Seismic modelling

Continuing the discussion of 4D seismic data for constraining simulation models, this section gives a brief overview of practical methods of seismic modelling. These methods were developed with the purpose of modelling single surveys rather than 4D differences of course, but today’s view at predicting 4D signal is based on differencing modelled data of individual surveys as it happens with the observed data: surveys are acquired independently (different logic applies in seismic inversion though where individual surveys can be treated as parts of a response of the same system – the reservoir, and processed simultaneously therefore).

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The objective of seismic modelling is to predict a seismogram that can be compared to the recorded seismic data in order to infer properties of the subsurface rocks. In the context of 4D seismic, the assumed changes in pressure and saturation will perturb our initial modelled seismogram which can be validated by comparing the synthetics to the observed data. Different methods of seismic modelling can be classified as direct methods, integral-equation methods and ray-tracing methods. For a detailed review of seismic modelling methods see Carcione et al, 2002; Margrave and Manning, 2004;

and Krebes, 2004.

In the direct methods, the wave field is simulated directly by solving the wave equations on a grid covering the full geological model. For this reason, these methods are also called full wave equation methods. The solution for the wave field can be very accurate at the expense of computation time which can be a very significant factor. Examples of direct methods, in order of increasing accuracy, are finite-difference, pseudospectral, and finite-element methods. The drawback of the finite-difference method is that the solution is distorted by the numerical dispersion on the grid. In the pseudospectral method the problem is partly alleviated by using optimum number of samples per wavelength due to working in the wavenumber domain. In the finite-element method, the wave equation is solved exactly (rather than using finite-difference derivative approximations) in a number of finite regions.

The integral-equation methods originate from integral representation of wave field and are based on Huygens principle which states that the wave field can be represented as a superposition of wave fields from volume point sources or boundary point sources. These two representations are addressed by volume integral equations and boundary integral equations methods. These methods are more restrictive than the direct methods but perform well on models with small inclusions, cracks or fractures in them.

The ray-tracing, or asymptotic, methods do not model the full wave field and therefore greatly benefit from the resulting modelling speed up. Such methods are based on representation of a seismogram as a superposition of reflection events having different arrival times and different amplitudes which is achieved by using an approximate solution to the wave equation. The method is capable of modelling any style of geological sections and produces accurate results for arrival times. The drawbacks of the ray tracing method are that it is not accurate near critical offsets and near caustic zones (where rays converge or focus), does not resolve thin beds well because the

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method is a high frequency approximation assuming that the medium properties change slowly within dominant wavelength. This leads to difficulties in modelling diffractions and subsequent migration of the traces.

It is also possible to model the target reservoir or other area of interest more accurately while leaving the rest of the model, e.g. the overburden, for the faster methods. This is called hybrid modelling which can combine finite-difference method with faster algorithms such as ray tracing (Lecomte, 1996; Hokstad et al, 1998; Gjøystdal et al,

2002).

When calculation of the seismogram requires many iterations not only to cover the volume of the geological model but also to generate many realisations of such seismic cubes, the speed of seismic modelling may become the main factor. For example, in seismic inversion, many realisations of seismic traces are generated during the fitting to the observed data. A particular case of a more general inverse problem is seismic history matching of simulation models which involves frequent rebuilding of the seismic predictions. In these situation, a popular method for seismic modelling is the simplest 1D convolution method. This has been used for calculating synthetic seismograms since the 1950s mainly to tie synthetic seismogram in the well to the observed seismic using the density and sonic logs.

The method is based on a convolutional model which is derived from the Green’s theorem stating that the seismogram is a convolution of a source waveform with the impulse response of Earth. The convolutional model is given by (Yilmaz, 2001):

( ) ( ) ( ) ( ) (2.13)

where x(t) is the recorded seismogram, w(t) is the basic seismic wavelet, r(t) is the Earth’s impulse response, n(t) is the random ambient noise, denotes convolution operator. The random noise in the equation originates from instrumental errors of the recorders, poor geophone coupling, environmental noises. A convolutional model of a seismogram with noise is illustrated in Figure 2.11.

1D convolution method assumes that that the earth is horizontally layered locally, and does not include multiples, converted waves, and attenuation effects, although, it offers excellent level of details in z direction (Margrave and Manning, 2004). This is why direct comparisons between 1D convolution and more precise methods such as finite- difference method indicate that the former lacks lateral coherency and horizontal

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resolution but provides good amplitude information (Figure 2.12, Figure 2.13). At the same time, modelling of the waterfloods shows that the internal multiples in the flooded zones partially subtract out making them less significant for the monitoring projects than for reservoir characterisation (Shahin et el, 2011). 1D convolution modelling has been widely used for conditioning simulation models (see for example Landa and Kumar, 2011), predicting time-lapse seismic effects from CO2 sequestration (Arts et al,

2007; Li et al, 2013), monitoring steam

chamber growth (Lerat et al, 2010), and so far, is the most popular seismic forward modelling method in closed- loop reservoir model updating workflows.

Figure 2.12. Comparison between observed seismic (a), synthetic by 1D convolution method (b), 2D elastic modelling and processing (c) for the Sleipner CO2 injection project. 4D seismic monitoring

aims to image CO2 plume at it migrates in the reservoir. Observed data shows a prominent multi-

tier signature, comprising a number of bright sub-horizontal reflections, growing with time, interpreted as arising from up to nine discrete layers of high saturation CO2, each up to a few

metres thick. Modelled 4D images show good agreement with the observed data on the main target features. After Arts et al, 2007.

Figure 2.11. Convolutional model of a seismogram. Asterisks denotes convolution operator. After Yilmaz, 2001.

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Figure 2.13. Comparison between real seismic data (A), synthetic by 1D convolution method (B) and by full wave equation (C) for Shuaiba reservoir, Bu Hasa carbonate field. 4D feasibility study of this carbonate reservoir shows applicability of the method for monitoring injection in this reservoir. Results of 1D convolution and full wave equation modelling mostly agree except for some details related to the peripheral water injection areas. After Marvillet et al, 2007.