Conceptual Framework of Finite Markov Chain Stochastic Processes and Their Application to Representative Geological Case Studies
Zahid A. Khan *
Directorate Geology & Mining, Lucknow, India.
R. C. Tewari *
Department of Geology, Sri Jai Narain Post Graduate College, Lucknow- 226001, India.
*Author to whom correspondence should be addressed.
Abstract
Stratigraphy is a key concept in understanding the depositional domain of a sedimentary unit and its prospects for the exploration of coal and hydrocarbon. Generally, stratigraphy analysis uses a convenient qualitative approach to interpret sedimentary environments and models are reconstructed based on visual description is well received among sedimentological researchers to conduct depositional and lithofacies analysis. The interpretations and environmental models derived from such generally subjective data are therefore wanting. New statistical tools in stratigraphy including semi-quantitative and quantitative models for categorical data analysis were introduced to compliment subjective stratigraphic analysis. These quantitative methods based on a simple probability theory are used to predict the successive units and corresponding process in a sedimentary sequence. The Finite Markovian statistics of vertical and lateral variability were made possible by integrating Walther’s Law, which states that the vertical succession of sedimentary facies in a geological column reflects the lateral changes and succession of depositional environments, thus simulating both lateral and vertical facies variability. Minor erosional breaks are regarded as probabilistic within the model. Based on stochastic methods it is observed that the early Permian coal bearing cycles are auto-cyclic in nature. Coal measure cyclothems or fluvial fining-upward cycles around the world are good examples of sedimentary succession laid down under the control of Markovian process.
The present contribution aims to highlight the scope of stochastic processes-- including Finite Markov Chain, Quasi-independence, Markov Reversibility and Marginal homogeneity. These quantitative methods enables more predictability in analysis of stratigraphic patterns and quantify sedimentary cycles for proper correlation within and between oil bearing and coal bearing successions and it’s bearing on exploration of coal and hydrocarbon and their development. The data used in this work comes from the vertical stratigraphic outcrop sections/subsurface borehole logs from the early Permian Gondwana stratigraphy of peninsular India.
Keywords: Stochastic finite Markov chain, quasi-independence, reversible Markov chain, coal bearing sequences, lithofacies analysis