Biaxial Anisotropy: Its Occurrence and Measurement With Multicomponent Induction Tools
Authors: Georgi Daniel Taylan , Schoen Jurgen Hans , Rabinovich Michael 
Published: September 2008

Abstract
For years petrophysicists interpreted logs as if all formations were locally homogenous and isotropic. They did this even though they well understood that sedimentary deposition occurs layer by layer, making transversely isotropic, TI, media the rule rather than the exception in sedimentary geology. With the widespread availability of multi-component induction logging technology petrophysicists are more willing to use TI resistivity models (i.e., Rh and Rv). Generally, sediments buried a few hundred meters or more are subject to a principal stress aligned with the overburden. In many environments, depending on mechanical properties, this leads to bedding perpendicular fractures, which can give rise to biaxial anisotropy in the macroscopic petrophysical properties.
We have developed resistivity models for laminated sand-sale formations in the presence of drilling- induced and natural fractures. The principal resistivities Rx, Ry, Rz are expressed as a function of horizontal and vertical resistivities of unfractured formation, fracture porosity and resistivity of fracture which is the resistivity of mud filtrate in the case of drilling induced fractures or the connate water resistivity and saturation in the case of natural fractures. These models allowed us to study the effect of fractures on the three principal resistivities as a function of laminated shale content and as a function of fracture porosity for both drilling induced and natural fractures. The combined three-step analysis of measured principal resistivities based on the modular concept allows a separation and quantification of the two anisotropy sources: fracturing and lamination. The analysis based on the three principal resistivities provides an estimate of fracture porosity, resistivity of the porous sand fraction (without fractures) and water saturations.
We have also developed an inversion algorithm for the biaxially anisotropic formations that reliably determines three principal resistivities from multi-component induction measurements. First, in the least-square inversion the multi-frequency focused magnetic field tensor is rotated to the principal coordinate system. This rotation is characterized by three angles: ?, ?, and ?; and three principal, diagonal magnetic field components Hxx, Hyy, and Hzz. After the three angles and three principal components are determined, we recover principal formation resistivities using a fast inversion algorithm based on a look-up table approach. To validate the theory and algorithms described above, we have simulated the multi-component induction measurements in thick biaxially anisotropic layers for various sets of parameters and used the software to process the data. The new approach was also applied to the real measurements acquired in the well with drilling induced fractures and as a result principal formation resistivities, fracture orientation, and fracture porosity were reliably determined.

Introduction
Early resistivity measurement pioneers realized that the earth is not homogenous and isotropic (C. Schlumberger 1920; Schlumberger et al., 1934). However, physics limited measurements primarily to those sensitive to Rh (e.g., in the plane normal to the borehole) and it was predicted that borehole effects would prevent resistivity anisotropy measurements (Moran and Gianzero 1979) and, even after the first successful measurements (Kriegshauser 2000) were reported concerns were raised about complicated borehole effects (Anderson 2002). Additional background information and review of multicomponent induction technology can be found in Rabinovich et al. (2007).