A geometry-independent effective model for the contact self-energies is proposed to calculate the quantum conductance of patterned graphene devices using Green's functions. A Corbino disk, being the simplest device where the contacts can not be modeled as semi-infinite ribbons, is chosen to illustrate this approach. This system's symmetry allows an analytical solution against which numerical calculations on the lattice can be benchmarked. The effective model perfectly describes the conductance of Corbino disks at low-to-moderate energies, and is robust against the size of the annular device region, the number of atoms on the edge, external magnetic fields, or electronic disorder. The contact model considered here affords an expedite, flexible, and geometry-agnostic approach easily allows the consideration of device dimensions encompassing several million atoms, and realistic radial dimensions of a few hundreds of nanometers.