A Pluri-Gaussian method is developed for facies variables in three dimensions to model vertical cyclicity related to facies ordering and rhythmicity. Cyclicity is generally characterised by shallowing-upward or deepening-upward sequences and rhythmicity by the repetition of facies at constant intervals along sequences. Both of these aspects are commonly observed in shallow-marine carbonate successions, especially in the vertical direction. A grid-free spectral simulation approach is developed, with a separable covariance allowing a dampened hole-effect to capture rhythmicity in the vertical direction and a different covariance in the lateral plane along strata, as in space-time models. In addition, facies ordering is created by using a spatial shift between two latent Gaussian functions in the Pluri-Gaussian approach. Rapid conditioning to data is performed via Gibbs sampling and kriging using the screening properties of separable covariances. The resulting facies transiograms can show complex patterns of cyclicity and rhythmicity. Finally, a three dimensional case study of shallow-marine carbonate deposits at outcrop shows the applicability of the modelling method.

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In siliciclastic and carbonate reservoirs, depositional facies are often described as being organized in cyclic successions that are overprinted by diagenesis. Most reservoir modeling workflows are not able to reproduce stochastically such patterns. Herein, a novel geostatistical method is developed to model depositional facies architectures that are rhythmic and cyclic, together with superimposed diagenetic facies. The method uses truncated Pluri-Gaussian random functions constrained by transiograms. Cyclicity is defined as an asymmetric ordering between facies, and its direction is given by a three-dimensional vector, called shift. This method is illustrated on two case studies. Outcrop data of the Triassic Latemar carbonate platform, northern Italy, are used to model shallowing-upward facies cycles in the vertical direction. A satellite image of the modern Bermuda platform interior is used to model facies cycles in the windward-to-leeward lateral direction. As depositional facies architectures are modeled using two Gaussian random functions, a third Gaussian random function is added to model diagenesis. Thereby, depositional and diagenetic facies can exhibit spatial asymmetric relationships. The method is applied in the Latemar carbonate platform that experiences syn-depositional dolomite formation. The method can also incorporate proportion curves to model non-stationary facies proportions. This is illustrated in Cretaceous shallow-marine sandstones and mudstones, Book Cliffs, Utah, for which cyclic facies and diagenetic patterns are constrained by embedded transition probabilities.

A Pluri-Gaussian method is developed for facies variables in three dimensions to model vertical cyclicity related to facies ordering and rhythmicity. Cyclicity is generally characterised by shallowing-upward or deepening-upward sequences and rhythmicity by the repetition of facies at constant intervals along sequences. Both of these aspects are commonly observed in shallow-marine carbonate successions, especially in the vertical direction. A grid-free spectral simulation approach is developed, with a separable covariance allowing a dampened hole-effect to capture rhythmicity in the vertical direction and a different covariance in the lateral plane along strata, as in space-time models. In addition, facies ordering is created by using a spatial shift between two latent Gaussian functions in the Pluri-Gaussian approach. Rapid conditioning to data is performed via Gibbs sampling and kriging using the screening properties of separable covariances. The resulting facies transiograms can show complex patterns of cyclicity and rhythmicity. Finally, a three dimensional case study of shallow-marine carbonate deposits at outcrop shows the applicability of the modelling method.