The joint physics and data-driven geometrical factor of array laterolog in layered formations

Abstract

Numerical geometrical factors (GFs) are developed for array laterolog measurements in one-dimensional (1D) cylindrically and planarly layered formations. The kernel of GFs assumes that laterolog responses can be expressed in terms of linear superposition of formation resistivities at different units. Two techniques, namely physics-driven modeling and data-driven approximation, have been employed to ensure both computational accuracy and efficiency of GFs. Physics-driven modeling utilizes a three-dimensional finite element algorithm to generate high-precision and fully labeled GFs data and library. Subsequently, these predetermined data are trained offline using classical neural network algorithm to establish an implicit relationship between formation parameters and GFs. The joint physics and data-driven GFs presents three main advantages. Firstly, it enables the direct representation of the spatial sensitivity of layered formations both laterally and horizontally in vertical and horizontal wells. Secondly, GFs can be used to rapidly calculate the array laterolog responses. The computation speed can be enhanced by thousands of times in comparison with traditional physic-driven modeling, allowing for real-time data processing. Another noteworthy aspect is that GFs forward modeling can handle formations with arbitrary layers, thereby resolving the fixed-layer modeling issue inherent in traditional data-driven methods. Thirdly, the GFs can be employed to approximate the derivative of logging response with respect to formation resistivity, significantly reducing the complexity of Jacobian matrix calculation for deterministic inversion. These numerical GFs not only serve as an excellent alternative simulator for array laterolog, but also will be helpful to accelerate the modeling and inversion of other logging measurement in layered structures.