Analytical Simulation Of Multi-component Induction Logging Responses In Anisotropic Formation

Thinly laminated layers,shale and fractured formations are important reservoirs.Electrical properties of these reservoirs always exhibit uniaxial anisotropy and even biaxial anisotropy,which makes the electrical logging responses more complex.Accurate identification and evaluation of anisotropic formations are keys to exploration and development of such thin interbedded reservoirs and fractured reservoirs.Multi-component induction logging tools,also called triaxial induction logging tools,are developed for the detection and evaluation of anisotropic formations.Their abundant information provide by EM components can be used to extract formation parameters such as resistivity and dipping angle.To study the impacts of uniaxial anisotropy and biaxial anisotropy on the responses of triaxial induction logging,the analytical solution of triaxial induction logging responses in layered uniaxial anisotropic formation is firstly derived based on the theory of Hertz potential function.The characteristics of triaxial induction logging responses in tool coordinate system and the formation coordinate system are studied.A phase curve named P_h is defined,and then an apparent resistivity curve called R_ha is given by means of calibration and interpolation according to the phase curve.The apparent resistivity curve R_ha is used for extracting the horizontal resistivity of the uniaxial anisotropic formation.Then,for solving the full responses of a triaxial induction tool in a homogeneous biaxial anisotropic medium,we derived analytical solution in the spectral domain for arbitrary direction magnetic dipole based on the triple Fourier transform.By taking advantages of contour integration,adaptive integral interval truncation technique and the periodicity property of EM field in spectral domain,fast and accurate integral method of triple inverse Fourier transform is readily available.The responses of triaxial induction logging in dipping biaxial formations are simulated and the effects of transverse and longitudinal anisotropy are investigated.Finally,based on the generalized reflection theory,the analytical solution of triaxial induction logging responses in a layered biaxial anisotropic formation is derived,and then the response characteristics in tool coordinate system and formation coordinate system are studied.Phase curves named P_xa and P_ya are defined based on the response characteristics,and the relationships between P_xa-R_x and P_ya-R_yis reflected in a calibration plates.The apparent resistivity curves called R_xa and R_ya of the formation are given by interpolation according to the plates.Finally a method to identify the biaxial anisotropic formation according to corners at the interfaces in the high-angle well is proposed.The results show that,in uniaxial anisotropic formation,the apparent resistivity R_ha obtained by combination of triaxial induction logging cross components has a high resolution,and it could accurately extract the boundary position and the horizontal resistivity.The lower frequency and the smaller source distance make the extraction more accurate.And the higher dip angle makes the higher resolution of R_ha,which can be applied into the identification of thin interbedded sequences in high-angle wells.In a homogeneous biaxial anisotropic medium,analytical solution in spectral domain are simplified because of the periodicity of spectral domain EM fields along angular direction,which makes the integral speed up for four times.By introducing an adaptive integral interval truncation method,orders of Gauss–Legendre quadrature are reduced without losing calculating precision.The larger the dip angle,the stronger effects of transverse anisotropy on the coplanar apparent resistivity are observed,whereas the coaxial components are more easily affected by longitudinal anisotropy.In layered biaxial anisotropic formations,the responses of triaxial induction logging components in tool coordinate system are compound influenced by the biaxial anisotropy.It is difficult to directly reveal the electric characteristics of the layers.But each component in the formation coordinate system is sensitive to specific parameters so that these components in formation coordinate system can be used for extracting these parameters.The electric characteristics are showed up based on the apparent resistivity curves R_xaand R_ya obtained from the components in medium coordinate system.The R_ya also could be used for obtaining the boundaries position,identifying thin layers and recognizing the biaxial anisotropy in high-angle wells by the corners at the interfaces of biaxial anisotropic formations.