A Targeted Quadrature Framework for Simulating Large-Scale 3D Anisotropic Electromagnetic Measurements

TL;DR

This work addresses fast, accurate forward modeling of frequency-domain borehole EM in 3D anisotropic formations for UDAR measurements. It introduces a targeted ROM framework built on a Lebedev finite-volume discretization and a block-quadrature reduced-order model, enabling real-time transfer-function evaluation $\mathcal{F}(\omega)$ and adjoint-based Jacobians for gradient-based inversion. Key innovations include two-scale effective-medium averaging, a self-adjoint block-Lanczos solver with Gauss and Gauss-Radau quadratures, and a two-frame coordinate mapping for curvilinear well paths. Numerical experiments on a North Sea–like synthetic reservoir demonstrate accurate nine-coupling UDAR responses across frequencies, monotone convergence, and strong HPC scalability, including a ~20 s adjoint Jacobian computation for large 3D problems. The framework supports near real-time 3D inversion and points to future enhancements such as GPU acceleration, randomized preconditioning, Krein–Nudelman quadratures, and uncertainty quantification.

Abstract

We develop a new, efficient, and accurate method to simulate frequency-domain borehole electromagnetic (EM) measurements acquired in the presence of three-dimensional (3D) variations of the anisotropic subsurface conductivity. The method is based on solving the quasi-static Maxwell equations with a goal-oriented finite-volume discretization via block-quadrature reduced-order modeling. Discretization is performed with a Lebedev grid that enables accurate and conservative solutions in the presence of any form of anisotropic electrical conductivity. Likewise, the method makes use of a new effective-medium approximation to locally account for non-conformal boundaries and large contrasts in electrical conductivity, especially in the vicinity of EM sources and receivers. The finite-volume discretization yields a large symmetric linear system of equations, which is reduced to a set of smaller structured problems via block Lanczos recursion. The formulation also enables the efficient calculation of the adjoint solution, which is necessary for gradient-based inversion of the measurements to estimate the associated spatial distribution of electrical conductivity, i.e., to solve the inverse problem. Specific applications and verifications of the new numerical simulation algorithm are considered for the case of borehole ultra-deep azimuthal resistivity measurements (UDAR) typically used for subsurface well geosteering and navigation. We verify the efficiency, robustness, and scalability of this approach using synthetic UDAR measurements acquired in a 3D formation inspired by North-Sea geology. The numerical experiments successfully verify the applicability of our modeling approach to real-time UDAR processing frameworks.

Figures (10)

• Figure 1: 3D view of the 4 Lebedev clusters shifted in space. Dotted gray lines are the primary grids and dashed gray lines the dual grid.
• Figure 2: Left: Block-transfer functions $\cal F$, including the nine UDAR coupling components. Right: 3D UDAR measurement system visualizing the three transmitter and receiver current densities.
• Figure 3: Slice of a layered resistivity model with a fault. The horizontal resistivity, $R_{\hbox{[}0.6]{\rm h}}$ in $[\Omega \cdot m]$, is shown on the left and the vertical resistivity, $R_{\hbox{[}0.6]{\rm v}}$, on the right.
• Figure 4: Example components of the adjoint Jacobian computation for $R_{\hbox{[}0.6]{\rm h}}$ (top-row) compared to their FD counterparts (bottom-row).
• Figure 5: Example components of the Adjoint Jacobian for $R_{\hbox{[}0.6]{\rm v}}$ computation (top-row) compared to their FD counterparts (bottom-row).