We consider a system of differential equations of Monge--Kantorovich type which describes the equilibrium configurations of granular material poured by a constant source on a network. Relying on the definition of viscosity solution for Hamilton--Jacobi equations on networks introduced in [P.-L. Lions and P. E. Souganidis, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), pp. 535--545], we prove existence and uniqueness of the solution of the system and we discuss its numerical approximation. Some numerical experiments are carried out.