In this work, we use the notion of modularity to define a new model of generation of random SAT instances with community structure, called Community Attachment. For high values of modularity (i.e., clear community structure), we realistically model pseudo-industrial random SAT formulas. This model also generates SAT instances very similar to classical random formulas using a low value of modularity. We also prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial SAT problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure. Finally, we use this generator to observe the connections between the modularity of the instance and some components of the solver, such as the variable branching heuristics or the clause learning mechanism.