Elastic wave propagation in heterogeneous, vertically layered elastic media plays a critical role in fields such as geophysics and acoustic (including sonic and ultrasound) oil well logging. Accurately modeling these interactions is challenging, particularly in complex geometries like perforations in cased-hole acoustic well-logging models. This study extends the Numerical Mode Matching (NMM) method to the field of elastic waves, reducing the original heterogeneous, vertically layered elastic wave problem into a series of two-dimensional (2D) waveguide eigenvalue problems coupled with a 1D layered medium problem. Additionally, it introduces a hybrid numerical method that combines NMM with the 3D Spectral Element Method (SEM) to address perforations in well-logging models. The hybrid NMM-SEM exploits NMM’s strengths for regular layered structures and SEM’s flexibility for irregular geometries. Numerical examples demonstrate significant improvements in both accuracy and efficiency compared to traditional numerical methods. This method offers an effective solution for acoustic well-logging applications and the potential for broader applications in geophysical modeling.

Determination of petrophysical parameters is regarded as a critical task for the exploration and production of oil and gas reservoirs. Traditionally, the joint inversion of electromagnetic (EM) and seismic data under petrophysics constraints can be exploited to reconstruct the distribution of reservoir petrophysical parameters. However, methods involved with such schemes face challenges when solving high-contrast nonlinear inverse scattering problems due to the complexity of the relationship between resistivity/velocity and petrophysical properties and the nonuniqueness of these inverse problems. Here, to resolve these challenges, we have developed a field data transformation-joint inversion method (FDT-JIS) to directly reconstruct the distribution of porosity and water saturation. Specifically, the chain rule to transform geophysical parameters into petrophysical parameters is leveraged, and the field data transformation module is subsequently utilized to transform the scattered field data generated under the test configuration into those under the training configuration. Finally, a joint network is adopted to establish the mapping relationship between EM and acoustics data and petrophysical parameters to attain the inversion of petrophysical parameters. With numerical examples, we demonstrate that FDT-JIS not only allows different transceiver configurations to be used in training and testing, but also accurately reconstructs petrophysical parameters of complex models with high contrast in noisy environments.

In elastic wave inverse scattering problems, the material-property reconstruction, e.g. distributions of mass density, compressional wave speed, and shear wave speed from a limited set of measurement data, has attracted considerable research interest. However, simultaneous inversion of multiple parameters endures high computation costs, and reconstructed compressional and shear speeds may become unrealistic if no physical constraint is imposed. Meanwhile, because large objects with high contrasts over the background medium tend to induce high nonlinearity during the inversion process, it is difficult to obtain high-quality high-contrast material properties. To overcome such difficulties, we have developed a multi-module deep learning scheme with physical constraint for multi-parameter elastic wave inversion of high-contrast objects in inhomogeneous media. This scheme consists of (1) a preliminary imaging module (PIM), in which a deep residual network (ResNet) is employed to convert the scattered field data into the preliminary inversion images, (2) an image-enhancement module (IEM), in which a U-Net is employed to further enhance the image quality, and (3) a convolutional neural network (CNN) that is employed as the physical constraint module (PCM) to allow elastic wave parameters to satisfy actual physical constraint. Numerical examples have demonstrated that the proposed scheme not only accurately achieves multi-parameter elastic wave inversion, but also has good generalizability. Finally, the scheme can be applied to complex objects with high contrasts in both noise-free and noisy environments.

This article first presents machine-learning (ML) inversion of resistivity profiles from multifrequency electromagnetic measurements on undulating terrain surfaces based on synthetic data training by the mixed spectral element method (MSEM). The inversion method combines several advanced technologies with various merits. A semiregular mesh generation method is designed and developed for adaption to complex undulating terrain and multifrequency measured data, and the proposed meshing technology is also suitable for modeling different training models under the same undulating terrain. By simulating the application scenarios of measurements, the apparent resistivity data at eight frequencies from 1 to 2048 Hz are simulated with the 2.5-D MSEM to ensure the accuracy and efficiency of the simulation of undulating terrains. Fast simulation of stochastic models for training datasets is achieved by twisting and extruding the initial model obtained by Bostick inversion. Since the unknown weight matrices are solved only once in the training process, the extreme learning machine (ELM) is used for ML inversion to reduce the training cost and obtain high-precision inversion results. Then it is applied to reconstruct a metallogenic model to verify the method’s validity and accuracy and to reconstruct the resistivity profile of underground ore bodies with actual measurements. The results show that the proposed method can be effectively used to detect metal ores at a depth of less than 3000 m underground.

This letter presents a fast and memory-efficient 3-D electromagnetic solver for logging-while-drilling (LWD) tools based on the flipped total field/scattered field-based discontinuous Galerkin frequency-domain (TF/SF DGFD) method. The new method inherits the Riemann transmission condition (RTC) with surface current sources from the mixed TF/SF DGFD method to couple nonconformal meshes as well as TF/SF solvers; it then extends to the moving-tool scenarios for the LWD application with a solver flipping technology and delivers a single global matrix of dramatically reduced dimension for all source positions, which can be solved directly on a normal-memory computer. By incorporating the tetrahedral/hexahedral mixed mesh and the curved domain decomposition method, the new solver is applied to two LWD cases and about 70% of unknowns can be saved for the flipped TF/SF DGFD remeshed regions.

An hp- and memory-adaptive discontinuous Galerkin time-domain algorithm is presented to efficiently model wave propagation in poroelastic media, with the incorporation of 3-D fully anisotropic intrinsic attenuation. From the perspective of physics, the attenuation from the triclinic rock frame, the loss in the pore fluid, and the friction for their interaction are all considered. From the perspective of implementation, a new frequency-domain constitutive equation is introduced, involving complex-valued poroelasticity matrix. A new Q value is defined as the ratio between its real and imaginary parts for every entry. Then, a generalized Maxwell body is adopted to approximate this frequency-dependent Q in the time domain. Mathematically speaking, the hyperbolicity is preserved for the new viscous poroelastic system. Our results corroborate that the intrinsic attenuation anisotropy makes tangible effects in fluid-saturated porous formations.

As one of the most important nondestructive characterization techniques, electromagnetic (EM) methods can be used in the subsurface fracture detection, especially for hydraulic fracture evaluation in unconventional petroleum exploration and development. The multiscale nature of long but extremely thin 3-D fractures is difficult for conventional EM modeling methods such as the finite element method (FEM) in numerical simulation. The problem becomes even more challenging when the effects of tilted borehole, casing, and planar stratified media need to be considered. So far, modeling a tilted borehole in layered media is still a major challenge for conventional methods. In this paper, we present the hybrid numerical mode-matching method with the stabilized biconjugate gradient fast Fourier transform method as the forward modeling algorithm that can efficiently model 3-D fractures in planar stratified media with a cased borehole environment. Numerical results validate the accuracy of the hybrid forward method and show orders of magnitude higher efficiency of this forward solver than the FEM.

In this paper, full mechanisms of dissipation and dispersion in poro-viscoelastic media are accurately simulated in time domain. Specifically, four Q values are first proposed to depict a poro-viscoelastic medium: two for the attenuation of the bulk and shear moduli in the solid skeleton, one for the bulk modulus in the pore fluid, and the other one for the solid-fluid coupling. By introducing several sets of auxiliary ordinary differential equations, the Q factors are efficiently incorporated in a high-order discontinuous Galerkin algorithm. Consequently, in the mathematical sense, the Riemann problem is exactly solved, with the same form as the inviscid poroelastic material counterpart; in the practical sense, our algorithm requires nearly negligible extra time cost, while keeping the governing equations almost unchanged. Parenthetically, an arbitrarily nonconformal-mesh technique, in terms of both h- and p-adaptivity, is implemented to realize the domain decomposition for a flexible algorithm. Furthermore, our algorithm is verified with an analytical solution for the half-space modeling. A validation with an independent numerical solver, and an application to a large-scale realistic complex topography modeling demonstrate the accuracy, efficiency, flexibility, and capability in realistic subsurface sensing.

Porous materials are ubiquitous in the subsurface formations of the earth where acoustic and seismic waves are used for remote sensing. However, it is not well understood how the dissipation and the dispersion of poroelastic waves are caused by the viscoelastic and viscous properties of the constituents such as solid grains and pore fluid and by the viscoelastic dissipation of the solid frame, as well as the viscodynamic coupling of the pore fluid to the solid frame due to its global and local flows relative to the solid grains. Such attenuation mechanisms have seldom been incorporated in subsurface sensing simulations, although they can be very important to applications. In this paper, we propose a complete attenuation model, including both full stiffness and viscodynamic dissipation, for poroelastic media in seismic wave simulations. Completely based on a generalized Zener model, the effects associated with physical dissipation and frequency-dependent dispersion are accurately simulated by a finite-difference time-domain algorithm. Verifications with analytical solutions show the accuracy, efficiency, and flexibility of our method. Numerical results demonstrate that the attenuation of Biot's model in the sediment of the seafloor has significant effects on acoustic wave scattering from complex geologic structures.

A high-order discontinuous Galerkin (DG) method with nonconformal meshes is developed to accurately simulate large-scale poroelastic wave propagation in 3-D isotropic media. An exact upwind flux is succinctly derived to serve as an accurate coupling solver for the DG algorithm. Specifically, the eigenvalue problem in the Riemann solution is effectively reduced from the rank of 13 to 4. Furthermore, this new numerical flux gives more explicit physical insight, which indicates three-type waves in poroelastic media: two P waves and one S wave. Validations and verifications with analytical/semianalytical numerical solutions demonstrate the accuracy, robustness, and flexibility of the proposed solver.

Rapid simulations of large-scale low-frequency subsurface electromagnetic measurements are still a challenge because of the low-frequency breakdown phenomenon that makes the system matrix extremely poor-conditioned. Hence, significant attention has been paid to accelerate the numerical algorithms for Maxwell's equations in both integral and partial differential forms. In this letter, we develop a novel 2.5-D method to overcome the low-frequency breakdown problem by using the mixed spectral element method with the divergence-free constraint and apply it to solve the marine-controlled-source electromagnetic systems. By imposing the divergence-free constraint, the proposed method considers the law of conservation of charges, unlike the conventional governing equation for these problems. Therefore, at low frequencies, the Gauss law guarantees the stability of the solution, and we can obtain a well-conditioned system matrix even as the frequency approaches zero. Several numerical experiments show that the proposed method is well suited for solving low-frequency electromagnetic problems.

We present an efficient nonconformal-mesh discontinuous Galerkin (DG) method for elastic wave propagation in viscous media. To include the attenuation and dispersion due to the quality factor in time domain, several sets of auxiliary ordinary differential equations (AODEs) are added. Unlike the conventional auxiliary partial differential equation-based algorithm, this new method is highly parallel with its lossless counterpart, thus requiring much less time and storage consumption. Another superior property of the AODE-based DG method is that a novel exact Riemann solver can be derived, which allows heterogeneous viscoelastic coupling, in addition to accurate coupling with purely elastic media and fluid. Furthermore, thanks to the nonconformal-mesh technique, adaptive hp-refinement and flexible memory allocation for the auxiliary variables are achieved. Numerical results demonstrate the efficiency and accuracy of our method.