We use Monte Carlo simulations to obtain thermodynamic functions and correlation functions in a lattice model we propose for sponge phases. We demonstrate that the surface-density correlation function dominates the scattering only along the symmetric-sponge (SS) to asymmetric-sponge (AS) phase boundary but not the boundary between the sponge-with-free-edges (SFE) and symmetric-sponge phases. At this second thermodynamic transition the scattering is dominated instead by an edge-density (or seam-density) correlation function. This prediction provides an unambiguous diagnostic for experiments in search of the SS-SFE transition.