In this article, we consider a nonlinear partial differential system describing two-phase transports and try to recover the source term and the nonlinear diffusion term when the state variable is known at different profile times. To this end, we use a POD-Galerkin procedure in which the proper orthogonal decomposition technique is applied to the ensemble of solutions to derive empirical eigenfunctions. These empirical eigenfunctions are then used as basis functions within a Galerkin method to transform the partial differential equation into a set of ordinary differential equations. Finally, the validation of the used method has been evaluated by some numerical examples.