U. S. Department of the Interior U. S. Geological Survey a Model for Grain Flow and Debris Flow

________________________________________________________5 INTRODUCTION_____________________________________________________5 RHEOLOGICAL MODELS________________________________________________9 Bagnold's Original Models for Grain Flows_______________________________ 9 Takahashi and Chen Models for Debris Flow and Debris Avalanche _______________ 10 McTigue, Savage, Goodman and Cowin Models for Dry Grain Flows _______________ 11 A Model for Macro-viscous and Inertial Flow _____________________________ 12 Macro Viscous Flow _________________________________________________12 Coupling to Migration of Interstitial Fluid ______________________________________12 Combined Macro Viscous and Inertial Flow ___________________________________14 EXPERIMENTS WITH GRAIN-FLOW AND DEBRIS-FLOW MATERIALS__________________16 Experimental Materials__________________________________________ 16 Properties of Sand-Water (Grain-Flow) Mixtures __________________________ 16 Behavior of Deforming Mixtures __________________________________________16 Sedimentation in Dispersion Gap __________________________________________17 Matching the Flow Curves ______________________________________________19 Comparison with Bagnold's Constants________________________________________19 Constants A2 and a _________________________________________________20 McTigue-Bagnold Numbers____________________________________________20 Properties of Sand-slurry Mixtures (Simple Debris)_________________________ 23 Reduction of Dispersion Gap ______________________ _____________________23 Bingham Viscosity and Strength of Simple Debris________________________________24 Inertial Viscosity of Simple Debris_________________________________________25 Transition Velocity Gradients ____________________________________________26 VELOCITY PROFILES FOR CHANNELIZED SAND-WATER GRAIN FLOWS AND SLURRY-SAND DEBRIS FLOWS _____________________________________________________26 Examples of Velocity Profiles _______________________________________ 26 Profiles According to Theoretical Analysis _______________________________ 26 Governing Equations__________________________________________________26 McTigue Solution __________________________________________________29 Undrained Flow_____________________________________________________29 Completely Drained Flow ______________________________________________32 CONCL USIONS ____________________________________________________38 A CKNO WLEDGMENTS _______________________________________________39 REFERENCES CITED 40