Stress impact on elastic anisotropy of triclinic porous and fractured rocks

Abstract

Understanding the stress dependence of elastic properties of rocks is important for reservoir characterization and seismic‐hazard monitoring. Several known approaches describing this dependence are the following: the nonlinear elasticity theory, effective‐medium theories for fractured rocks with stress‐dependent crack densities, and the piezosensitivity approach (also called the porosity deformation approach). Here I propose a generalization of the piezosensitivity approach to triclinic rocks. I assume the isotropy of the tensor describing sensitivity of elasticity to small strains of the pore space, and generalize known linear and exponential stress dependencies of compliances. This generalization is capable of describing the effect of loads on elastic properties of anisotropic rocks when the principal stresses are not necessarily aligned with the symmetrical axes of the unstressed anisotropic material. For example, the generalization describes how monoclinic anisotropy changes under isostatic stress or pore pressure, and how tilted transverse isotropy changes to monoclinic anisotropy due to a pseudo‐triaxial (or a uniaxial) load. The results are expected to be valid up to several hundred megapascals. This theory is closely related to the two other approaches mentioned above. On the one hand, for unloaded rocks, the theory is consistent with the noninteracting scalar‐crack approximation. On the other hand, the theory's predictions of mutual relations between isotropic third‐order elastic moduli is in good agreement with literature data on corresponding laboratory measurements. Thus, using the piezosensitivity approach, the physical nonlinearity of rocks can quantitatively be rather well explained by the strain of compliant pores.

Rock elasticity as a function of the uniaxial stress: laboratory measurements and theoretical modelling of vertical transversely isotropic and orthorhombic shales

“…On the other hand, the linear tensor in the porosity deformation approach (PDA) approach is anisotropic. Moreover, Shapiro and Kaselow () use an anisotropic piezo‐sensitivity tensor (resulting also in anisotropic non‐linear tensor), under the restrictions of stress application perpendicular to the symmetry planes and the elastic symmetry being orthorhombic or higher, whereas Shapiro () considered the linear anisotropy of the medium increased up to a triclinic (arbitrary anisotropy and no restrictions regarding the direction of the stress application). In this case, the isotropic piezo‐sensitivity tensor seems to be sufficient for the description of the experimental results.…”

“…In the case of an orthorhombic medium, which is uniaxially loaded along the symmetry axis x 1 (Fig. ), we use the following equation set derived from equation (79) of Shapiro (): …”

“…The parameters B ik and F c indicate the coefficients of the exponential function of the stress and describe the effect of deformation of the compliant pore space (for further details, see Shapiro and Kaselow ; Mayr et al . ; Shapiro . In the case of uniaxial loading along the symmetry axis, x 1 , the coefficients B 22 and B 33 are included (subsumed) in the fitting coefficient of the reference state parameter .…”

“…For the vertical transversely isotropic (VTI) medium under uniaxial loading along the symmetry axis x 3 (Fig. ), the equation set derived from equation (79) of Shapiro () has the following form: …”

“…The stress sensitivity tensor is assumed to be symmetric and independent of any spatial rotation and of any plane reflection (i.e. it is assumed to be isotropic), for more details see Shapiro (). This quantity is defined by equation : where is the bulk compressibility of the Swiss cheese model and corresponds to …”