We are interested in the simulation of shallow dense avalanches in which the snow is considered as a compressible viscoplastic fluid. Such models can be derived by taking the shallow water asymptotics of a 3D Navier-Stokes-Bingham system with free surface. From the numerical analysis point of view, such equations imply the construction of new schemes which are able to compute accurately stationary states. We here present a well-balanced finite volume/augmented Lagrangian method which couples well-balanced finite volume scheme for the spatial discretization with the augmented Lagrangian method to treat the optimization problem stem- ming from the Bingham nature of the flow equations considered here, namely a system close to the shallow water equations but with an equation on the velocity which includes the yield limit (due to the Bingham stress tensor).