Abstract
This paper deals with the problem of estimating fracture planes, given only the data at borehole intersections with fractures. We formulate an appropriate model for the problem and give a solution to fitting the planes using a Markov chain Monte Carlo (MCMC) implementation. The basics of MCMC are presented, with particular emphasis given to reversible jump, which is required for changing dimensions. We also give a detailed worked example of the MCMC implementation with reversible jump since our implementation relies heavily on this new methodology. The methods are tested on both simulated and real data. The latter is a unique data set in the form of a granite block, which was sectioned into slices. All joints were located and recorded, and the joint planes obtained by stacking strike lines.
This work is important in the risk assessment for the underground storage of hazardous waste. Problems and extensions are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baecher GB, Lanney NS, Einstein HH (1977) Statistical descriptions of rock properties and sampling. In: Proceedings of the 19th symposium on rock mechanics. American Institute of Mining, Metallurgical and Petroleum Engineers, New York, pp 56–65
Billaux D, Chiles JP, Hestir K, Long J (1989) Three-dimensional statistical modelling of a fractured rock mass—an example from the Fanay-Augéres mine. Int J Rock Mech Min Sci Geomech 26(3):281–299
Chiles JP (1988) Fractal and geostatistical methods for modelling a fracture network. Math. Geol. 20(6):631–654
Chiles JP (1989) Modélisation géostatistiques de réseaux des fractures. In: Armstrong M (ed) Geostatistics (Proceedings of the 3rd international geostatistics congress), vol. 1. Kluwer Academic Publishers, Dordrecht, pp 57–76
Chiles JP, de Marsily G (1993) Stochastic models of fracture systems and their use in flow and transport modelling. In: Bear J, Tsang CF, de Marsily G (eds) Flow and contaminant transport in fractured rock. Academic Press, San Diego, pp 169–236
Committee on Fracture Characterization and Fluid Flow—U.S. National Committee for Rock Mechanics (1996) In: Rock fractures and fluid flow—contemporary understanding and applications. National Academy Press, Washington, pp 337–368
Dershowitz WS, Einstein HH (1988) Characterizing rock joint geometry with joint system models. Rock Mech Rock Eng 21:21–51
Dershowitz W, LaPointe P (1994) Discrete fracture approaches for oil and gas applications. In: Nelson PP, Laubach SE (eds) Proceedings of the Northern American rock mechanics symposium, Austin, TX. Balkema, Rotterdam, pp 19–30
Dershowitz W, Follin S, Eiben T, Andersson J (1999) Alternative models project—discrete fracture network modelling for performance assessment of Aberg. Technical report R-99-43, Swedish nuclear fuel and waste management Co., Stockholm
Einstein HH (2003) Uncertainty in rock mechanics and rock engineering—then and now, In: Proceedings of the 10th international congress of the ISRM. The South African institute of mining and metallurgy symposium series S33, vol. 1, pp 281–293
Farmer CL (1988) The generation of stochastic fields of reservoir parameters with specified geostatistical distributions. In: Mathematics of oil production. Oxford science publications. Clarendon Press, Oxford, pp 235–252
Green PJ (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4):711–732
Guardiano FB, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics Troia 1992, vol. 1. Kluwer Academic, Dordrecht, pp 133–144
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109
Kulatilake PHSW, Wathugala DN, Stephansson O (1993) Joint network modelling with a validation exercise in Stripa mine, Sweden. Int J Rock Mech Min Sci Geomech 30(5):503–526
Long JCS, Billaux DM (1987) From field data to fracture network modelling: an example incorporating spatial structure. Water Resour Res 23(7):1201–1216
Long JCS, Witherspoon PA (1985) The relationship of the degree of interconnection to permeability in fracture networks. J Geophys Res B 90(4):3087–3098
Mardia KV, Jupp PE (1999) In: Directional statistics. Chichester, Wiley, p 429
Martin JA, Dowd PA, Mardia KV, Fowell RJ, Xu C (2005) A three-dimensional data set of the fracture network of a granite. http://www.leeds.ac.uk/StochasticRockFractures/
Matheron G (1975) In: Random sets and integral geometry. Wiley, New York, p 261
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1091
Odling NE (1992) Permeability and simulation of natural fracture patterns, in Structural and Tectonic modelling and its application to petroleum geology. Nor Pet Soc Spec Publ 1:365–380
Ripley BD (1992) Stochastic models for the distribution of rock types in petroleum reservoirs. In: Walden AT, Guttorp P (eds) Statistics in the environmental and earth sciences. Arnold, London, pp 247–282
Sahu S (2000) Tutorial lectures on MCMC, p 21. www.maths.soton.ac.uk/staff/Sahu/utrecht/mcmc.pdf
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd edn. Chichester, Wiley
Wang L (1995) Modelling complex reservoir geometries with multiple-point statistics. SCRF report
Wen R, Sinding-Larsen R (1997) Stochastic modelling and simulation of small faults by marked point processes and Kriging. In: Baafi EY, Schofield NA (eds) Geostatistics Wollongong ’96, vol. 1. Kluwer Academic, Dordrecht, pp 398–414
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mardia, K.V., Nyirongo, V.B., Walder, A.N. et al. Markov Chain Monte Carlo Implementation of Rock Fracture Modelling. Math Geol 39, 355–381 (2007). https://doi.org/10.1007/s11004-007-9099-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11004-007-9099-3