Soil Erosion:Sediment in Lake

When lakes are included in a GSSHA model along with sediment transport the lake interacts with the overland sediment transport and the channel sediment transport. Within the lake sediments settle and may exit the lake as overland or channel flow.

10.3.1 Lake and overland sediment interaction

When overland flow encounters an overflow cell that is currently part of lake, the fate of the sediments depends on its classification as either larger than sand or smaller than sand. Sand size and larger particles are deposited immediately into the overland cell on the edge of the lake within the lake. This sediment is added to the deposited fraction of sediments in that overland flow cell. For sediments smaller than sand, the sediments are added to the amount of sediments considered suspended in the lake. The fate of these sediments will be discussed in Section 10.3.3. When a lake overtakes a overland flow cell the sediments in that overland flow cell have the same fate as sediments flowing into the reservoir.

10.3.2 Lake and channel sediment interaction

Sediments in the channel network that encounter a lake have a similar fate. In this case the bedload, sand size and larger particles, are assumed to be deposited in the channel within the lake. All sand size and larger particles are immediately removed from the channel cell. The amount of bed load deposited in the reservoir is accounted for but the volume of the reservoir is not reduced in future calculations. Fines in channel flow entering a lake are added to the suspended lake sediments, discussed in the following section. Sediments in channel cells overtaken by the lake have the same fate as those entering the lake by flow.

10.3.3 Fate of suspended sediments in lakes

Within the lake the lake volume is treated as a completely mixed reactor with a single value of suspended sediments throughout the lake. As additional sediment come into the lake they are assumed to instantly mix throughout the lake, changing the single value of concentration in the lake. The lake is assumed to be quiscent, such that particles in suspension can settle. For each class of particle size, the fall velocity, also referred as terminal velocity or settling velocity, is calculated as described in the following section. Provided the settled volume does not exceed the volume of suspended volume of sediments for a particular size class the total settled volume of each particle size (k) of suspended sediments is defined as:

ss_vol_lake_settled[k] = ss_conc_lake[k] * fall_vel[k] * dt* lake_area (113)

where: k is the size class of suspended particles, ss_vol_lake_settled[k] is the settled volume of size class k, ss_conc_lake[k] is the concentration of suspended sediments of size class k, fall_vel[k] is the terminal velocity of the k size class, dt is the routing time step, and lake_area is the current time lake wetted area. The settled volume is uniformly distributed over all overland cells currently with the lake. The concentration of sediments in the lake is calculated as the total amount of sediments in the lake divided by the lake volume.

ss_conc_lake[k] = ss_vol_lake[k] / lake_vol (114)

The settled particles are subtracted from the total in the lake.

In addition to settling, particles may flow back onto the overland flow and into the channels. In either case, the particles are added to the overland flow cell or channel cell and subtracted from the lake total. The concentration of particles in the lake accounts for all sources and sinks and changes in lake volume.

10.3.4 Terminal velocity derivation

10.3.4.1 Terminal velocity in the presence of buoyancy force:

When the buoyancy effects are taken into account, an object falling through a fluid under its own weight can reach a terminal velocity (settling velocity) if the net force acting on the object becomes zero. When the terminal velocity is reached the weight of the object is exactly balanced by the upward buoyancy force and drag force. That is

W=F_b+D (115)

where: W = weight of the object, F_b = buoyancy force acting on the object, and D = drag force acting on the object. If the falling object is spherical in shape, the expression for the three forces is given below:

W = π/6d³*ρ_sg (116)

F_b = π/6d³ρg (117)

D = C_d/2ρV²A (118)

where: d = diameter of the spherical object g = gravitational acceleration, ρ = density of the fluid, ρ_s = density of the object, A = πd²/4 = projected area of the sphere, C_d= drag coefficient, and V = characteristic velocity (taken as terminal velocity, V_t). Substitution of Equations 116–118 into Equation 115 and solving for terminal velocity, V_t to yields the following expression:

V_t = sqrt(4gd/3/C_d(ρ_s-ρ)/ρ) (119)

10.3.4.2 Terminal velocity in creeping flow

Fig. Creeping flow past a sphere: stream lines, drag force F_d and force by gravity F_g.

For very slow motion of fluid, the inertia forces of the fluid are negligible (assumption of massless fluid) in comparison to other forces. Such flows are called creeping flows, determined from the Reynolds Number (Re). The equation of motion for creeping flow (simplified Navier-Stokes equation) is given by:

▼p = μ▼²v (120)

where: v= velocity vector field p = pressure field μ = fluid viscosity The analytical solution for creeping flow around a sphere was first given by Stokes in 1851. From Stokes' solution, the drag force acting on the sphere can be obtained as

D = 3πμdV or C_d = 24/Re (121)

where the Reynold's number, Re = ρdv/μ. The expression for the drag force given by Equation 121 is called Stokes law. When the value of C_d is substituted in the Equation 119, we obtain the expression for terminal velocity of a spherical object moving under creeping flow conditions:

V_t = gd²/18/μ(ρ_s-ρ) (122)

Equation 122 is used in GSSHA to calculate the fall velocity of the sediment in lakes.