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

Abstract

To simulate debris-flow behaviour from initiation to deposition, we derive a depth-averaged, twophase 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, v s . 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 ∇ · v s . Calculation of the dilation rate involves the effects of an elastic compressibility and an inelastic dilatancy angle proportional to m − m eq , where m eq 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 m − m eq 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.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

“…The equations derived in our companion paper [1] are also distinct from those of previous debris-flow models in other regards. The role of pore-fluid pressure in our model is unlike that in the models mentioned above, because it is one of the evolving dependent variables.…”

“…We use bed-normal coordinates, where x and y are local orthogonal directions tangential to the bed, and z is the bed-normal direction in which quantities are depth averaged (see fig. 6 in [1]). We adopt conventional shallow-flow assumptions and thereby assume that the depth of the flow is small relative to the tangential length scales of the flow (e.g.…”

“…7 in Iverson & George [1]), u and v are the depth-averaged flow velocities in the x and y directions, respectively, m is the depth-averaged solid volume fraction and p b denotes the pore-fluid pressure at the basal boundary. The equations can be written as…”

“…In a companion paper, we derive a new depth-averaged model aimed at seamlessly simulating debris-flow motion from initiation to deposition [1]. Here, we describe mathematical properties of the model, our numerical solution technique, and tests of numerical predictions against experimental data.…”

“…The role of pore-fluid pressure in our model is unlike that in the models mentioned above, because it is one of the evolving dependent variables. In our model evolving granular dilatancy relates the evolution of excess pressure and solid volume fractions, and closure is provided by a Darcy drag law for solid-fluid interactions [1].…”