Porous materials are one of the most common classes of sound absorbers for acoustic treatments. However, thin layers of these classical materials are not efficient at absorbing low-frequency sound waves, which is a practical shortcoming. Low-frequency absorption can be improved by perforated screens in combination with an air gap or a classical porous absorber, since they increase the overall tortuosity of the combined system. A less investigated, but in principle similar alternative to achieve high-tortuosity absorbers is perforating initially closed-cell foams. Yet, at high sound pressure levels (SPL), non-linearities of the surface impedance arise due to flow-separation in the vicinity of the perforations. Therefore, it is necessary to adapt existing porous material models for SPL-dependency which is illustrated here for the case of micro-perforated mineral foams. The proposed investigations are carried on experimentally, a foam sample is tested for flow-resistivity as well as for sound absorption at normal-incidence using impedance tube measurements. We furthermore observe and predict the change of effective fluid properties and Johnson-Champoux-Allard (JCA) parameters with respect to SPL. The most significant of them is the increase of static air-flow resistivity, which drastically changes the equivalent density of the porous medium, and has a negative effect on the sound absorption. The proposed model accurately predicts the change in acoustic absorption of rigidly-backed perforated porous treatments.