A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis

To simulate debris-flow behaviour from initiation to deposition, we derive a depth-averaged, two-phase model that combines concepts of critical-state soil mechanics, grain-flow mechanics and fluid mechanics. The model's balance equations describe coupled evolution of the solid volume fraction, m, basal pore-fluid pressure, flow thickness and two components of flow velocity. Basal friction is evaluated using a generalized Coulomb rule, and fluid motion is evaluated in a frame of reference that translates with the velocity of the granular phase, vs. Source terms in each of the depth-averaged balance equations account for the influence of the granular dilation rate, defined as the depth integral of vs. Calculation of the dilation rate involves the effects of an elastic compressibility and an inelastic dilatancy angle proportional to mmeq, where meq is the value of m in equilibrium with the ambient stress state and flow rate. Normalization of the model equations shows that predicted debris-flow behaviour depends principally on the initial value of mmeq and on the ratio of two fundamental timescales. One of these timescales governs downslope debris-flow motion, and the other governs pore-pressure relaxation that modifies Coulomb friction and regulates evolution of m. A companion paper presents a suite of model predictions and tests.