Elastic parameter inversion of Longmaxi Formation shale based on the least squares method

Abstract

Elastic parameters are important parameters in the study of shale anisotropy, exploration, and development, and their accuracy is of great significance. In this study, shale in the lower Silurian Longmaxi Formation of southeastern Chongqing was investigated by obtaining the acoustic velocity of the partial bedding angle through an acoustic transmission experiment based on the elastic wave theory of the VTI medium and the least squares method, thus establishing a shale elastic parameter inversion method for improved accuracy. Results found that (1) the stratification angle was negatively correlated with the wave velocity; (2) velocity results calculated using the inversion-obtained elastic parameter were consistent with the measured results, with a maximum relative error of less than 1.52%, demonstrating that elastic parameters obtained by the proposed inversion method are valid; and (3) Longmaxi Formation shale has strong anisotropy, with the most significant direction of shear wave splitting being the direction in which the bedding angle is 0°. The results of this study provide a working foundation for using acoustic logging data to more accurately obtain the elastic parameters of shale and to study shale anisotropy characteristics, which have important theoretical value and practical significance.

Appendix

The general elastic wave equation is:

$$ P\frac{\partial^2}{\partial {t}^2}U=L\left(C{L}^TU\right) $$

where ρ is the medium density; U= (u_x, u_y, u_z) ^T is the displacement vector; L is the partial derivative operator matrix; and C is the elastic tensor. L as shown below:

$$ L=\left[\begin{array}{ccc}\frac{\partial }{\partial x}& 0& 0\\ {}0& \frac{\partial }{\partial y}& 0\\ {}0& 0& \frac{\partial }{\partial z}\end{array}\kern1.25em \begin{array}{ccc}0& \frac{\partial }{\partial z}& \frac{\partial }{\partial y}\\ {}\frac{\partial }{\partial z}& 0& \frac{\partial }{\partial x}\\ {}\frac{\partial }{\partial y}& \frac{\partial }{\partial x}& 0\end{array}\right] $$

The elasticity tensor C of VTI medium is:

$$ C=\left[\begin{array}{ccc}{C}_{11}& {C}_{12}& {C}_{13}\\ {}{C}_{12}& {C}_{22}& {C}_{23}\\ {}{C}_{13}& {C}_{23}& {C}_{33}\\ {}0& 0& 0\\ {}0& 0& 0\\ {}0& 0& 0\end{array}\kern1.25em \begin{array}{ccc}0& 0& 0\\ {}0& 0& 0\\ {}0& 0& 0\\ {}{C}_{44}& 0& 0\\ {}0& {C}_{55}& 0\\ {}0& 0& {C}_{66}\end{array}\right] $$

Among them, the elastic parameter satisfies the following relationship: C₂₂ = C₁₁,C₂₃ = C₁₃,C₅₅ = C₄₄,${\mathrm{C}}_{66}=\frac{1}{2}\left({\mathrm{C}}_{11}-{\mathrm{C}}_{12}\right)$.

When the relationship C₁₁ = C₂₂ = C₃₃ = λ + 2μ,C₁₂ = C₁₃ = C₂₃ = λ,C₄₄ = C₅₅ = C₆₆ = μ is satisfied, C is the elastic tensor of the isotropic medium, where λ and μ are the Lame coefficients.