D025 Geostatistical Stochastic Elastic Iinversion - An Efficient Method for Integrating Seismic and Well Data Constraints

Transcription

1 D025 Geostatistical Stochastic Elastic Iinversion - An Efficient Method for Integrating Seismic and Well Data Constraints P.R. Williamson (Total E&P USA Inc.), A.J. Cherrett* (Total SA) & R. Bornard (CGGVeritas) SUMMARY We present a new stochastic algorithm for the inversion of multi-angle seismic reflection data and well logs. The probability distribution of P- and S-impedances in the subsurface is characterised by a set of representative and mutually-independent samples, or realisations. Each of these realisations not only fits the observed data to within defined tolerances, but also satisfies lateral and vertical spatial correlations from geostatistics. Thanks to a judicious choice of the form of the geostatistical constraints, and a number of optimisations that this form makes possible, this algorithm is extremely rapid. A typical inversion, generating 100 realisations over a study area, takes around one day of computation on a modest workstation. Nevertheless, the algorithm allows for complex stratigraphic geometries, which normally require more expensive inversion schemes.

2 Introduction A successful reservoir characterisation workflow requires integration of available data sources across all geoscience disciplines. As these different data are often highly dissimilar in resolution (both vertical and horizontal), in signal-to-noise ratio, and are more or less sensitive to different properties of the subsurface, a probabilistic approach seems a natural route towards integration. Bayes theorem provides a simple means to combine the individual probability distribution functions (PDFs) corresponding to each data type, and hence to derive an estimate of a given property, and its uncertainties, given all available data. In addition to its advantages in data integration, this probabilistic approach is of great value in risk analysis, as it yields estimates of uncertainty which are commensurate with the reliability and resolution of each data source. Furthermore, it allows new data constraints to be added to the model as these data become available, and is of great importance in downstream workflows (for example, in the construction of reservoir flow models which are consistent with seismic data, or in other words which fit the constraints due to those data to within acceptable limits). Of course, for practical reasons, most models will require a very large number of parameters, and hence it will be necessary (or at least simpler) to work with a population of representative samples of this PDF ( realisations ), rather than storing the distribution itself (Figure 1). Figure 1: A set of P-impedance realisations generated by geostatistical stochastic inversion of seismic and well data. Each realisation fits the data to a prescribed accuracy. The study from which this figure is taken is described by Escobar et al. (2006). The method we present here is an efficient means to generate realisations of P- and, optionally, S-impedances over an area of interest. It operates directly in a stratigraphic grid, with horizontal sampling equal to that of the seismic data, and irregular, geological-scale vertical sampling in the seismic time domain. Each realisation fits available seismic and well data to within user-supplied constraints, in addition to an a priori impedance distribution and spatial correlations from geostatistics. The realisations are mutually independent, and hence are representative samples of the correct a posteriori PDF. Single-trace inversion The acoustic inversion of seismic data at a single trace location serves as both an introduction to the method, and as a fundamental unit of the full, multi-trace algorithm. The model consists of a vertical sequence of cells (with geometry defined in time, but of arbitrary sampling), and each cell has a logarithm of P-impedance, stored in vector y. The advantage of working in logs of impedance, rather than directly in impedance itself, is that (within the scope of convolutional modelling) the synthetic zero-offset seismic trace s becomes a linear function of y, and can be written as follows:

3 s = W y, where the matrix W is a function of the seismic wavelet, and also depends upon the (time) position and thickness of the cells within the model. This scheme can be extended to the elastic case by grouping seismic traces of different angles in the vector s, and adding logarithms of S-impedances to y. In the elastic case, the construction of W will also rely upon the choice of a linearised AVA expression in terms of log impedances, and incorporate the corresponding AVA coefficients. If we assume a Gaussian prior distribution of log impedances, and a Gaussian noise model for the seismic data, the combined probability distribution of y due to both seismic data and the a priori model is also Gaussian, and is given by P(y seismic) = exp[-½[(wy-s 0 ) T C S -1 (Wy-s 0 ) + (y-μ) T C M -1 (y-μ)]] constant, where s 0 contains the observed seismic data, C S is the seismic data covariance matrix (often diagonal in practice), μ is the prior mean and C M the model covariance matrix (which depends on prior variances and any vertical correlations we wish to impose). If this trace is intersected by a logged well, an additional term may be added to this expression (which will require another covariance matrix to characterise the uncertainties on the log measurements, and due to upscaling issues). This Gaussian distribution may also be written in terms of a posterior mean π and covariance C T : P(y seismic) exp[-½(y-π) T C T -1 (y-π)] constant C T -1 = W T C S -1 W + C M -1 π = C T (W T C S -1 s 0 + C M -1 μ) At this point, the only remaining task is to sample log impedance traces y i from this multivariate Gaussian distribution. This can be done by eigenvector decomposition of C T -1, which provides an orthogonal basis along which the sampling may be carried out as a series of 1-D Gaussian operations, the eigenvalues giving the inverse variances associated with each axis. For reasons of performance, we may also use an alternative scheme based on the Cholesky decomposition of C T -1. However, both these operations take O(N 3 ) time in the number of model parameters to be sampled simultaneously. It should be noted that, in the special case of regular sampling of impedances in time (and hence regular, parallel layering across the region of investigation), W becomes a Toeplitz matrix, and the costly eigenvector or Cholesky decomposition of C T -1 can be replaced by a Fourier transform (Buland et al. 2003). However, the method we present in this paper retains the complete decomposition strategy, as although relatively computationally expensive, it is much less restrictive, and allows a wider range of geological cases to be treated. Multi-trace inversion In multi-trace geostatistical inversion, we impose lateral correlation between neighbouring traces in the impedance realisations. For this reason, the extension of the above single-trace algorithm to generate impedance realisations for an entire stratigraphic grid requires certain manipulations in order to be tractable. It is common for the number of traces in such an inversion to number in the millions. Although it is possible to write the joint PDF for all traces in the same manner as for one, including lateral correlations (variograms) into the model covariance matrix, the resulting system would be too unwieldy for the subsequent operations required of it, and in particular would preclude the use of eigenvector or Cholesky decomposition.

4 The problems posed by simultaneous inversion of all traces can be avoided by use of a trace-by-trace Sequential Gaussian Simulation (SGS), as for example used by Haas and Dubrule (1994). This technique allows the generation of impedance realisations by visiting each trace in turn, but modifying the prior model term to account for the conditional probability distribution due to the previously sampled traces. This conditional probability is computed through the use of Kriging techniques, and it should be noted that the resulting distribution is a function of each particular realisation, rather than identical for all. The order in which traces are visited within the SGS is theoretically arbitrary (Figure 2). However, in practical applications the Kriging is performed with only a subset of previouslysimulated data points, in order to reduce computational cost, enhance numerical stability and to permit efficient parallelisation of the algorithm. Usually, a small number of nearby neighbours (for example, eight), chosen to have maximum correlation with the current trace, are sufficient to provide an excellent approximation to the true Kriged distribution. However, this approximation can sometimes lead to poor results, typically when a nearby data value with low uncertainty is outside the selected subset of neighbours (but which themselves have high uncertainties). A practical solution to this problem is to prefer to start the SGS by visiting the traces which will exhibit low posterior variances in other words, those with the strongest data constraints, such as intersecting well logs, followed by zones of high seismic signal quality, leaving zones without seismic coverage to last. This precaution is sufficient to ensure that the realisations follow the desired levels of fit to geostatistics and all available well and seismic data. Figure 2: The mean and standard deviation of P-impedance after inversion. The choice of random path for the SGS (or in the third case, an ordered list of traces) is arbitrary. Within the scheme of visiting traces in descending order of data constraint, many possibilities exist for different paths to be taken in the SGS. Certain geometries may be desirable, for example, to aid parallelisation by delineating compartments of the study area. Typically, a randomised path is chosen where it will not interfere with implementation constraints. Haas and Dubrule (1994) take the additional precaution of a different random path for each realisation, as their algorithm generates a single realisation per SGS. In this method, we choose to share the same random path between realisations, as it allows for improved computational efficiency.

5 Conclusions The geostatistical stochastic inversion scheme described here is an efficient means to generate large numbers of impedance realisations which fit seismic and well data to desired levels of accuracy. A typical elastic inversion, with e.g. 100 realisations, can be carried out in a day with a dual-processor workstation. This efficiency is made possible by the imposition of certain constraints on the form of the prior geostatistics, which render the posterior distribution of log impedances Gaussian. In both speed and generality, this method lies somewhere between the more general, but significantly more expensive techniques based on simulated annealing (Debeye et al., 1996), Markov Chain Monte Carlo simulation (Contreras et al., 2005) and Monte Carlo with a posteriori comparison to seismic (Haas and Dubrule, 1994); and the fast but geometrically restrictive method of Buland et al. (2003). An additional advantage of our method is that it allows a precise level of fit to each data source, and thus the fidelity (or lack thereof) is respected. For example, the scheme naturally accommodates spatially-varying levels of seismic signal-to-noise ratios (Figure 3). The variation between the realisations generated by the inversion characterise the inherent uncertainties in impedance, and hence the knowledge of the subsurface that we cannot probe with the band-limited seismic and spatially-sparse well data available. An integrated geomodelling chain can take these impedance realisations as a starting point for depth conversion and a stochastic population of geological and dynamic properties. Figure 3: A comparison between real and synthetic seismic data for a single realisation, and the corresponding residuals. The horizontal bands of noise are from the buffer zones required for seismic inversion, and are outside the zone of interest. For this test, a lower signal-tonoise ratio was specified on the right, and this is reflected in higher amplitude residuals. References Buland, A., Kolbjørnson, O. and Omre, H., Rapid spatially coupled AVO inversion in the Fourier domain. Geophysics 68, no. 3, Contreras, A., Torres-Verdin, C., Kvein, K., Fasnacht, T. and Chesters, W., AVA Stochastic Inversion of Pre-Stack Seismic Data and Well Logs for 3D Reservoir Modeling. 67 th Ann. Mtg.: Eur. Assoc. of Geosci. And Eng., Expanded Abstracts, F014. Debeye, H.W.J., Sabbah, E. and van der Made, P.M., Stochastic inversion. 66 th Ann. Internat. Mtg.: Soc. of Expl. Geophys., Escobar, I., Williamson, P., Cherrett, A., Doyen, P.M., Bornard, R., Moyen, R. and Crozat, T., Fast geostatistical stochastic inversion in a stratigraphic grid. 76 th Ann. Internat. Mtg.: Soc. of Expl. Geophys., Haas, A. and Dubrule, O., Geostatistical inversion A sequential method of stochastic reservoir modelling constrained by seismic data. First Break 12, no. 11,