Incorporating spatial distribution into stochastic modelling of fractures: Multifractals and Lévy-stable statistics

Journal of Structural Geology

Volumn 20, Issue 4, 1998, Pages 473-486

  Belfield, William C ^a  

^a BP EXPLORATION   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; FRACTALS; FRACTURE; FUNCTIONS; MATHEMATICAL MODELS; RANDOM PROCESSES; STATISTICAL METHODS; STRAIN;

FRACTAL DIMENSION; LEVY-STABLE STATISTICS; MULTIFRACTALS; SPACING DISTRIBUTION FUNCTIONS; SPATIAL CLUSTERING;

TECTONICS;

FRACTAL ANALYSIS; FRACTURE MODELLING; LEVY-STABLE STATISTICS;

EID: 0032052363     PISSN: 01918141     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0191-8141(97)00118-1     Document Type: Article

References (52)

1. Barton, C. C. (1995) Fractal analysis of scaling and spatial clustering of fractures. In Fractals in the Earth Sciences, ed. C. C. Barton and P. R. LaPointe, pp. 141-178. Plenum Press, New York.
2. Barton, C. C. and Larson, E. (1985) Fractal geometry of two-dimensional fracture networks at Yucca Mountain, southwestern Nevada. In Proceedings of the International Symposium on Fundamentals of Rock Joints, ed. O. Stephannson, pp. 77-84, Bjorkliden, Sweden.
3. Becker, A. and Gross, M. R. (1996) Mechanism for joint saturation in mechanically layered rocks: an example from southern Israel. Tectonophysics 257, 223-237.
4. Belfield, W. C. (1992) Simulation of subseismic faults using fractal and multifractal geometry, (paper SPE 24751). Proceedings of the SPE Annual Technical Conference, 903-911.
5. Belfield, W. C. (1994) Multifractal characteristics of natural fracture apertures. Geophysical Research Letters 21, 2641-2644.
6. Belfield, W. C. and Sovich, J. (1995) Fracture statistics from horizontal wellbores. Journal of Canadian Petroleum Technology 34, 47-50.
7. Brooks, B. A., Allmendinger, R. W. and de la Barra, Garrido I. (1996) Fault spacing in the El Teniente Mine, central Chile: evidence for nonfractal fault geometry. Journal of Geophysical Research 101, 13633-13653.
8. Cacas, M. C., de Marsily, G., Tillie, B., Barbeau, A., Durand, E., Feuga, B. and Peaudecerf, P. (1990) Modeling fracture flow with a stochastic discrete fracture network: calibration and validation I. The flow model. Water Resources Research 26, 479-489.
9. Castaing, C., Halawani, M. A., Gervais, F., Chilès, J. P., Genter, A., Bourgine, B., Ouillon, G., Brosse, J. M., Martin, P., Genna, A., Janjou, D. (1996) Scaling relationships in intraplate fracture systems related to Red Sea rifting. Tectonophysics 261, 291-314.
10. Childs, C., Walsh, J. J. and Watterson, J. (1990) A method for estimation of the density of fault displacements below the limits of seismic resolution in reservoir formations. In North Sea Oil and Gas Reservoirs II, ed. A. T. Buller, E. Berg, O. Hjelmeland, J. Kleppe, O. Torsaeter and J. O. Aasen, pp. 309-318. Graham & Trotman, London.
11. Chilès, J. P. (1988) Fractal and geostatistical methods for modeling of a fracture network. Mathematical Geology 20, 631-654.
12. Chilès, J. P. and de Marsily, G. (1993) Stochastic models of fracture systems and their use in flow and transport modeling. In Flow and Contaminant Transport in Fractured Rock, ed. J. Bear, C. F. Tsang and G. de Marsily, pp. 169-231. Academic Press, San Diego.
13. Cowie, P. A., Sornette, D. and Vanneste, C. (1995) Multifraclal scaling properties of a growing fault population. Geophysical Journal International 122, 457-469.
14. Davis, A., Marshak, A., Wiscombe, W. and Cahalan, R. (1994) Multifractal characterizations of nonstationarity and intermittency in geophysical fields: observed, retrieved, or simulated. Journal of Geophysical Research 99, 8055-8072.
15. Einstein, H. H. and Baecher, G. B. (1983) Probabilistic and statistical methods in engineering geology specific methods and examples. Rock Mechanics and Rock Engineering 16, 39-72.
16. Evertsz, C. J. G. and Mandelbrot, B. B. (1992) Multifractal measures. In Chaos and Fractals, ed. H. O. Peitgen, H. Jurgens and D. Saupe, pp. 921-953. Springer, Berlin.
17. Feller, W. (1971) An Introduction to Probability Theory and Its Applications. J. Wiley & Sons, New York, Vol. II, p. 669.
18. Gauthier, B. D. M. and Lake, S. D. (1993) Probabilistic modeling of faults below the limit of seismic resolution in the Pelican Field, North Sea, offshore U.K. Bulletin of the American Association of Petroleum Geologists 77, 761-777.
19. Gillespie, P. A., Howard, C. B., Walsh, J. J. and Watterson, J. (1993) Measurement and characterisation of spatial distribution of fractures. Tectonophysics 226, 113-141.
20. Godderij, R. R. G. G., Chessa, A. G., Bruining, J. and Kreft, E. (1995) Conditional simulation of subseismic faults (paper SPE 30620). Proceedings of the SPE Annual Technical Conference, 939-951.
21. Gross, M. R. and Engelder, T. (1995) Strain accommodated by brittle failure in adjacent units of the Monterey Formation, U.S.A.: scale effects and evidence for uniform displacement boundary conditions. Journal of Structural Geology 17, 1303-1318.
22. Heffer, K. J. and Bevan, T. G. (1990) Scaling relationships in natural fractures - data, theory, and applications (paper SPE 20981). Proceedings of the Eurpoean Petrology Conference 2, 367-376.
23. Hestir, K., Chiles, J. P., Long, J. and Billaux, D. (1987) Three dimensional modeling of fractures in rock: from data to a regionalized parent-daughter model. In Flow and Transport Through Unsaturated Fractured Rock, ed. D. D. Evans and T. J. Nicholson. Geophysical Monograph, 42, pp. 133-140. American Geophysical Union, Washington, DC.
24. Hewett, T. A. (1986) Fractal distributions of reservoir heterogeneity and their influence on fluid transport (paper SPE 15386). Proceedings of the SPE Annual Technical Conference, 843-850.
25. Jackson, P. and Sanderson, D. J. (1992) Scaling of fault displacements from the Budajoz Cordoba shear zone, SW Spain. Tectonophysics 210, 179-190.
26. Knott, S. D., Beach, A., Brockbank, P. J., Brown, J. L., McCallum, J. E. and Welbon, A. I. (1996) Spatial and mechanical controls on normal fault populations. Journal of Structural Geology 18, 359-372.
27. Laubach, S. E. (1991) Fracture patterns in low-permeability gas reservoir rocks in the Rocky Mountain region, (paper SPF 21853). Proceedings of the SPE Rocky Mountain Low-Permeability Reservoir Symposium, pp. 501-510.
28. Long, J. C. S., Remer, J. S., Wilson, C. R. and Witherspoon, P. A. (1982) Porous media equivalents for networks of discontinuous fractures. Water Resources Research 18, 645-658.
29. Long, J. C. S. and Billaux, D. M. (1987) From field data to fracture network modeling: an example incorporating spatial structure. Water Resources Research 23, 1201-1216.
30. Main, I. (1996) Statistical physics, seismogenesis, and seismic hazard. Reviews of Geophysics 34, 433-462.
31. Mandelbrot, B. B. (1982) The Fractal Geometry of Nature. W.H. Freeman, New York.
32. Mantegna, R. N. (1994) Fast, accurate algorithm for numerical simulation of Lévy-stable stochastic processes. Physical Review (E) 49, 4677-4683.
33. Mantegna, R. N. and Stanley, H. E. (1995) Scaling behaviour in the dynamics of an economic index. Nature 376, 46-49.
34. Marrett, R. and Allmendinger, R. (1991) Estimates of strain due to brittle faulting: sampling of fault populations. Journal of Structural Geology 13, 735-738.
35. Montroll, E. W. and Shlesinger, M. F. (1982) On 1/f noise and other distributions with long tails. Proceedings of the National Academy of Sciences 79, 3380-3383.
36. Munthe, K. L., Orme, H., Holden, L., Damsleth, E., Heffer, K., Olsen, T. S. and Watterson, J. (1993) Subseimic faults in reservior description and simulation (paper SPE 26500). Proceedings of the SPE Animal Technical Conference, 843-850.
37. Narr, W. and Suppe, J. (1991) Joint spacing in sedimentary rocks. Journal of Structural Geology 13, 1037-1048.
38. Nicol, A., Walsh, J. J., Watterson, J. and Gillespie, P. A. (1996) Fault size distributions-are they really power-law. Journal of Structural Geology 18, 191-197.
39. Odling, N. E. (1992) Network properties of a two-dimensional fracture pattern. Pure and Applied Geophysics 138, 95-114.
40. Odling, N. E. (1997) Scaling and connectivity of joint systems in sandstones from western Norway. Journal of Structural Geology 19, 1257-1271.
41. Painter, S., Paterson, L. and Boult, P. (1995) Improved technique for the stochastic interpolation of reservoir properties (paper SPE 30599). Proceedings of the SPE Annual Technical Conference, 750-745.
42. Peacock, D. C. P. (1996) Field examples of variations in fault patterns at different scales. Terra Nova 8, 361-371.
43. Peacock, D. C. P. and Sanderson, D. J. (1994) Strain and scaling of faults in the Chalk at Flambrough Head, U.K. Journal of Structural Geology 16, 97-107.
44. Pollard, D. D. and Segall, P. (1987) Theoretical displacements and stresses near fractures in rock: with applications to faults, joints, veins, dikes, and solution surfaces. In Fracture Mechanics of Rocks, pp. 277-349, ed. B. K. Atkinson. Academic Press, London.
45. Rives, T., Razack, M., Petit, J-P. and Rawnsley, K. D. (1992) Joint spacing: analogue and numerical simulations. Journal of Structural Geology 14, 925-937.
46. Schlische, R. W., Young, S. S., Ackermann, R. V. and Gupta, A. (1996) Geometry and scaling relations of a population of very small rift-related normal faults. Geology 24, 683-686.
47. Tchalenko, J. S. (1970) Similarities between shear zones of different magnitudes. Bulletin of the Geological Society of America 81, 1625-1640.
48. Walsh, J. J. and Watterson, J. (1993) Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies. Journal of Structural Geology 15, 1509-1512.
49. Watterson, J., Walsh, J. J., Gillespie, P. A. and Easton, S. (1996) Scaling systematics of fault sizes on a large-scale range fault map. Journal of Structural Geology 18, 199-214.
50. Wu, H., Willemse, E. J. M. and Pollard, D. (1995) Prediction of fracture density and drainage distance in layered rock masses using borehole data. In Proceedings of Eurorock '94: Rock Mechanics in Petroleum Engineering, pp. 487-494, Delft. The Netherlands.
51. Yielding, G., Walsh, J. J. and Watterson, J. (1992) The prediction of small-scale faulting in reservoirs. First Break 10, 449-460.
52. Yielding, G., Needham, T. and Jones, H. (1996) Sampling of fault populations using sub-surface data: A Review. Journal of Structural Geology 18, 135-146.