Full waveform inversion procedures with irregular topography

Full waveform inversion (FWI) is a form of seismic inversion that uses data residual, found as the misfit, between the whole waveform of field acquired and synthesized seismic data, to iteratively update a model estimate until such misfit is sufficiently reduced, indicating synthetic data is generated from a relatively accurate model. The aim of the thesis is to review FWI and provide a simplified explanation of the techniques involved to those who are not familiar with FWI. In FWI the local minima problem causes the misfit to decrease to its nearest minimum and not the global minimum, meaning the model cannot be accurately updated. Numerous objective functions were proposed to tackle different sources of local minima. The ‘joint deconvoluted envelope and phase residual’ misfit function proposed in this thesis aims to combine features of these objective functions for a comprehensive inversion. The adjoint state method is used to generate an updated gradient for the search direction and is followed by a step-length estimation to produce a scalar value that could be applied to the search direction to reduce the misfit more efficiently. Synthetic data are derived from forward modelling involving simulating and recording propagating waves influenced by the mediums’ properties. The ‘generalised viscoelastic wave equation in porous media’ was proposed by the author in sub-chapter 3.2.5 to consider these properties. Boundary layers and conditions are employed to mitigate artificial reflections arising from computational simulations. Linear algebra solvers are an efficient tool that produces wavefield vectors for frequency domain synthetic data. Regions with topography require a grid generation scheme to adjust a mesh of nodes to fit into its non-quadrilateral shaped body. Computational co-ordinate terms are implemented within wave equations throughout topographic models where a single point in the model in physical domain are represented by cartesian nodes in the computational domains. This helps to generate an accurate and appropriate synthetic data, without complex modelling computations. Advanced FWI takes a different approach to conventional FWI, where they relax upon the use of misfit function, however none of their proponents claims the former can supplant the latter but suggest that they can be implemented together to recover the true model.Open Acces

Seismic reverse-time migration in viscoelastic media

Seismic images are key to exploration seismology. They help identify structures in the subsurface and locate potential reservoirs. However, seismic images suffer from the problem of low resolution caused by the viscoelasticity of the medium. The viscoelasticity of the media is caused by the combination of fractured solid rock and fluids, such as water, oil and gas. This viscoelasticity of the medium causes attenuation of seismic waves, which includes energy absorption and velocity dispersion. These two attenuation effects significantly change the seismic data, and thus the seismic imaging. The aim of this thesis is to deepen the understanding of seismic wave propagation in attenuating media and to further investigate the method for high-resolution seismic imaging. My work, presented in this dissertation, comprises the following three parts. First, the determination of the viscoelastic parameters in the generalised viscoelastic wave equation. The viscoelasticity of subsurface media is succinctly represented in the generalised wave equation by a fractional temporal derivative. This generalised viscoelastic wave equation is characterised by the viscoelastic parameter and the viscoelastic velocity, but these parameters are not well formulated and therefore unfavourable for seismic implementation. The causality and stability of the generalised wave equation are proved by deriving the rate-of-relaxation function. On this basis, the viscoelastic parameter is formulated based on the constant Q model, and the viscoelastic velocity is formulated in terms of the reference velocity and the viscoelastic parameter. These two formulations adequately represent the viscoelastic effect in seismic wave propagation. Second, the development of a fractional spatial derivatives wave equation with a spatial filter. This development aims to effectively and efficiently solve the generalised viscoelastic wave equation with fractional temporal derivative, which is numerically challenging. I have transferred the fractional temporal derivative into fractional spatial derivatives, which can be solved using the pseudo-spectral implementation. However, this method is inaccurate in heterogeneous media. I introduced a spatial filter to correct the simulation error caused by the averaging in this implementation. The numerical test shows that the proposed spatial filter can significantly improve the accuracy of the seismic simulation and maintain high efficiency. Moreover, the proposed wave equation with fractional spatial derivatives is applied to compensate for the attenuation effects in reverse-time migration. This allows the dispersion correction and energy compensation to be performed simultaneously, which improves the resolution of the migration results. Finally, the development of reverse-time migration using biaxial wavefield decomposition to reduce migration artefacts and further improve the resolution of seismic images. In reverse-time migration, the cross-correlation of unphysical waves leads to large artefacts. By decomposing the wavefield both horizontally and vertically, and selecting only the causal waves for cross-correlation, the artefacts are greatly reduced, and the delicate structures can be identified. This decomposition method is also suitable for reverse-time migration with attenuation compensation. The migration results show that the resolution of the final seismic image is significantly improved, compared to conventional reverse-time migration.Open Acces

Multiscale Method for Elastic Wave Propagation in the Heterogeneous, Anisotropic Media

Seismic wave simulation in realistic Earth media with full wavefield methods is a fundamental task in geophysical studies. Conventional approaches such as the finite-difference method and the finite-element method solve the wave equation in geological models represented with discrete grids and elements. When the Earth model includes complex heterogeneities at multiple spatial scales, the simulation requires fine discretization and therefore a system with many degrees of freedom, which often exceeds current computational abilities. In this dissertation, I address this problem by proposing new multiscale methods for simulating elastic wave propagation based on previously developed algorithms for solving the elliptic partial differential equations and the acoustic wave equation. The fundamental motivation for developing the multiscale method is that it can solve the wave equation on a coarsely discretized mesh by incorporating the effects of fine-scale medium properties using so-called multiscale basis functions. This can greatly reduce computation time and degrees of freedom compared with conventional methods. I first derive a numerical homogenization method for arbitrarily heterogeneous, anisotropic media that utilizes the multiscale basis functions determined from a local linear elasticity equation to compute effective, anisotropic properties, and these equivalent elastic medium parameters can be used directly in existing elastic modeling algorithms. Then I extend the approach by constructing multiple basis functions using two types of appropriately defined local spectral linear elasticity problems. Given the eigenfunctions determined from local spectral problems, I develop a generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media in both continuous Galerkin (CG) and discontinuous Galerkin (DG) formulations. The advantage of the multiscale basis functions is they are model-dependent, unlike the predefined polynomial basis functions applied in conventional finite-element methods. For this reason, the GMsFEM can effectively capture the influence of fine-scale variation of the media. I present results for several numerical experiments to verify the effectiveness of both the numerical homogenization method and GMsFEM. These tests show that the effectiveness of the multiscale method relies on the appropriate choice of boundary conditions that are applied for the local problem in numerical homogenization method and on the selection of basis functions from a large set of eigenfunctions contained in local spectral problems in GMsFEM. I develop methods for solving both these problems, and the results confirm that the multiscale method can be powerful tool for providing accurate full wavefield solutions in heterogeneous, anisotropic media, yet with reduced computation time and degrees of freedom compared with conventional full wavefield modeling methods. Specially, I applied the DG-GMsFEM to the Marmousi-2 elastic model, and find that DG-GMsFEM can greatly reduce the computation time compared with continuous Galerkin (CG) FEM

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Quantitative seismic interpretation in thin-bedded geology using full-wavefield elastic modelling

Refleksjonsseismikk brukes til å lage seismiske «bilder» av den øverste delen av jordskorpen, blant annet med tanke på leting etter reservoarer for olje, gass, karbonlagring og geotermisk energi. I tillegg til å gi grunnlag for en strukturell tolkning, kan de seismiske dataene brukes til å kvantifisere egenskapene til det faste materialet og væskeinnholdet i bergartene. Et viktig verktøy i slik kvantitativ seismisk tolkning er analyse av såkalt AVO: amplitudenes variasjon med avstanden mellom kilde og mottaker (offset). Tynne geologiske lag gir utfordringer for AVO-modellering og tolkning, fordi lagtykkelsen vil kunne være mindre enn oppløsningen i de seismiske dataene. En problemstilling som tas opp i denne avhandlingen er nettopp hvordan man kan gjøre nøyaktig seismisk (forover) modellering i medier med tynne lag. En konvensjonell tilnærming innen AVO- modellering og inversjon er å bruke såkalt konvolusjonsmodellering. Denne metoden tar imidlertid bare hensyn til de primære seismiske refleksjonene og er derfor unøyaktig når modellene har tynne lag. To bedre alternativer er endelig-differanse-modellering og reflektivitetsmetoden. Reflektivitetsmetoden er en delvis analytisk modelleringsmetode for horisontalt lagdelte medier og er beregningsmessig billigere enn endelig-differansemodellering, der beregningene er basert på et tett samplet rutenett (grid). Jeg viser i avhandlingen at reflektivitetsmetoden er godt egnet for AVO-modellering i lagdelte medier. Seismiske data har en båndbegrenset karakter. En konsekvens er at beregning av reservoaregenskaper fra seismiske data generelt ikke er entydig, noe som særlig kommer til uttrykk for lagdelt geologi med tynne lag. Probabilistiske inversjonsmetoder, som for eksempel bayesianske metoder, tar hensyn til denne flertydigheten ved å forutsi sannsynligheter, noe som gjør det mulig a kvantisere usikkerheten. I avhandlingen kombinerer jeg seismisk modellering med bayesiansk klassifisering og inversjon. Modelleringen er utført med reflektivitetsmetoden og er basert på det komplette elastiske bølgefeltet. Formålet er å adressere to konkrete kvantitative seismiske tolkningsproblemer: 1) kvantifisering av usikkerhet i bayesiansk porevæske-klassifisering i nærvær av tynne lag med høy impedans, forårsaket av kalsittsementering i sandstein, og 2) estimering av reservoaregenskapene til turbiditt-reservoarer karakterisert ved alternerende lag av sandstein og skifer. I den første anvendelsen viser jeg i en modelleringsstudie at kalsitt-sementerte lag kan gi en detekterbar refleksjonsrespons, noe som kan påvirke amplituden målt ved reservoartoppen og dermed forstyrre AVO-målingen. Den observerte effekten øker usikkerheten ved porevæske-klassifisering basert på AVO-attributter, som jeg har demonstrert i en case-studie. Følgelig øker sannsynligheten for en falsk hydrokarbon-indikasjon betydelig i nærvær av kalsittsementerte lag. I den andre anvendelsen presenterer jeg en bayesiansk inversjon som tar AVO-skjæringspunktet og gradienten målt på toppen av et reservoar som inngangsdata og estimerer sannsynlighetstetthetsfunksjonen til forholdstallene «net-to-gross» og «net-pay-to-net». Metoden ble anvendt på syntetiske data og AVO-attributtkart fra Jotunfeltet på norsk kontinentalsokkel. Det ble funnet at AVO-gradienten korrelerer med reservoarets net-togross forhold, mens AVO-skjæringspunktet er mest følsomt for typen porevæske. Etter inversjon genererte jeg kart over de mest sannsynlige verdiene av forholdene net-to-gross og net-pay-to-net, samt kart over net pay og usikkerhetene. Disse kartene kan bidra til å identifisere potensielle soner med høy reservoarkvalitet og hydrokarbonmetning.Reflection seismics is used to image the subsurface for the exploration of oil and gas, geothermal or carbon storage reservoirs, among others. In addition to the structural interpretation of the resulting seismic images, the seismic data can be interpreted quantitatively with the goal to obtain rock and fluid properties. An essential tool in quantitative seismic interpretation is the analysis of the amplitude variation with offset (AVO). Thin-bedded geology below the seismic resolution poses challenges for AVO modelling and interpretation. One problem addressed in this thesis is accurate seismic forward modelling in thin-bedded media. Primaries-only convolutional modelling, commonly used in conventional AVO modelling and inversion, is prone to failure in the presence of thin beds. Better alternatives are finite-difference modelling or the reflectivity method. The reflectivity method is a semi-analytic modelling method for horizontally layered media and is computationally cheaper than finite-difference modelling on densely sampled grids. I show in this thesis that the reflectivity method is well-suited for the AVO modelling of layered media. The band-limited nature of seismic data is one reason for the non-unique estimation of reservoir properties from seismic data, especially in thin-bedded geology. Probabilistic inversion methods, such as Bayesian methods, honour this non-uniqueness by predicting probabilities that allow the uncertainty to be quantified. In this thesis, I integrate full-wavefield elastic seismic modelling by the reflectivity method with Bayesian classification and inversion. The objective is to address two concrete quantitative seismic interpretation problems: 1) the uncertainty quantification of Bayesian pore-fluid classification in the presence of thin high-impedance layers caused by calcite cementation in sandstone, and 2) the estimation of reservoir properties of turbidite reservoirs characterised by sand-shale interbedding. In the first application, I show through a modelling study that calcite-cemented beds lead to detectable reflection responses that can interfere with the target reflection at the reservoir top and thereby perturb the AVO behaviour. The observed effect increases the uncertainty of pore-fluid classification based on AVO attributes, as demonstrated by a case study. Consequently, the probability of a false hydrocarbon indication is significantly increased in the presence of calcite-cemented beds. In the second application, I present a Bayesian inversion that takes the AVO intercept and gradient measured at the top of a reservoir as input and estimates the probability density function of the net-to-gross ratio and the net-pay-to-net ratio. The method was applied to synthetic data and AVO attribute maps from the Jotun field on the Norwegian Continental Shelf. It was found that the AVO gradient correlates with the net-to-gross ratio of the reservoir, while the AVO intercept is most sensitive to the type of pore fluid. After inversion, maps of the most-likely values of the net-to-gross ratio, net-pay-to-net ratio, net pay and the uncertainty could be generated. These maps help to identify potential zones of high reservoir quality and hydrocarbon saturation.Doktorgradsavhandlin

Institute for Computational Mechanics in Propulsion (ICOMP)

The Institute for Computational Mechanics in Propulsion (ICOMP) is a combined activity of Case Western Reserve University, Ohio Aerospace Institute (OAI) and NASA Lewis. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1991 are described