Elastic moduli of a solid with spherical pores: New self-consistent method

Abstract

It is well known that the elastic moduli of most rocks are much lower than the moduli of their constituent minerals. Adams and Williamson [1] accounted for this by conjecturing the presence of voids which serve to increase the compliance of the rock; the extent to which a particular void lowers the elastic moduli depends primarily on its shape. Brace [2] showed that the compressibilities of low-porosity rocks decreased with pressure and then levelled off to values consistent with those of their mineral components, implying that the porosity existed in the form of thin cracks which closed up under pressure. King [3] observed that a preferred orientation of open cracks had a marked effect on seismic velocities, with the major reduction in velocity being perpendicular to the plane of open fractures. For a higher porosity rock such as a sandstone, however, the compressibility can be quite high compared to its major constituent (quartz) even at high stresses, reflecting the presence of nonclosable 'equi-dimensional' pores. Although the actual shapes of voids in rocks are usually quite irregular, two prototypical voids, the spherical pore and the elliptical crack, have each been the subject of extensive mathematical treatment with regard to their precise effect on elastic moduli. The mathematical problem of predicting the elastic moduli of a material permeated with voids of a particular shape is in fact a special case of the more general problem of predicting the elastic moduli of multicomponent media. If a valid method of predicting the effect of inclusions on elastic moduli were found it might be possible, for example, to distinguish between oilsaturated and water-saturated rocks on the basis of their elastic moduli (as derived from seismic data). Possibly the most promising use of such a theory in the long-term would be the inversion of seismic data in order to determine details of the pore structure of a rock, such as crack density and aspect-ratio distribution, which could then be used to predict other rock properties such as fluid permeability, thermal conductivity or electrical resistivity.