Debris flow equations ESP 2022 Humphrey

Shear stresses in a debris flow

We use a coordinate system where x is the flow direction (parallel to slope a) and z is in the up direction (perpendicular to slope). The equation for the shear stress at any height z from the base plane of the debris flow, (the stress is on the plane parallel to the surface of the flow, in an approx. uniform flow, in the x direction): Note that z is not vertical, but perpendicular to flow and that z is equal to 0 at the bed, z is equal to the thickness of the viscous layer (D_viscous ) at the plug/shearing layer boundary and z is equal to the depth of the flow ( D_total ) at the surface. The stress is given by:

where D is the total depth of flow and a is the slope angle. Note this form of the shear stress equation is universal for cases where you need to measure from the bed up, instead of the surface down.

If a flow, that was moving, has stopped, then the shear stress at the bed is approximately the critical shear stress:

The critical shear stress is also related to the thickness of the relatively non-shearing plug flow region:

Velocities in the flow

If the effective viscosity and the thickness of the viscous zone are known, the velocity at any level in the flow is given by:

….. valid from 0 < z < D_viscous

….. valid from D_viscous < z < D_total (note this is a constant velocity)

or, in terms of the total thickness and the critical shear stress:

…… likewise valid from 0 < z < D_viscous

The thickness of the viscous zone is given by:

The surface velocity, or the velocity of the plug is given by:

Note there are typically several unknown quantities: m, D_viscous, D_plug, velocity etc. However; you can often find a quantity from combining or rearranging the above equations. For example, the effective viscosity can be calculated if the other variables are known: