Capturing Geological Realism in Stochastic Simulations of Rock Systems with Markov Statistics and Simulated Annealing

Abstract

Simulated annealing is a numerical algorithm that can be used to impose statistical structures on numerical grids representing heterogeneous rock or sediment. In this paper, we use the flexibility of simulated annealing to generate grids with Markov statistical structures. Our purpose is to transmit the rich geological information captured in Markov statistics into stochastic grids while maintaining the flexibility of annealing to honor field data. Performance issues that compromise annealing grids imbued with Markovian properties include scales of bedding or rock bodies relative to grid size, and the amount of geological complexity in the embedded Markov structures. The remedies to these issues include proper selection of grid size, careful choice of annealing type, and consideration of an alternative annealing stopping rule based on a chi-squared test statistic. If performance issues are overcome, complex stratal patterns such as higherorder dependency, cyclicity, and directionality can be replicated in grids by this method. In addition, accounting for variations in depositional rate allows for transference of Markov structures obtained from vertical boreholes to the horizontal dimension when other information is lacking. A field example using borehole data collected at the Gloucester special waste site near Ottawa, Canada, as well as synthetic examples, demonstrate the technique and performance issues.

Markov Models for Linking Environments and Facies in Space and Time (Recent Arabian Gulf, Miocene Paratethys)

“…Markov chains and weighted digraphs provided the opportunity to forecast changes in facies by manipulating transition likelihoods obtained from lateral patterns in a living landscape. This allows translating knowledge of facies transitions in core or outcrop (= time) for the reconstruction of ancient landscapes (= space) and vice versa (Doveton, 1994;Parks et al, 2000;Elfeki & Dekking, 2001 and the frequency of facies that can be expected in any time slice. Thus, the estimation of the lateral dimensions of reservoir rocks can be improved, for example.…”

“…Quantifi cation of the overall contribution of facies to the composition of the landscape and the spatial relationships among facies is attempted. More complex Markov models can be applied to use the information gained by our approach to predict facies outside the known outcrop (Parks et al, 2000;Elfeki & Dekking, 2001.…”

“…In numerous previous studies, Markov chains have shown that conditional probabilistic relationships (knowledge of an event/facies will defi ne probability of a following event/facies) exist among vertical facies successions (Krumbein, 1967;Krumbein & Greybull, 1967;Gingerich, 1969;Doveton, 1971Doveton, , 1994Miall, 1973;Carle & Fogg, 1997;Wilkinson et al, 1997Wilkinson et al, , 2003Lehrmann & Rankey, 1999;Parks et al, 2000;Davis, 2002). Stages of ecological community and landscape successions have Markovian properties (Pielou, 1969;Horn, 1975;Usher, 1979;Turner et al, 2001;Urban & Wallin, 2002), which can be expressed using graphs (Roberts, 1976;Cantwell & Forman, 1993;Bunn et al, 2000;Urban & Keitt, 2001;Urban, 2005).…”

“…The presence of comparable facies frequencies and conditional transition probabilities (i.e. the Markov property) in both systems would suggest a quantitative link between the two time-slices and between vertical and lateral succession of facies -since from Walther's Law it may be deduced that one should exist (Doveton, 1994;Parks et al, 2000;Elfeki & Dekking, 2001. Using discrete mathematical tools the expression of a quantitative link of temporal and spatial dynamics will be attempted in order to explore whether the relationship among facies in a landscape can be reduced to a simple graph or matrix encapsulating a model of the system's functioning that can then be used to predict changes in the spatial characteristics of facies through time, for example in changing envir onments.…”

Geostatistical modelling of cyclic and rhythmic facies architectures

“…The representation of asymmetry and facies ordering is, however, straightforward in one dimension with Markov chains (Carle and Fogg 1996;Parks et al 2000;Li 2007;Purkis et al 2012) or renewal processes (Matheron 1968), which are based on probabilities of transition between facies, but are difficult to generalise to two or three dimensions. Rhythmicity is another important aspect observed in vertical facies successions.…”