Recovery of the parameters of a focal zone by the data of earthquakes

We present an approach for solving the inverse kinematic problem of seismic with internal sources, based on the method of multidimensional data approximation on irregular grids. The times of arrival of elastic waves to the seismic stations are considered as known. The hodographs from earthquake to the stations are approximated for further determining the velocities of longitudinal and transverse waves using the eikonal equation. The ratio of these velocities determines the Poisson’s ratio, and the other elastic parameters of the medium can be found in units of the density. The results of implementation of the approach, based on the real data, are presented.

Development and Application of Fast Simulation Based on the PSS Pressure as a Spatial Coordinate

Recent studies have shown the utility of the Fast Marching Method and the Diffusive Time of Flight for the rapid simulation and analysis of Unconventional reservoirs, where the time scale for pressure transients are long and field developments are dominated by single well performance. We show that similar fast simulation and multi-well modeling approaches can be developed utilizing the PSS pressure as a spatial coordinate, providing an extension to both Conventional and Unconventional reservoir analysis. We reformulate the multi-dimensional multi-phase flow equations using the PSS pressure drop as a spatial coordinate. Properties are obtained by coarsening and upscaling a fine scale 3D reservoir model, and are then used to obtain fast single well simulation models. We also develop new 1D solutions to the Eikonal equation that are aligned with the PSS discretization, which better represent superposition and finite sized boundary effects than the original 3D Eikonal equation. These solutions allow the use of superposition to extend the single well results to multiple wells. The new solutions to the Eikonal equation more accurately represent multi-fracture interference for a horizontal MTFW well, the effects of strong heterogeneity, and finite reservoir extent than those obtained by the Fast Marching Method. The new methodologies are validated against a series of increasingly heterogeneous synthetic examples, with vertical and horizontal wells. We find that the results are systematically more accurate than those based upon the Diffusive Time of Flight, especially as the wells are placed closer to the reservoir boundary or as heterogeneity increases. The approach is applied to the Brugge benchmark study. We consider the history matching stage of the study and utilize the multi-well fast modeling approach to determine the rank quality of the 100+ static realizations provided in the benchmark dataset against historical data. The multi-well calculation uses superposition to obtain a direct calculation of the interaction of the rates and pressures of the wells without the need to explicitly solve flow equations within the reservoir model. The ranked realizations are then compared against full field simulation to demonstrate the significant reduction in simulation cost and the corresponding ability to explore the subsurface uncertainty more extensively. We demonstrate two completely new methods for rapid reservoir analysis, based upon the use of the PSS pressure as a spatial coordinate. The first approach demonstrates the utility of rapid single well flow simulation, with improved accuracy compared to the use of the Diffusive Time of Flight. We are also able to reformulate and solve the Eikonal equation in these coordinates, giving a rapid analytic method of transient flow analysis for both single and multi-well modeling.

Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

We consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .

Rectifiability of entropy defect measures in a micromagnetics model

We study the fine properties of a class of weak solutions u of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in [T. Rivière and S. Serfaty, Limiting domain wall energy for a problem related to micromagnetics, Comm. Pure Appl. Math. 54 2001, 3, 294–338]. In particular, we prove that the entropy defect measure associated to u is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of u.

Homotopy solutions of the acoustic eikonal equation for strongly attenuating transversely isotropic media with a vertical symmetry axis

Many applications in seismology involve the modeling of seismic wave traveltimes in anisotropic media. We present homotopy solutions of the acoustic eikonal equation for P-waves traveltimes in attenuating transversely isotropic media with a vertical symmetry axis. Instead of the commonly used perturbation theory, we use the homotopy analysis method to express the traveltimes by a Taylor series expansion over an embedding parameter. For the derivation, we first perform homotopy analysis of the eikonal equation and derive the linearized ordinary differential equations for the coefficients of the Taylor series expansion. Then, we obtain the homotopy solutions for the traveltimes by solving the linearized ordinary differential equations. Results on approximate formulae investigations demonstrate that the analytical expressions are efficient methods for the computation of traveltimes from the eikonal equation. In addition, these formulas are also effective methods for benchmarking approximated solutions in strongly attenuating anisotropic media.

An Inverse Problem Involving a Viscous Eikonal Equation with Applications in Electrophysiology

In this work we discuss the reconstruction of cardiac activation instants based on a viscous Eikonal equation from boundary observations. The problem is formulated as a least squares problem and solved by a projected version of the Levenberg–Marquardt method. Moreover, we analyze the well-posedness of the state equation and derive the gradient of the least squares functional with respect to the activation instants. In the numerical examples we also conduct an experiment in which the location of the activation sites and the activation instants are reconstructed jointly based on an adapted version of the shape gradient method from (J. Math. Biol. 79, 2033–2068, 2019). We are able to reconstruct the activation instants as well as the locations of the activations with high accuracy relative to the noise level.

Regularity and global structure for Hamilton–Jacobi equations with convex Hamiltonian

We consider the multidimensional Hamilton–Jacobi (HJ) equation [Formula: see text] with [Formula: see text] being a constant and for bounded [Formula: see text] initial data. When [Formula: see text], this is the typical case of interest with a uniformly convex Hamiltonian. When [Formula: see text], this is the famous Eikonal equation from geometric optics, the Hamiltonian being Lipschitz continuous with homogeneity [Formula: see text]. We intend to fill the gap in between these two cases. When [Formula: see text], the Hamiltonian [Formula: see text] is not uniformly convex and is only [Formula: see text] in any neighborhood of [Formula: see text], which causes new difficulties. In particular, points on characteristics emanating from points with vanishing gradient of the initial data could be “bad” points, so the singular set is more complicated than what is observed in the case [Formula: see text]. We establish here the regularity of solutions and the global structure of the singular set from a topological standpoint: the solution inherits the regularity of the initial data in the complement of the singular set and there is a one-to-one correspondence between the connected components of the singular set and the path-connected components of the set [Formula: see text].

LOCALIZATION OF MICROSEISMIC EVENTS USING PHYSICS-INFORMED NEURAL NETWORK SOLUTION TO THE EIKONAL EQUATION

The paper demonstrates an algorithm for using physics-informed neural networks in workflow of processing microseismic data regarding the problem of localization of microseismic events. The proposed algorithm involves the use of a physics-informed neural network solution to the eikonal equation to calculate the traveltimes of the first arrivals. As a result, the network solution is compared with the observed arrival times to solve the inverse kinematic problem to determine the coordinates of the event locations. Using a synthetic 3D example, it was shown that the average absolute error of the arrival time misfit was less than 0.25 ms, and the average localization error did not exceed 4.5 meters.