Vladimir Puzyrev

Dispersion-optimized quadrature rules for isogeometric analysis: Modified inner products, their dispersion properties, and optimally blended schemes

Abstract This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize the Pythagorean eigenvalue theorem of Strang and Fix. The proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non-uniform meshes as well as for different polynomial orders and continuity of the basis. The optimally-blended schemes improve the convergence rate of the method by two orders with respect to the fully-integrated Galerkin method. The proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems.

Quadrature blending for isogeometric analysis

Abstract We use blended quadrature rules to reduce the phase error of isogeometric analysis discretizations. To explain the observed behavior and quantify the approximation errors, we use the generalized Pythagorean eigenvalue error theorem to account for quadrature errors on the resulting weak forms [28]. The proposed blended techniques improve the spectral accuracy of isogeometric analysis on uniform and non-uniform meshes for different polynomial orders and continuity of the basis functions. The convergence rate of the optimally blended schemes is increased by two orders with respect to the case when standard quadratures are applied. Our technique can be applied to arbitrary high-order isogeometric elements.

Three-Dimensional Modeling of the Casing Effect in Onshore Controlled-Source Electromagnetic Surveys

The presence of steel-cased wells and other infrastructure causes a significant change in the electromagnetic fields that has to be taken into consideration in modeling and interpretation of field data. A realistic and accurate simulation requires the borehole casing to be incorporated into the modeling scheme, which is numerically challenging. Due to the huge conductivity contrast between the casing and surrounding media, a spatial discretization that provides accurate results at different spatial scales ranging from millimeters to hundreds of meters is required. In this paper, we present a full 3D frequency-domain electromagnetic modeling based on a parallel finite-difference algorithm considering the casing effect and investigate its applicability on the borehole-to-surface configuration of the Hontomín CO2 storage site. To guarantee a robust solution of linear systems with highly ill-conditioned matrices caused by huge conductivity contrasts and multiple spatial scales in the model, we employ direct sparse solvers. Different scenarios are simulated in order to study the influence of the source position, conductivity model, and the effect of the steel casing on the measured data. Several approximations of the real hollow casing that allow for a large reduction in the number of elements in the resulting meshes are studied. A good agreement between the modeled responses and the real field data demonstrates the feasibility of simulating casing effects in complex geological areas. The steel casing of the well greatly increases the amplitude of the surface electromagnetic fields and thus improves the signal-to-noise ratio and the sensitivity to deep targets.

Time-lapse full waveform inversion of vertical seismic profile data: Workflow and application to the CO2CRC Otway project

Vertical seismic profile (VSP) is one of the technologies for monitoring hydrocarbon production and CO2 geosequestration. However quantitative interpretation of time-lapse VSP is challenging due to its irregular distribution of source-receiver offsets. One way to overcome this challenge is to use full waveform inversion (FWI), which does not require regular offsets. We present a workflow of elastic FWI applied to offset vertical seismic profile data for the purpose of identification and estimation of time-lapse changes introduced by injection of 15,000 tonnes of CO2-rich gas mixture at 1.5 km depth. Application of this workflow to both synthetic and field data shows that elastic FWI is able to detect and quantify the time-lapse anomaly in P wave velocity with the magnitude of 100-150 m/s.

Generalization of the Pythagorean Eigenvalue Error Theorem and Its Application to Isogeometric Analysis

This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules.

Spectral approximation properties of isogeometric analysis with variable continuity

Abstract We study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis. Such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh, which allows for large computational savings during the solution of the resulting linear system. The continuity reduction results in extra degrees of freedom that modify the approximation properties of the method. The convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis. We show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra, such as stopping bands and outliers, and present a unified description of these effects in finite element method, isogeometric analysis, and refined isogeometric analysis. Accuracy of the refined isogeometric analysis approximations can be improved by using non-standard quadrature rules. In particular, optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence, as it occurs in standard isogeometric analysis.

Full Waveform Inversion of Time-lapse Offset VSP Data - CO2CRC Otway Project Case Study

Summary Vertical Seismic Profile (VSP) is a useful tool for time-lapse monitoring. We conduct elastic 2D Full Waveform Inversion (FWI) on offset Vertical Seismic Profile (VSP) data to detect and quantify the time-lapse anomaly introduced by the CO2 geosequestration. Two datasets are being studied: a 2D synthetic dataset and the field dataset acquired during the Stage 2C of the CO2CRC Otway project. FWI proves capable of detecting the time-lapse anomaly on both datasets, however, the strength of the anomaly in the field data inversion results is lower than expected from theoretical predictions. We attribute this to 3D effects, which are not taken into account. In the end, FWI proves to be an instrument that fits the monitoring problem well. The inversion of the monitor dataset is significantly quicker than the baseline inversion, which enables the time-lapse anomaly identification shortly after the data acquisition.

Electromagnetic response prediction for conductive, permeable, heterogeneous steel-cased wells

Summary Steel-cased wells can be used as galvanic sources to detect and estimate variations in subsurface electrical conductivity. Most of the modeling approaches are based on approximations to the casing and ignore the variations in magnetic permeability. In this paper, we investigate the impact of varying electrical conductivity and magnetic permeability on the electromagnetic response of steel-cased wells. As an example, we consider a full three-dimensional realistic representation of a steel-cased well in the surface-to-borehole and borehole-to-surface configurations. We show how the variable conductivity and permeability along the length of the casing can affect the electromagnetic measurements.