Acoustic metamaterials can be designed by inserting along the path of a sound wave periodically spaced side resonators. An example of efficient design was recently proposed consisting of a perforated stacking of flat annular cavities (the pancake resonator), the perforation allowing the propagation of sound waves. The pancake resonator is used in absorber mode and the theoretical description of sound absorption can be achieved with the help of the theory of sound propagation in fluid saturated porous media in which two porosities are considered: the main porosity associated with the perforation and a porosity associated with the flat cavity volumes. Considering a perforation diameter and flat cavity thickness ranging from submillimetric values to a few millimeters allows a wide range of material permeabilities and permeability contrasts between main pore and stacking of cavities. The relatively small values of diameter and cavity thickness also results in the existence of viscous and thermal boundary layers in the main pore (the perforation) and in the flat cavities. This metamaterial makes simultaneous use of viscothermal losses and periodicity in order to achieve low frequency sound absorption for an overall small absorber thickness. Experimental results are also presented for the validation of the model.