A differential model for growing sandpiles on networks. In this paper the authors 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, recently introduced by P.-L. Lions and P. Souganidis [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 27, No. 4, 535–545 (2016; Zbl 1353.35111)], they prove the existence and uniqueness of the solution of the system under consideration. In the second part of the paper the authors consider a finite difference approximation scheme to compute the solution. Moreover, in the last section of the work, they also present some numerical experiments, carried on using SPNET (Sand Piles on NETworks), an easy-to-use program written in C they developed for the numerical approximation of the sandpile problem.