

Median value of settling velocity during scouring (m/s)

\(\Delta a\) :

Scoured or silted area at \(\Delta x\) in \(\Delta t\) (m2)

\(\Delta h\) :

Depth of deposition during \(\Delta t\) (m)

\(\Delta S\) :

Concentration of sediment supplement from bed material (kg/m3)

\(\Delta {{S}_{h}}\) :

Sediment concentration corresponding to disturbed thickness of bed material taken part in scouring and sorted, but not scoured (kg/m3)

\(\Delta {{S}_{m}}\) :

Sediment concentration corresponding to the amount of disturbed bed material (kg/m3)

\(\Delta t\) :

Time increment(s)

\(\Delta {{t}_{i}}\) :

Time increment from instant \({{t}_{i-1}}\) to \({{t}_{i}}\) (s)

\(\Delta {{V}_{i.j}}\) :

Volume of deposit at interval \(\Delta {{x}_{j}}\) during \(\Delta {{t}_{i}}\) (m3)

\(\Delta {{x}_{{}}}\) :

Space interval along flow direction (m)

\(\Delta {{x}_{j}}\) :

Space interval along flow direction from \({{x}_{j-1}}\) to \({{x}_{j}}\) (m)

\(a\) :

Scoured or silted area (m2)

a(x, t):

Equation of total area of erosion or deposition in period t (m2)

\(A\) :

Cross-sectional area of flow (m2)

\(A(x,z,t )\) :

Equation of cross-sectional area (m2)

\(B\) :

Cross-sectional width of flow (m)

\(B(x,z,t )\) :

Equation of cross-sectional width (m)

\({{B}_{k}}\) :

Stable width of cross section (m)

\({{D}_{50}}\) :

Median size of deposit or bed material (mm)

\({{D}_{l}}\) :

Particle size of lth size group (mm)

\(g\) :

Gravitational acceleration (m/s2)

\(h\) :

Average depth of flow (m)

\(H\) :

Water level (m)

\({{h}_{0}}(x )\) :

Equation of the water surface (m)

\({{h}_{1}}(x )\) :

Equation of the bed surface (m)

\(i\) :

Subscript, which is the parameter that indicates mean value from instant \({{t}_{i-1}}\) to \({{t}_{i}}\)

\(j\) :

Subscript, which is the parameter indicates mean value from \({{x}_{j-1}}\) to \({{x}_{j}}\) or at position x j

\({{J}_{f}}\) :

Energy slope

K :

Coefficient of sediment-carrying capacity

\(L\) :

Distance from inlet section to outlet section (m)

\({{L}_{4\cdot l}}\) :

Mean step length of suspended particle (m)

n :

Manning’s coefficient of roughness (s/m1/3)

m :

Coefficient of sediment-carrying capacity

\({{m}_{l}}\) :

Total number of size groups of sediment

\({{P}_{1\cdot l}}\) :

Size distribution of bed material

\({{P}_{4\cdot l}}\) :

Size distribution of suspended load

\({{P}_{1\cdot l\cdot 0}}\) :

Size distribution of bed material at initial instant

\({{P}_{4\cdot l\cdot 0}}\) :

Size distribution of suspended load at inlet section

\(P_{4\cdot l}^{*}\) :

Size distribution of sediment-carrying capacity

\(P_{4\cdot l\cdot 0}^{*}\) :

Size distribution of sediment-carrying capacity at inlet section

\(\tilde{P}_{4\cdot l}^{*}\) :

Size distribution of sediment supplement

\(\tilde{P}_{4\cdot l\cdot 0}^{*}\) :

Size distribution of sediment supplement when \({{\lambda }^{*}}=1\)

\(q\) :

Flow discharge of unit width (m2/s)

Q :

Flow discharge (m3/s)

\({{q}_{4\cdot l}}\) :

Sediment discharge of unit width (kg/ms)

\(S\) :

Sediment concentration or sediment concentration at outlet section (kg/m3)

\({{S}_{0}}\) :

Sediment concentration at inlet section (kg/m3)

\({{S}_{l}}\) :

Sediment concentration of lth size group (kg/m3)

\({{S}^{*}}\) :

Sediment-carrying capacity (kg/m3)

\(S_{0}^{*}\) :

Sediment-carrying capacity at inlet section (kg/m3)

\(S_{l}^{*}\) :

Sediment-carrying capacity at outlet section (kg/m3)

\(\bar{S}\) :

Mean concentration along vertical direction of total suspended load (kg/m3)

\({{\bar{S}}_{l}}\) :

Mean concentration along vertical direction of lth size group (kg/m3)

\(S(x,y )\) :

Concentration of total suspended load at point (x, y) (kg/m3)

\({{S}_{l}}(x,y )\) :

Concentration of lth size group at point (x, y), kg/m3

\({{S}^{*}}(l )\) :

Sediment-carrying capacity of uniform sand with particle size, \({{D}_{l}}\) (kg/m3)

\({{t}_{4\cdot 0\cdot l}}\) :

Time period for incipient suspension of a rest particle, i.e. time period for a particle transformed from rest to suspension to get off the bed surface (s)

\(V\) :

Average velocity of flow (m/s)

\({{V}_{x}}{{,}_{{}}}{{V}_{y}}\) :

Mean flow velocity in x and y direction, respectively (m/s)

\(W\) :

Total weight of deposit (kg)

\({{W}_{l}}\) :

Deposited weight of lth size group (kg)

\({{W}_{m}}\) :

Amount of total bed material before scouring (kg)

\({{W}_{m\cdot l}}\) :

Amount of lth size group of bed material before scouring (kg)

\({{{W}'}_{{}}}\) :

Weight of deposits in the river channel (kg)

\({W}''\) :

Weight of deposits on the flood plain (kg)

z :

Elevation (m)

z k :

Elevation corresponding to the stable width of section (m)

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  1. Sediment Research Department, China Institute of Water Resources and Hydroelectric Power Research, 20, Chegongzhuang West Road, 100044, Beijing, China

    Qiwei Han & Mingmin He

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    Lawrence K. Wang

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Han, Q., He, M. (2015). Mathematic Modelling of Non-Equilibrium Suspended Load Transport, Reservoir Sedimentation, and Fluvial Processes. In: Yang, C., Wang, L. (eds) Advances in Water Resources Engineering. Handbook of Environmental Engineering, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-11023-3_4

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