Investigation of generalized Markov chain performance in simulation of discrete variables in a case study

Geological modeling of heterogeneous facies plays an important role in the detection of stratigraphic uncertainty. In this research, three methods, Indicator kriging (IK), Sequential Indicator Simulation (SIS), and Generalized Coupled Markov Chain (GCMC) were applied to predict geological categories at unknown locations. Then the results of all three methods were compared.

There are various methods for estimating and determining the spatial variation of categorical variables using geological data and exploratory wells. One of the best of these methods is geostatistical methods. As new Geostatistical methods, the GCMC algorithm, one of the Markov chain models, has been used in the earth sciences to simulate categorical variables of sedimentary deposits. This method is based on the calculation of transition probability matrixes with respect to the direction and spatial variations between classes. Due to the realistic results and easy implementation, the GCMC method is a suitable tool for the initial predicting and modeling of categorical variables in sedimentary environments.

Methodology and Approaches:

In this study, one of the drilling profiles in block C of the Bostanabad Areshtenab limestone deposit was selected for modeling. At this point, three carbonate units can be distinguished from the 5 exploratory boreholes dataset. To build the prediction models, after transforming the coordinates into a stratigraphic coordinates system (unfolding the strata by vertical transformation), the vertical and horizontal variability and continuity structure of the three existing classes were modeled with indicator variograms and transition probabilities. Then the mentioned geostatistical prediction techniques were applied to generate the spatial variability models.

In general, this study suggested the application of three geostatistical prediction methods for constructing realistic subsurface models of the categorical variables. According to the results, the IK result represented the general occurrence trend better. However, the spatial variability structure could not be reflected sufficiently and clearly. Although, in the SIS results fine and subtle variations were reflected, the produced patterns were more scattered. As the result of this study, the GCMC method can reproduce the global statistics, spatial structural functions (transiograms), and more realistic subsurface models, especially with sparse data in sedimentary systems.