Other Underflow Model Considerations

ERDC TN-DOER-N7

August 2000

The transition between low- and high-concentration is less distinct than the laminar-turbulent

transition, and depends on the grain-size and cohesive characteristics of the material. A break can

occur at about 160-220 g/L where yield stress and viscosity increase sharply with increased

concentration (see Teeter 1992a). High-concentration underflow solids are supported by grain

interactions or cohesion (Lowe 1982). Individual grains become frozen in a suspension when the

shear stress on particles is less than that which permanently deforms the visco-elastic suspension.

Van Kessel and Kranenburg (1997) found a concentration breakpoint at about 200 g/L for the

underflow behavior of their kaolinite experiments, with laminar flow occurring at higher concen-

trations.

The bed shear stress τb can be derived from the steady, quasi-uniform flow by means of a momentum

equation for the direction of flow and assuming a linear vertical shear-stress distribution (Van Kessel

and Kranenburg 1997):

τb = ρ - ρw gh cos θ tan θ -

(3)

where

ρw = the water density

g = the acceleration of gravity

θ = the bed slope

x = the direction of flow

It appears that both the bed slope and h/ x terms can be important in field situations. The local

mud-flow rate q can be derived by integrating the velocity profile over both the plug and shear-flow

regions of the laminar profile yielding:

τ bh2 2

ξ 1- ξ 3

(4)

where ξ = 1 - τ y τb .

Other Underflow Model Considerations. Pipeline discharges generally last hours or days at

a location, adding another dimension to the problem of predicting underflow spreading extent. As

noted earlier, the final deposit builds by deposition from the underflow. Thus, bed slope θ can

change appreciably during the disposal operation.

Even after underflow deposits become dense enough to resist entrainment by the overlying water

column, they can still be eroded particle-by-particle under the action of waves and currents. A

discussion of various erosion modes is given in Teeter (1992b). The erodibility of channel mud

after being slurried and allowed to settle has been observed in the laboratory to increase by a factor

of 4 or more over that of the original channel material (see Chou et al. 1998; Johnson et al. 1999).