Journal of Central South University

, Volume 25, Issue 6, pp 1399–1408

# Oil-gas reservoir lithofacies stochastic modeling based on one- to three-dimensional Markov chains

• Zhi-zhong Wang (王志忠)
• Xiang Huang (黄翔)
• Yu-ru Liang (梁玉汝)
Article

## Abstract

Stochastic modeling techniques have been widely applied to oil-gas reservoir lithofacies. Markov chain simulation, however, is still under development, mainly because of the difficulties in reasonably defining conditional probabilities for multi-dimensional Markov chains and determining transition probabilities for horizontal strike and dip directions. The aim of this work is to solve these problems. Firstly, the calculation formulae of conditional probabilities for multi-dimensional Markov chain models are proposed under the full independence and conditional independence assumptions. It is noted that multi-dimensional Markov models based on the conditional independence assumption are reasonable because these models avoid the small-class underestimation problem. Then, the methods for determining transition probabilities are given. The vertical transition probabilities are obtained by computing the transition frequencies from drilling data, while the horizontal transition probabilities are estimated by using well data and the elongation ratios according to Walther’s law. Finally, these models are used to simulate the reservoir lithofacies distribution of Tahe oilfield in China. The results show that the conditional independence method performs better than the full independence counterpart in maintaining the true percentage composition and reproducing lithofacies spatial features.

## Key words

independence assumption Markov chain reservoir lithofacies small-class underestimation transition probability

# 基于一到三维马尔科夫链的油气储层岩相随机模型

## References

1. [1]
SALOMÃO M C, REMACRE A Z. The use of discrete Markov random fields in reservoir characterization [J]. Journal of Petroleum Science and Engineering, 2001, 32(2): 257–264.
2. [2]
LIANG Y, WANG Z, GUO J. Reservoir lithology stochastic simulation based on Markov random fields [J]. Journal of Central South University, 2014, 21(9): 3610–3616.
3. [3]
STREBELLE S. Conditional simulation of complex geological structures using multiple-point statistics [J]. Mathematical Geology, 2002, 34(1): 1–21.
4. [4]
KOLBJØRNSEN O, STIEN M, KJØNSBERG H, FJELLVOLL B, ABRAHAMSEN P. Using multiple grids in Markov mesh facies modeling [J]. Mathematical Geosciences, 2014, 46(2): 205–225.
5. [5]
CAO J H, YANG J C, WANG Y C, WANG D, SHI Y C. Extreme learning machine for reservoir parameter estimation in heterogeneous sandstone reservoir [J]. Mathematical Problems in Engineering, DOI: 10.1155/2015/287816.Google Scholar
6. [6]
CAO R, MA Y Z, GOMEZ E. Geostatistical applications in petroleum reservoir modelling [J]. Journal of the Southern African Institute of Mining and Metallurgy, 2014, 114(8): 625–631.Google Scholar
7. [7]
SOLEIMANI M, SHOKRI B J. 3D static reservoir modeling by geostatistical techniques used for reservoircharacterization and data integration [J]. Environmental Earth Sciences, 2015, 74(2): 1403–1414.
8. [8]
EIDSVIK J, MUKERJI T, SWITZER P. Estimation of geological attributes from a well log: An application of hidden markov chains [J]. Mathematical Geology, 2004, 36(3): 379–397.
9. [9]
LI Jun, XIONG Li, ZHANG Li, BIAN Guo, LIU Jian. Facies controlled stochastic modeling based on Markov chain models [J]. Progress in Geophysics, 2010, 25(1): 298–302. (in Chinese)Google Scholar
10. [10]
CARLE S F, FOGG G E. Transition probability-based indicator geostatistics [J]. Mathematical Geology, 1996, 28(4): 453–476.
11. [11]
NIKOOGOFTAR H, MEHRGINI B, BAHROUDI A, TOKHMECHI B. Optimization of the Markov chain for lithofacies modeling: An Iranian oil field [J]. Arabian Journal of Geosciences, 2015, 8(2): 799–808.
12. [12]
CAO G, KYRIAKIDIS P C, GOODCHILD M F. Combining spatial transition probabilities for stochastic simulation of categorical fields [J]. International Journal of Geographical Information Science, 2011, 25(11): 1773–1791.
13. [13]
HUANG X, WANG Z, GUO J. Prediction of categorical spatial data via Bayesian updating [J]. International Journal of Geographical Information Science, 2016, 30(7): 1426–1449.
14. [14]
VISTELIUS A B. On the question of the mechanism of formation of strata [J]. Doklady Akademii Nauk SSSR, 1949, 65(2): 191–194.Google Scholar
15. [15]
CARLE S F, FOGG G E. Modeling spatial variability with one and multidimensional continuous-lag Markov chains [J]. Mathematical Geology, 1997, 29(7): 891–918.
16. [16]
ELFEKI A, DEKKING M. A Markov chain model for subsurface characterization: theory and applications [J]. Mathematical Geology, 2001, 33(5): 569–589.
17. [17]
LI Jun, YANG Xiao, ZHANG Xiao, XIONG Li. Lithologic stochastic simulation based on the three-dimensional Markov chain model [J]. Acta Petrolei Sinica, 2012, 33(5): 846–853. (in Chinese)Google Scholar
18. [18]
LI W. Markov chain random fields for estimation of categorical variables [J]. Mathematical Geology, 2007, 39(3): 321–335.
19. [19]
LIU Zhen, HAO Tian, FANG Hui. Modeling of stochastic reservoir lithofacies with Markov chain model. Acta Petrolei Sinica, 2005, 26(5): 57–60. (in Chinese)Google Scholar
20. [20]
HUANG X, WANG Z, GUO J. Theoretical generalization of Markov chain random field from potential function perspective [J]. Journal of Central South University, 2016, 23(1): 189–200.
21. [21]
PAN H, LUO M, ZHANG Z, FAN Z. Lateral contrast and prediction of carboniferous reservoirs using logging data in Tahe oilfield, Xinjiang, China [J]. Journal of Earth Science, 2010, 21(4): 480–488.
22. [22]
YANG Kai, AI Di, GENG Jian. A new geostatistical inversion and reservoir modeling technique constrained by well-log, crosshole and surface seismic data [J]. Chinese Journal of Geophysics, 2012, 55(8): 2695–2704. (in Chinese)Google Scholar

© Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

## Authors and Affiliations

• Zhi-zhong Wang (王志忠)
• 1
• Xiang Huang (黄翔)
• 2
• Yu-ru Liang (梁玉汝)
• 3
1. 1.School of Mathematics and StatisticsCentral South UniversityChangshaChina
2. 2.Data Center (Beijing)Agricultural Bank of ChinaBeijingChina
3. 3.Department of Railway Locomotive and Electromechanical EquipmentShandong PolytechnicJi’nanChina