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Seismic reservoir characterization focuses on the prediction of reservoir properties based on the available geophysical and petrophysical data. The inverse problem generally includes continuous properties, such as petrophysical and elastic attributes, and discrete properties, such as lithology/fluid classes. We have developed a joint probabilistic inversion methodology for the prediction of petrophysical and elastic properties and lithology/fluid classes that combined statistical rock physics and Bayesian seismic inversion. The elastic attributes depend on continuous petrophysical variables, such as porosity and clay content, and discrete lithology/fluid classes, through a nonlinear rock-physics relationship together. The seismic model relates the elastic attributes, such as velocities and density, to their seismic response (reflectivity, traveltime, and amplitudes). The advantage of our integrated approach is that the inversion method accounts for the uncertainty associated to each step of the modeling workflow. The lithology/fluid classes are assigned by a Markov random field prior model to capture vertical continuity and vertical sorting of the lithology/fluid classes. Because rock and fluid properties are in general not Gaussian, a spatially coupled Gaussian mixture prior model based on the lithology/fluid classes is constructed. The forward geophysical operator includes a lithology-/fluid-dependent rock physics model and a linearized seismic model based on the convolution of the seismic wavelet with the reflectivity coefficient series. The solution of the inverse problem consists of the posterior distributions of petrophysical and elastic properties and lithology/fluid classes. We proposed an efficient Markov chain Monte Carlo algorithm to sample from the posterior models and assess the uncertainty. Our methodology is demonstrated on a seismic cross section from a survey in the Norwegian Sea, and it shows promising results consistent with well-log data measured at the well location as well as reliable prediction uncertainties.

Efficient assessment of convolved hidden Markov models is discussed. The bottom-layer is defined as an unobservable categorical first-order Markov chain, while the middle-layer is assumed to be a Gaussian spatial variable conditional on the bottom-layer. Hence, this layer appear as a Gaussian mixture spatial variable unconditionally. We observe the top-layer as a convolution of the middle-layer with Gaussian errors. Focus is on assessment of the categorical and Gaussian mixture variables given the observations, and we operate in a Bayesian inversion framework. The model is defined to make inversion of subsurface seismic AVO data into lithology/fluid classes and to assess the associated elastic material properties. Due to the spatial coupling in the likelihood functions, evaluation of the posterior normalizing constant is computationally demanding, and brute-force, singlesite updating Markov chain Monte Carlo algorithms converges far too slow to be useful. We construct two classes of approximate posterior models which we assess analytically and efficiently using the recursive Forward-Backward algorithm. These approximate posterior densities are used as proposal densities in an independent proposal Markov chain Monte Carlo algorithm, to assess the correct posterior model. A set of synthetic realistic examples are presented. The proposed approximations provides efficient proposal densities which results in acceptance probabilities in the range 0.10-0.50 in the Markov chain Monte Carlo algorithm. A case study of lithology/fluid seismic inversion is presented. The lithology/fluid classes and the elastic material properties can be reliably predicted. * torstein.fjeldstad@ntnu.no 1 arXiv:1710.06613v1 [physics.geo-ph]

A one-step approach for Bayesian prediction and uncertainty quantification of lithology/fluid classes, petrophysical properties and elastic attributes conditional on prestack 3D seismic amplitude-versus-offset data is presented. A 3D Markov random field prior model is assumed for the lithology/fluid classes to ensure spatially coupled lithology/fluid class predictions in both the lateral and vertical directions. Conditional on the lithology/fluid classes, we consider Gauss-linear petrophysical and rock physics models including depth trends. Then, the marginal prior models for the petrophysical properties and elastic attributes are multivariate Gaussian mixture models. The likelihood model is assumed to be Gauss-linear to allow for analytic computation. A recursive algorithm that translates the Gibbs formulation of the Markov random field into a set of vertical Markov chains is proposed. This algorithm provides a proposal density in a Markov chain Monte Carlo algorithm such that efficient simulation from the posterior model of interest in three dimensions is feasible. The model is demonstrated on real data from a Norwegian Sea gas reservoir. We evaluate the model at the location of a blind well, and we compare results from the proposed model with results from a set of 1D models where each vertical trace is inverted independently. At the blind well location, we obtain at most a 60 % reduction in the root mean square error for the proposed 3D model compared to the model without lateral spatial coupling.