SEPARATION OF GEOMETRICAL AND MECHANICAL FACTORS FOR A NON-LINEAR THEORY OF POROELASTIC MEDIA

In this work a non-linear theory of saturated poroelastic media has been developed. This theory generalizes the Biot approach. In non-linear media the problem of separation of geometrical factors which are defined by finite deformations of the skeleton and liquid, and mechanical factors which are defined by a non-linear dependency of the stress tensor on the medium deformations tensor, is often significant. Geometrical factors are also determined by the condition that the initial stress tensor is defined as a ratio between force and deformed area, and not between force and initially unstressed area. Geometrical non-linearities are accounted for through transition from Cauchy stress tensor to Piola-Kirchhoff tensor. An additional tensor which is a ratio between force and undeformed area is introduced to define physical non-linearities. This tensor is defined as a derivative of deformation potential. This procedure we have developed completely for a two-component “solid body – liquid” system. Dynamic equations for non-linear deformation of the skeleton a liquid have been derived. Non-linear effective stress tensor which plays the main role in the theory of poroelastic media theory has been developed. The link between the effective stress tensor and liquid pressure in a non-linear case has been established. This link is defined by Biot-Willis parameter which is generalized for a non-linear case and is dependant on initial stress in the skeleton and initial liquid pressure. Geometrical and mechanical non-linearities in Biot-Willis parameter have been separated. Through Biot-Willis parameter it is possible to determine higher pressure zones, compaction zones and zones with stressed rocks. The link between Biot-Willis parameter and tensors of non-linear elastic constants and elastic waves’ velocities in a pre-stressed medium has been also established in this work. This ratio generalizes the expression of this parameter containing linear elasticity moduli which was derived in the paper by Nur and Bayerlee. We have also considered the issue of separation of geometrical and physical factors in poroelasticity equations which are linearized in the locality of stressed condition.