Manifestations of the so-called structure induced nonlinearity are considered for the case of a granular medium, the latter being a generally accepted model of nonconsolidated rocks in seismics. The consideration is carried out using the medium model in the form of the 'ideal' random packing of spherical elastic granules in which the interparticle space can be filled with a fluid. A physical equation of such a medium is derived; the dependencies of nonlinear parameters on the grain material elastic moduli, the fluid compressibility and the initial medium strain are analyzed. The influence of defects in nonideal grain packings (that is, the presence of a fraction of unloaded intergranular contacts) upon the nonlinear properties of the medium is investigated. It is shown that the packing nonideality has the stronger effect on higher-order nonlinear properties. It is demonstrated that the nonlinear parameters may be used in exploration seismology as a much more sensitive and informative characteristic compared with conventionally used linear moduli.