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GoINTRODUCTION

The information that seismic processing aims to extract from the data is: (1) an estimate of the structural map of the earth, and (2) estimates about the mechanical properties of the target (possible hydrocarbon reservoirs). The process of estimating the earth’s material properties (porosity, velocity, density, etc.) is called inversion. Ultimately, this information is interpreted to deduce the geological structure, size, and type of possible hydrocarbon accumulations. In this abstract, we focus on methods to estimate the earth’s subsurface model given the best fit to the recorded data in the sense of minimizing the data misfit using a specific metric (e.g., the L2 norm). Most of these methods use iterative schemes, in which the model is updated based on a search direction computed from a gradient of a cost function. These types of optimization problems have been given a lot of attention in seismic exploration in recent years; one example is full waveform inversion (FWI). The FWI theory was originally developed by Tarantola (1984, 1988); its most general formulation involves a quadratic objective function measuring the differences (in terms of dynamics and kinematics) between model data and measured data. The inverted model, which generates a realization of model data that minimizes the objective function, is the output of FWI. The goal of FWI is to invert for a model that closely describes the actual model (earth) that produced the measured data (Crase et al., 1990; Ikelle et al., 1986; Sirgue et al., 2008; Vigh and Starr, 2008). The final solution to a geophysical inverse problem like FWI might differ significantly from any ideal solution due to data incompleteness, errors in the model parameterization, violation of assumptions (e.g., assuming an acoustic model to invert elastic measured data), noise, intrinsic issues with the specific mathematical description of the problem (e.g., under-determined problems), etc. Seismic inverse problems are, in general, ill-posed and ill-conditioned; thus, their null space can be significantly large and many different solutions (models) can exist that fit a given dataset equally well.

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Technology: Information Technology > Artificial Intelligence > Representation & Reasoning > Optimization (0.73)

Acoustic finite-difference modeling is playing an increasingly important role in seismic imaging (e.g. in reverse time migration) but the additional cost of elastic finite-difference modeling restricts its use in commercial imaging technology. The cost of full elastic finite-difference modeling can exceed the cost of acoustic modeling in the same velocity model by two orders of magnitude or more. A technique is described that corrects an acoustic finite-difference simulation for elastic effects. It is based on calculating the errors in the elastic wave equation using the acoustic simulation as an approximate solution. The errors are used to generate an effective source field for an additional acoustic simulation that calculates a correction to the wavefield produced in the original acoustic simulation. The cost of this approach is greater than that of an acoustic simulation but much less than that of a full elastic simulation.

For many applications in imaging and reservoir characterization (reverse time migration, waveform inversion, etc.), we require accurate simulations of seismic wave propagation. To realistically model the Earth, these are needed for elastic, anisotropic and anelastic models. The finite-difference method is widely used in this context as it is robust, simple to implement, and offers a good balance between accuracy and efficiency. However, it is still a computational challenge to perform elastic, anisotropic finite-difference simulations in three dimensions, so approximate calculations are often performed in an equivalent acoustic model. Even for P waves, the amplitudes of the first arrivals in the acoustic medium differ from those in the elastic medium. The objective of this paper is to describe a scheme whereby the acoustic wavefield can be partially corrected for elastic effects without incurring the cost of the full elastic computation. Consider two models, one acoustic and the other elastic, designed so that the density and acoustic/P-wave velocity fields match. For a pressure source, only P waves will be excited, so the solutions in the acoustic medium and for P waves in the elastic medium are expected to be very similar, at least in a limited time window around the first arrivals. The most significant differences will occur in the amplitudes of reflected and transmitted P waves from interfaces (or pseudointerfaces where properties vary rapidly). In the regions away from interfaces, properties are either homogeneous or varying slowly and smoothly and the coupling between P and S waves is insignificant. The objective is to correct the acoustic solution for elastic effects at interfaces, without incurring the cost of the full elastic solution. This paper describes a method to correct acoustic simulations for some of the effects of elasticity. We hope to correct the amplitudes of the P-wave arrivals for the effects of elasticity, particularly those caused by reflection and transmission coefficients at interfaces, at a cost considerably less than the cost of full elastic simulations. We do not expect to simulate the shear waves generated at interfaces. If necessary, the process can be applied iteratively to improve the accuracy of the correction.

acoustic equation, acoustic finite-difference simulation, acoustic simulation, acoustic solution, amplitude, elastic correction, elastic effect, elastic equation, elastic finite-difference correction, equation, error, interface, method, model, particle, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, solution, source, Upstream Oil & Gas, Wave

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

Özbek, Ali (Schlumberger) | Vassallo, Massimiliano (WesternGeco) | Özdemir, Kemal (WesternGeco) | Eggenberger, Kurt (WesternGeco) | van Manen, Dirk-Jan (WesternGeco)

basis function, component, crossline, crossline direction, Crossline Wavefield Reconstruction, generalized matching pursuit, GMP, joint interpolation, multi-component streamer, particle, pursuit, reconstruction, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, separation, signal, Upstream Oil & Gas, wavefield

SPE Disciplines: Reservoir Description and Dynamics > Reservoir Characterization > Seismic processing and interpretation (1.00)

In seismic data reconstruction, algorithms tend to fall into one of two categories, being rooted in either signal processing or the wave equation. Examples of the former include Spitz (1991), G¨ul¨unay (2003), Liu and Sacchi (2004), Hennenfent and Herrmann (2006), and Naghizadeh and Sacchi (2007), while examples of the later include Stolt (2002), Chiu and Stolt (2002), Trad (2003), Ram´ırez et al. (2006), and Ram´ırez and Weglein (2009). SPDR2 belongs to the family of wave equation based methods for data reconstruction. It differs from previous efforts in its parameterization of model space, being based on shot-profile migration (e.g. Biondi, 2003) and de-migration operators. Additionally, it relies on data fitting methods such as those used in Trad (2003), rather than direct inversion and asymptotic approximation which are used in, for example, Stolt (2002). A challenge in data reconstruction is alias. In particular, when aliased energy is present and interferes with signal, their separation becomes challenging (but, not impossible). A recent example of data reconstruction is Naghizadeh and Sacchi (2007). They use the non-aliased part of data to aid in the reconstruction of the aliased part of data. An alternative approach is to transform data via some operator that maps from data space to some model space, and such that in that model space, the corresponding representation of signal and alias are separable. This is a common approach in many signal processing methods, and is also the approach that we take in SPDR2. In particular, the SPDR2 model space is the sum of constant velocity shot-profile migrated gathers (i.e. a sum of common shot image gathers). This means that the SPDR2 model space is a representation of the earth’s reflectors parameterized by pseudo-depth (i.e. depth under the assumption of a constant migration velocity model) and lateral position. We will show that under the assumption of limited dips in the earth’s reflectors, the SPDR2 model space allows for the suppression of alias while preserving signal, thus allowing for the reconstruction of aliased data. We begin with a description of shot-profile migration and demigration built from the Born approximation to the acoustic wave-field and constant velocity Green’s functions. We apply shot-profile migration to an analytic example in order to illustrate its mapping of signal and alias from data space (shot gathers) to model space.

dimensional shot-profile, geophysics, interpolation, list, mapping, Ramirez, reconstruction, Reservoir Characterization, reservoir description and dynamics, Sacchi, SEG Denver, seismic processing and interpretation, seismic record, seismic trace interpolation, shot-profile migration data reconstruction, Stolt, Upstream Oil & Gas, Weglein

Wave equation solutions based on finite-differences is a standard technique and has been widely used for seismic forward modeling and reverse-time migration. However, the time step for the explicit method is restricted by the stability condition and to obtain good results both the spatial and time derivatives need to be computed with accurate operators. This can be achieved using higher order finite-difference schemes or very fine computational grids. However, both approaches increase the computational cost. On the other hand, numerical dispersion normally appears in the finite-difference results and can contaminate the signals of interest. Numerical dispersion noise is a very well known problem in finite-difference methods and several algorithms have being proposed to obtain seismic modeling sections and migration results free from this noise. In this paper, we propose to use the finite-difference technique together with a predictor-corrector method to obtain an efficient algorithm for seismic modeling and reverse time migration. First, we derive a new wave equation which we call the anti-dispersion wave equation. Then, we present some numerical results to demonstrate that the finite difference scheme based on this new anti-dispersion wave equation can be used as a new tool for seismic modeling and migration, producing little numerical dispersion compared with the original wave equation but requiring slightly more computational cost.

The finite-difference method, one of the most popular methods of numerical solution of partial differential equations, has been widely used in seismic modeling (Alford et al., 1974) and migration (Claerbout, 1985). The numerical evaluation of the derivatives appearing in the wave equation are obtained by a Taylor series expansion. The accuracy of these derivative operators is dependent on the order of approximation, i.e., the number of terms used in the Taylor series. Also for a given order, the accuracy of the derivatives calculation depends on the grid spacing. The finite-difference solution of the wave equation, even in an isotropic medium, has numerical phase and group velocity that are different for those of the true medium. The spatial frequency components propagating at different velocities produce a distortion of the waveform as propagation proceeds. This numerical error, also called numerical dispersion, can be attenuated by either reducing the grid size or using a higher order finite difference schemes (Liu et al., 2010; Stoffa and Pestana, 2009; Pestana and Stoffa, 2009). A small grid size helps to increase the accuracy but results in a larger number of grid points to represent the model. Thus, there is a substantial increase in computational cost. However, a higher order operator allows a coarser grid but also brings an increase in computation saving compared to a lower order scheme. In this paper we are proposing a new approach to mitigate the numerical dispersion by using a modified wave equation. The new wave equation appears using the predictor-corrector method or Charlie’s method Dey and Dey (1983). The objective is to attenuate the numeric dispersion that appears in the seismic modeling when we use the traditional wave equation.

acoustic wave equation, anti-dispersion wave equation, application, equation, explicit predictor-corrector solver, geophysics, Leveille, list, predictor-corrector method, rapid expansion method, Reservoir Characterization, reservoir description and dynamics, Reverse Time Migration, SEG, SEG Denver, seismic modeling, seismic processing and interpretation, Upstream Oil & Gas

SPE Disciplines:

Wave extrapolation in time plays an important role in seismic imaging (reverse-time migration), modeling, and full waveform inversion. Conventionally, extrapolation in time is performed by finite-difference methods (Etgen, 1986). Spectral methods (Tal-Ezer et al., 1987; Reshef et al., 1988) have started to gain attention recently and to become feasible thanks to the increase in computing power. The attraction of spectral methods is in their superb accuracy and, in particular, in their ability to suppress dispersion artifacts (Chu and Stoffa, 2008; Etgen and Brandsberg-Dahl, 2009). Theoretically, the problem of wave extrapolation in time can be reduced to analyzing numerical approximations to the mixeddomain space-wavenumber operator (Wards et al., 2008). In this paper, we propose a systematic approach to designing wave extrapolation operators by approximating the space-wavenumber matrix symbol with a lowrank decomposition. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. The optimized separable approximation or OSA (Song, 2001) was previously employed for wave extrapolation (Zhang and Zhang, 2009; Du et al., 2010) and can be considered as another form of lowrank decomposition. However, the decomposition algorithm in OSA is significantly more expensive, especially for anisotropic wave propagation, because it involves eigenfunctions rather than rows and columns of the original extrapolation matrix. Our algorithm can also be regarded as an extension of the interpolation algorithm of Etgen and Brandsberg-Dahl (2009), with optimally selected reference velocities and weights. Another related method is the Fourier finite-difference (FFD) method proposed by Song and Fomel (2010). FFD may have an advantage in efficiency, because it uses only one pair of forward and inverse Fast Fourier Transforms per time step. However, it does not offer flexible controls on the approximation accuracy.

The algorithm does not require, at any step, access to the full matrix W, only to its selected rows and columns. Once the decomposition is complete, it can be used at every time step during the wave extrapolation process.

Our next example (Figure 3) corresponds to wave extrapolation in a 2-D smoothly variable isotropic velocity field. As shown by Song and Fomel (2010), the classic finite-difference method tends to exhibit dispersion artifacts with the chosen model size and extrapolation step, while spectral methods exhibit high accuracy. The wavefield snapshot (Figure 5) confirms the ability of our method to handle complex models and sharp velocity variations.

algorithm, approximation, Artificial Intelligence, column, correspond, decomposition, equation, extrapolation, lowrank, lowrank approximation, lowrank symbol approximation, matrix, method, representative, Reservoir Characterization, reservoir description and dynamics, SEG, SEG SEG Denver, seismic processing and interpretation, seismic wave extrapolation, transform, Upstream Oil & Gas, Wave, wave extrapolation, wavefield

INTRODUCTION

For the development of hydrocarbon as well as geothermal reservoirs fluid injections through a borehole into the surrounding rock are frequently used. Such operations accomplished for enhancing hydrocarbon recovery (Economides and Nolte, 2000) or for creation of Enhanced Geothermal Systems (Majer et al., 2007) are accompanied by microseismic activity. This is because large parts of the Earths crust are in a sub-critical state where small, local stress changes may trigger microearthquakes. Introducing the so-called SBRC-approach (seismicity-based reservoir characterization) and assuming that the pore-fluid pressure perturbation induced by a fluid-injection obeys the law of linear diffusion Shapiro et al. (2002) show that the order of magnitude of the field-scale hydraulic permeability tensor can be estimated from the spatio-temporal distribution of observed microseismicity. The estimation of the hydraulic transport properties is done with the concept of the so-called ’triggering front’ providing an approximate outermost envelope of the distances between event locations and the injection point r as function of the time t elapsed since beginning of injection. However from field and laboratory experiments one has understood that the assumption of linear pore-fluid pressure diffusion is not always valid (Shapiro and Dinske, 2009). Pore-fluid pressure can strongly impact fluid transport properties. Lie et al. (2009) show a significant influence of effective pressure on permeability of several tight gas sandstone samples as well as for a granite rock. Yilmaz et al. (1994) simulate 1-D pore-fluid pressure profiles with a pressure-dependent permeability. Based on laboratory studies their suggested model shows an exponential relation of permeability on pore-fluid pressure. On that basis Hummel and M¨uller (2009) simulate and analyse synthetic clouds of 1-D and 2-D microseismicity based on nonlinear pore-fluid pressure diffusion. We use the triggering front (equation 1) to obtain heuristic estimates of hydraulic diffusivity from microseismic clouds.

analysis, diffusion, diffusivity, equation, estimate, exponential, flow in porous media, Fluid Dynamics, front, good agreement, hydraulic diffusivity, hydraulic transport, injection, interaction, microseismicity, nonlinear fluid-rock, permeability, profile, property, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, Shapiro, spatiotemporal pore-fluid, synthetic cloud, Upstream Oil & Gas

SPE Disciplines:

Li, Yandong (Research Institute of Petroleum Exploration and Development, PetroChina Co. Ltd) | Liu, Wei (Research Institute of Petroleum Exploration and Development, PetroChina Co. Ltd) | Zhang, Yan (Research Institute of Petroleum Exploration and Development, PetroChina Co. Ltd)

analysis, class, class II well, curvature, dissolution, distribution, dolomite, dolomite reservoir, Dolomite reservoir delineation, formation, fracture, paleotopography, porous dolomite reservoir, reservoir, Reservoir Characterization, reservoir description and dynamics, seismic processing and interpretation, target formation, Upstream Oil & Gas, well

Oilfield Places:

- North America > United States > Texas > Permian Basin (0.98)
- North America > United States > New Mexico > Permian Basin (0.98)

clay, clay mineral, clay-bound water, conductivity, electrical conductivity, equation, estimate, formation evaluation, free water, function, mineral, model, porosity, Reservoir Characterization, reservoir description and dynamics, resistivity, resistivity transform, sand, SEG SEG Denver, seismic processing and interpretation, shale, transform, Upstream Oil & Gas, water

analytic signal, anomaly, co sin, contact, deconvolution, dip, estimate, gravity, Hilbert transform, magnetic contact, magnetic profile, magnetic sheet, method, multiple-source werner deconvolution, Reservoir Characterization, reservoir description and dynamics, sheet, sin, source, structural dip, tan, Upstream Oil & Gas

Thank you!