STUDY ON 3D FRACTAL DISTRIBUTION LAW OF THE SURFACE NUMBER IN ROCK MASS

4 4 Vol.4 No.4 005 Chinese Journal of Rock Mechanics and Engineering Feb. 005 1 1 1 (1. 03004. 1008) ( ) D S N S D N D = D S 1 D N ST SP N N S N = kn S k TU 45 A 1000 6915(005)04 0601 09 STUDY ON 3D FRACTA DISTRIBUTION AW OF THE SURFACE NUMBER IN ROCK MASS FENG Zeng-chao 1 ZHAO Yang-sheng 1 WEN Zai-ming 1 (1. Mining Technology Institute Taiyuan University of Technology Taiyuan 03004 China. Department of Mining China University of Mining and Technology Xuzhou 1008 China) Abstract Based on numerical simulation experiment the fractal distribution law of surface number in rock mass is proved. According to large amount of calculate and theoretical inducement the two important fractal parameters for fractured face are achieved. Through a great deal of calculation the relationship between D S and D fractal dimension D of trace is obtained which is D =D S 1 and D is not related to the other parameters including N S dip angle ST azimuth SP of 3D surface. The relationship of N S and the initial value of trace number N is obtained. There are projection relation between N and ST and SP and proportional relation between N and Ns which is N = kn S where k depends on surface projection. These relations provide the foundation for studying number and scale of surface in rock masses. Key words mining engineering surface in rock mass fractal law surface trace section plane of rock mass ( ) 1 003 08 05 003 10 4 (50404017 50134040 50174040) (1971 ) 001 E-mail zc-feng@163.com

60 005 (1) 0 ( 0 [1 ] ) 0 N( 0 ) () 0 8 δ 1 = 0 / 1 δ1 0 / N( 0 /) k (3) δ k = 0 / 3k 0 δ k 50 N( 0 / k ) [3 4] Fisher (4) 0 / k Bingham [5] N( 0 / k ) [6] (5) lg-lgn [7] Monte Carlo DS N( δ ) = N Sδ (1) [8 9] Monte Carlo N S [10] 1 D S 3 [11 14] (1) () (3) n r ST SP 1 3.1 0 = 1 N S = 1 D S =.40 3

4 4. 603 Y 3.3 C 0 = 1 n r 1 D S1 =.3 N S1 = 1 ST 1 = 30 SP 1 = ST B 75 D S =.6 N S = 1 ST = 60 SP = D 15 3 SP Z 1 O A X ( 3) 1 Fig.1 Relation of dip angle azimuth and normal vector 3 Table 3 The fractal parameters of trace on various section 1 plane with grouping-distributed 3D surface ( 1) 0 = 1 D N R 0 0 / 0 /4 0 /8 0 /16 0 /3 1 XOY 1.00.00 6.06 15.08 36.81 89.78 1.34 7 0.96 0 0.998 (D S =.40 N S = 1) Table 1 The fractal parameters of trace on various section plane with fully-random distributed 3D surface XOZ 0.45 0.96.95 7.5 18.3 47.41 1.364 9 0.41 9 0.999 (D S =.40 N S = 1) XOY 0.43 1.53 4.89 15.19 44.13 143.47 1.661 6 0.461 5 0.999 0 = 1 0 0 / 0 /4 0 /8 0 /16 0 /3 D N R XOY 0.75 1.95 5.36 14.8 39.43 104.78 1.431 8 0.739 0.999 YOZ 0.76.0 5.47 14.98 39.66 104.17 1.43 4 0.761 1.000 XOZ 0.77.00 5.45 14.93 39.69 104.18 1.4 5 0.76 0.999 ( 8.6%) ( 0.99) 3 (.9%.96%) 3. 0 = 1 N S = 1 D S =.4 ST = 45 SP = 75 3 1 4 D 3D ( ) D (D S =.40 N S = 1 ST = 45 SP = 75 ) D D S Table The fractal parameters of trace on various section D plane with partly -random distributed 3D surface (D S =.40 N S = 1 ST = 45 SP = 75 ) 0 = 1 0 0 / 0 /4 0 /8 0 /16 0 /3 D N R XOY 0.99.50 6.79 18.60 50.4 19.91 1.417 9 0.965 0.999 YOZ 0.76 1.90 5.1 13.71 36.73 93.09 1.397 1 0.745 0.999 XOZ 0.73 1.83 4.91 13.68 34.81 99.06 1.417 7 0.706 0.999 YOZ 0.91 1.8 5.47 13.44 3.4 80.99 1.318 0 0.84 6 0.998YOZ 3.01 8.99 8.0 83.54 49.90 1.595 7 0.997 7 1.000 XOZ 0.90.69 8.1 5.5 78.51 39.84 1.615 1 0.885 4 1.000 ( 0.9) D S 4.1 D S D D S ST SP XOZ YOZ XOY ( 1 ) D D S

604 005 D S 3 D D D S D S D Fig. Relationship between 3D fractal dimension of surface number D S and fractal dimensions D and their mean of various orthogonal section plane of fully-random distribution XOZ YOZ XOY 3 ( 1 ) 1.54% D 4 D D S D D =.005 1D 0.991 8 () S 1 + D D ST = 45 ST = 85 3 D S =.405 ST SP D Fig.3 The relationship between azimuth SP dip angle ST and fractal dimension D of D when D S =.405 D S R = 0.996 5 D S 4. D S D 4 D S D D S (N S = 1 ST = 45SP = 60 ) Fig.4 The relationship between 3D fractal dimension of surface ST SP number D S and fractal dimensions D of various XOZ YOZ XOY orthogonal section plane of partly-random distribution ( 1 ) (N S = 1 ST = 45SP = 60 ) D ST SP ST 4 SP D D D S D ST SP D ( 3) 5 D 3D D ST SP 5.1 N ST SP

4 4. 605 D S N SP ST ST SP SP XOZ YOZ XOY ST N ( 1 1 ) N = N ST SP 0.971 9 0.988 9sin ( ST )cos ( SP) (5) R = 0.984 6 4 XOZ N SP N ST ST SP N 5. D 3D N = 0.941 6sin(ST) + 0. 06 (3) D S R = 0.997 4 ST SP 5 YOZ N S N SP ST XOZ YOZ XOY SP ( 1 1 ) ST N N = 0.970 1 0.987 sin ( ST )sin ( SP) (4) R = 0.980 6 6 XOY N = kn S N N S 5 D S ST SP N N S 4 XOZ N (D S =.4) Table 4 The initial value of trace number N in XOZ section plane with different dip angle ST and azimuth angle SP(D S =.4) ST/( ) 0 10 0 30 40 50 60 70 80 90 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 0 0.000 00 15 0.61 8 0.3 0 0.06 4 0.10 0.175 1 0.169 3 0.187 7 0.16 6 0.34 7 0.56 7 0.15 05 30 0.50 6 0.493 7 0.453 7 0.407 6 0.387 0 0.383 9 0.40 1 0.460 5 0.507 5 0.505 8 0.45 4 45 0.713 5 0.71 4 0.711 0.665 5 0.635 1 0.66 0.651 9 0.715 5 0.715 5 0.701 0 0.684 78 60 0.871 5 0.886 1 0.870 6 0.89 0 0.89 6 0.870 6 0.87 6 0.871 5 0.879 7 0.850 7 0.875 79 75 0.976 9 0.958 7 0.979 0 0.950 3 0.967 8 0.963 7 0.95 0 0.979 0 0.978 1 0.969 8 0.967 53 90 0.96 5 0.975 9 0.981 0 0.985 0 0.967 0.975 9 0.973 9 0.977 9 0.961 7 0.975 9 0.973 69 5 YOZ N (D S =.4) Table 5 The initial value of trace number N in YOZ section plane with different dip angle ST and azimuth angle SP(D S =.4) ST/( ) 0 10 0 30 40 50 60 70 80 90 0 0.981 0 0.960 5 0.969 0.985 0 0.967 0.975 9 0.973 9 0.96 5 0.96 5 0.960 5 15 0.981 0 0.973 9 0.973 9 0.981 0 0.956 1 0.956 1 0.969 8 0.956 1 0.945 0 0.663 7 30 0.96 5 0.967 0.973 9 0.979 0 0.931 9 0.935 4 0.913 1 0.908 7 0.88 7 0.683 45 0.96 5 0.967 0.956 1 0.948 6 0.916 7 0.849 3 0.796 6 0.751 0 0.733 0 0.590 7 60 0.96 5 0.973 9 0.956 7 0.901 9 0.839 1 0.691 0.560 8 0.504 5 0.49 5 0.46 3 75 0.981 0 0.965 1 0.945 5 0.88 7 0.791 6 0.659 8 0.46 9 0.308 8 0.7 0 0.55 5 90 0.97 9 0.979 0 0.950 6 0.871 5 0.777 5 0.649 7 0.505 8 0.349 7 0.175 7 0.000 0

606 005 6 XOY N (D S =.4) Table 6 The initial value of trace number N in XOY section plane with different dip angle ST and azimuth angle SP(D S =.4) ST/( ) 0 10 0 30 40 50 60 70 80 90 0 0.983 0 0.969 0.96 5 0.981 0 0.969 0.967 0.969 0.969 0.975 9 0.969 15 0.958 7 0.940 4 0.960 7 0.954 1 0.979 0 0.964 8 0.981 0 0.981 0 0.960 5 0.969 30 0.873 6 0.870 6 0.893 8 0.915 1 0.935 4 0.93 8 0.954 1 0.969 8 0.983 0 0.976 9 45 0.71 0 0.734 0.757 5 0.811 0.840 0.890 8 0.945 5 0.951 6 0.981 0 0.96 5 60 0.499 7 0.480 9 0.496 4 0.554 3 0.701 7 0.850 9 0.911 0 0.971 8 0.965 1 0.96 5 75 0.57 5 0.5 0 0.316 9 0.458 1 0.660 7 0.798 7 0.881 7 0.956 7 0.973 9 0.967 90 0.000 0 0.185 3 0.349 7 0.498 7 0.633 9 0.763 3 0.873 6 0.946 5 0.96 7 0.985 0 N S 5 XOZ N N S (D S =.4 SP = 0 ) Fig.5 Relationship between N of XOZ section plane and the initial value of surface number N S (D S =.4 SP = 0 ) and the mean of N 5.3 D S N 0 0 7 ST SP N S = 1 7 D S Table 7 The number of surface and trace on fractal XOZ YOZ XOY observation ( 1 ) N D S D S 6 D S N 0.755 3.54 N D S N 0.755 0.707 6 D 3D 6 D S N Fig.6 Relationship between D S of fully-random distribution 0 1 1 1 1 0 / 4 8 4 0 /4 16 4 64 16 0 /8 64 8 51 64 0 /16 56 16 4 096 56 0 /3 1 04 3 3 768 1 04 0 /64 4 096 64 6 144 4 096 0 /18 16 384 18 097 15 16 384 (1) 0 1 N( δ ) = δ

4 4. 607 D S N S 1 1 N = sin( ST ) 7 XOY n 1 n 1 N( δ ) = δ D = 1 N = k k 1 N ( 0 / ) = ( 0 / ) (7) D S = D = D S 1 N = 1 5.1 () 1 ( N = 1 sin ( ST ) cos (π ) = 1 ) 7 YOZ n ncos(st) n 3 N( δ ) = δ k k 1 D S = 3 N S = 1 N ( 0 / ) = cos( ST )( 0 / ) (8) N = cos(st) 5.1 7 N = 1 sin ( ST )sin (π ) = cos( ST ) N( δ ) = δ D = N = 1 D S D D S = 3 D = D S 1 D = D S 1 N S = 1 (3) 0 1 0 0 ST SP = N S 1 90 ( 7) N S N( δ ) = δ 7 Y C B 8 100 # 100 # 30 590 m Z 37 A 3 C 8 SP D 3D ST O D X 7 Fig.7 The projection of single surface XOZ n nsin(st) n (1) k k 1 () N ( 0 / ) = sin( ST )( 0 / ) (6)

608 005 8 100 # Table 8 The observation and forecast result of rock crack of drill No.100 in Jiexiu /m D N D N D S N S ST/( ) 316 1.5 1 1.46 7 1.47 1.5 1.36 1.609 5 4.6 61.6 K4 3 1.850 1.068 0 1.888 8 1.36 1.870 1.716 66.3 4.6 37 1.051 0 1.708 1.181 0 1.174 4.116 1.893 5.1 56.8 398 1.0 8 1.450 0 1.64 1 1.758 7.33 1.908 4 4.6 3.7 40 1.69 3.48 3 1.78 0.383 8.710.456 4 8.4 33. 503 1.606 4.617 0 1.639 9 1.764 3.63 3.130 0 61.6 5.1 577 1.31 9 1.088 4 1.313 6 1.044 9.73 1.187 8 3.7 80.5 615 1.536 0.816 1 1.589 9 1.41 8.563 3.003 56.8 71.1 (References) [1] Kwaśiewski M A Wang J A. Surface roughness evolution and mechanical behavior of rock joints under shear[j]. Int. J. Rock Mech. 8 Min. Sci. 1997 34(3/4) 709. [] Zhang Z X Yu J Kou S Q et al. On study of influences of loading rate on fractal dimension of fractal surface in gabbro[j]. Rock Mech. Rock Engng. 001 34(3) 35 4. (1) [s. l.] [s. n.] 1953. 95 305. orientation date[j]. Math. Geology 1976 8 9 3. Geomech. Abstr. 1981 18 183 197. () 14 110 10. D D S 1 N N S 365. N [8] ST SP (3) Advance of Rock Mechanics 1988 5 5 7.(in Chinese)) (4) Geomech. Abstract. 1993 1 1 3. [10] [3] Fisher S R. Dispersion on a sphere[a]. In Proc. Roy. Soc. and.[c]. [4] Shanley R J Mahtab M A. Delineation and analysis of clusters in [5] Priest S D Hudson J A. Estimation of discontinuity spacing and trace length using scan line surveys[j]. Int. J. Rock Mech. Sci. and [6] Zanbak C. Statistical interpretation of discontinuity contour diagram[j]. Int. J. Rock Mech. Min. Sc. Geomech. Abstract 1997 [7] Zaitsev Y V Wittmann F H. Simulation of crack propagation and failure of concrete[j]. Materiaux et Construction 1981 14 357. [J]. 1988 5 5 7. (Pan Bietong Jing anru. Simulation and application of probability model of rock mass structure[j]. Recent [9] Kulatilake P H S W. Joint network modeling with a validation exercise in stripe mine Sweden[J]. Int. J. Rock Mech. Sci. and.

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