A multivariate method for identifying structural domain boundaries in a rock mass
ORIGINAL PAPER
A multivariate method for identifying structural domain boundaries in a rock mass
Shengyuan Song ?Qing Wang ?Jianping Chen ?
Yanyan Li ?Qi Zhang ?Chen Cao
Received: 27 August 2014 / Accepted: 7 October 2014 ó Springer-Verlag Berlin Heidelberg 2014
Abstract The mechanical and hydraulic behavior of rock masses vary from one domain to another because of the presence of discontinuities.Hence,structural domain boundaries should be determined according to the proper-ties of these discontinuities.In this work,a new method for identifying structural domain boundaries based on discon-tinuity orientation and trace length is proposed.The new method is performed in three steps.First,Schmidt plots are divided into equal-area patches on the hemisphere.Second,the discontinuities occurring in each patch are ranked according to their trace lengths.Third,the Wald–Wolfo-witz runs test is used to analyze the sequence of disconti-nuities.The results obtained by applying the new method to four simulated discontinuity data sets were more rigor-ous than Miller’s method in that Miller’s method only considers discontinuity orientation whereas the new method considers both orientation and trace length.Both the proposed method and Miller’s method gave the same,reliable,results when they were applied to discontinuity data collected from four adjacent adits at the Songta dam site.Only PD232and PD252corresponded to the same structural domain according to the two methods,because the discontinuity data collected from PD232and PD252had not only similar orientation distributions but also similar trace length distributions.The other advantage of the new method is that it can be used to evaluate the homogeneity of structural populations with small sample sizes.
Keywords Structural domains áDiscontinuity
orientation áDiscontinuity trace length áWald–Wolfowitz runs test áSchmidt plot
Introduction
Cracks or fractures are widely distributed in rock masses.The presence of such discontinuities can strongly affect the mechanical and hydraulic behavior of rock masses (Bu-detta et al.2001;Baghbanan and Jing 2007).Hence,the properties of a discontinuous rock mass vary from one domain to another.In 3-D discontinuity network simula-tion,each structural domain should be represented by a separate discontinuity geometry model.Therefore,it is essential to identify domain boundaries before generating a model (Chen et al.1995).
A structural domain is also known as a statistically homogeneous region.Hence,discontinuities in the region should have similar distributions for orientation,trace length,spacing,roughness,and ?lling.However,discon-tinuity orientation is a key in?uence on the properties of rock masses.Traditionally,only the orientation is consid-ered when determining structural domain boundaries.Pi-teau and Russell (1971)adopted a cumulative sums technique to determine discontinuity orientation trends,patterns of discontinuity behavior,and whether disconti-nuity information collected from one region can be extrapolated to another,thus indicating whether the two regions belong to the same structural domain.However,this method requires information about dominant discon-tinuity set orientations and treats discontinuity dip and strike separately.
Discontinuity orientations are commonly represented by hemisphere projection plots of discontinuity poles
S.Song áQ.Wang áJ.Chen (&)áY.Li áQ.Zhang áC.Cao College of Construction Engineering,Jilin University,Changchun 130026,China e-mail:chenjpwq@https://www.wendangku.net/doc/673080374.html,
Bull Eng Geol Environ (2015)74:1407–1418DOI 10.1007/s10064-014-0686-5
(Schmidt1925;Priest1993;Jimenez-Rodriguez and Sitar 2006).Such a plot is visual and combines discontinuity dip and strike.Based on this technique,the boundaries of a structural domain can be identi?ed by comparing hemi-sphere projection plots of discontinuities collected from different regions of a rock mass.This method is very subjective because the similarity of the hemisphere pro-jection plots that are compared is often gauged based on geologic experience.Furthermore,when the hemisphere projection plots are complex and discontinuity poles are scattered,a visual comparison makes it dif?cult to deter-mine whether discontinuity populations are collected from the same structural domain.Thus,a mathematical and statistical method that can identify domain boundaries objectively must be developed.
The?rst statistical method for identifying structural domain boundaries was proposed by Miller(1983).In this method,Schmidt plots are divided into equal-area patches on a hemisphere.Each patch is considered a cell in the contingency table and the chi-square test is used to evaluate the homogeneity of structural populations by comparing the frequencies of discontinuity poles corresponding to patches.This method provides a quantitative index for comparing Schmidt plots.However,there are some limi-tations on the application of the chi-square test.First,a minimum sample size of150discontinuities is required. Second,discontinuity poles corresponding to patches on the hemispheres need to meet Lancaster’s criterion(1969). Mahtab and Yegulalp(1984)divided each hemisphere into 100equal-area squares and used a random test of Poisson distribution to determine structural domain boundaries. Kulatilake et al.(1996)proposed a modi?ed version of Miller’s method.In this modi?ed method,the p value is used as an index to evaluate the tested regions with respect to homogeneity strength.Based on Mahtab and Yegulalp’s method,Martin and Tannant(2004)employed a correlation coef?cient to evaluate the degree of similarity between different regions.These methods are similar because the discontinuity orientations are considered in the Schmidt plot when determining the homogeneity of structural populations.
Kulatilake et al.(1997)used the box fractal dimension as a measure to determine structural domain boundaries in a tunnel at the Three Gorges dam site in China.The results indicated that the box fractal dimension represents the combined impact of discontinuity density and discontinu-ity-size distribution on the structural domains of rock masses.Escuder Viruete et al.(2001)employed a geosta-tistical analysis method to extrapolate discontinuity density from a known region to a peripheral region.Zhang et al. (2011)used the chi-square test to determine structural domain boundaries according to discontinuity orientation, trace length,roughness,and aperture.However,this method has disadvantages because it counts dip directions and dip angles separately.
In the work reported in this paper,a new method for identifying structural domain boundaries in a jointed rock mass that considers both discontinuity orientation and discontinuity trace length is presented.This method is performed in three steps.First,Schmidt plots are divided into equal-area patches using Miller’s method.Second, discontinuities corresponding to each patch are ranked according to their trace lengths.Third,the Wald–Wolfo-witz runs test is used to evaluate the homogeneity of structural populations.
Methodology for identifying structural domains Processing of orientation data
The hemisphere projection plot of discontinuity poles combines dip direction and dip angle.Hence,the method proposed by Miller was applied to process discontinuity orientation data.In Miller’s method,the lower hemisphere is divided into patches that have equal surface areas.The determination of equal-area patches on a hemisphere is achieved by dividing the lower hemisphere into a certain number of dip bands.In order to minimize the potential bias when comparing several hemisphere projection plots, the number of dip bands and the number of patches in each band are selected such that the resulting patches on the same hemisphere have near-equal areas(Miller1983). Example networks that have equal-area patches are pre-sented in https://www.wendangku.net/doc/673080374.html,ler indicated that the38-patch scheme provides a more reasonable treatment of discontinuity clusters with near-vertical dip angles than the34-patch scheme(Miller1983),noting that relatively few high-angle discontinuities are displayed on the subsequent contrasting hemispheres.Therefore,the34-patch scheme was utilized to process the discontinuity orientation data in this work. The number of discontinuity poles within each patch on the hemispheres is shown in Table1.
Wald–Wolfowitz runs test
The Wald-Wolfowitz runs test is a nonparametric test that can be used to determine if the difference between two sample distributions is signi?cant(Wald and Wolfowitz 1940).The null hypothesis H0is that the two samples have the same distribution,whereas the alternative hypothesis H1is that the two samples have different distributions. Suppose there are two independent data samples:X and Y.Sample X has n1data points and sample Y has n2data points.The procedure of the Wald–Wolfowitz runs test is as follows(Siegel1956;Sharp2006)Fig.2:
1408S.Song et al.
1.
Two data samples are merged into a single combined data set Z with the total number of observations n ?n 1tn 2,and the values are ranked in this set.2.Rank i is assigned to the i th smallest observation in the whole set Z .
3.
Observations in Z are assigned as follows:
T i ?
0If the i th observation Z i belongs to sample X
1If the i th observation Z i belongs to sample Y
(
e1T
4.
If T i ?T i t1?T i t2?ááá?T i tl ?T i tl t1,
the sequence T i t1;T i t2;...;T i tl eTis considered a run.The length of this run is l .The Wald–Wolfowitz runs test statistic U is the total number of runs in data set T .U values range from 2to k .The greater the value of U ,the more similar the distributions of sample X and sample Y are.The k value is calculated using
k ?2?min n 1;n 2eTt1:
e2T5.
For samples with small numbers of data points (i.e.,
n 1B 20and n 2B 20),the p value can be obtained from a Wald–Wolfowitz runs test table.When sample sizes are greater than 20,U can be considered to have an approximately normal distri-bution and the p value can be calculated according to the normal distribution.The mean and the standard deviation of U are as follows (Wald and Wolfowitz 1940
):
Table 1Number of poles within each patch on each Schmidt plot Schmidt plot Patch 1Patch 2…
Patch 341n 1,1n 1,2n 1,342n 2,1n 2,2n 2,34::::r
n r ,1
n r ,2
n r ,34
A multivariate method for identifying structural domain boundaries 1409
l U?
2n1n2
n1tn2
t1e3T
r U?
???????????????????????????????????????????????
2n1n2e2n1n2àn1àn2T
en1tn2Ten1tn2à1T
s
:e4T
6.The calculated p value should be compared with the
signi?cance level a,which was taken to be0.05in this work.If the p value is smaller than the signi?cance level,the null hypothesis H0is rejected and the difference between the two sample distributions is signi?cant.The samples being compared are therefore considered to be different.Otherwise,the samples are considered to be statistically similar.
Determining the rank value in the Wald–Wolfowitz runs test is important for the work presented here.When two structural populations are compared,their similarity is evaluated by comparing the distributions of discontinuity orientation and trace length.First,the discontinuity orien-tations of the regions being compared are divided into several patches according to Miller’s method.Discontinu-ity poles occurring in the same patch make different con-tributions to the structural domains because these discontinuities with similar orientations have different trace lengths.Therefore,discontinuities that occur in each corresponding patch on any two of the Schmidt plots that are compared are ranked according to their trace lengths. Table2presents an example of how the discontinuities of compared regions are ranked.
Application to arti?cial data
Arti?cial data sets
To verify the rationality and validity of the proposed method,the Wald–Wolfowitz runs test was applied to arti?cial data sets.In this work,four discontinuity data sets were generated randomly.In each simulated data set,the discontinuity orientations obeyed the bivariate normal distribution and the discontinuity trace lengths obeyed the normal distribution(Zanbak1977;Chen et al.1995; Marcotte and Henry2002).To study the performance of the proposed method,Miller’s method was also applied to determine structural domains.A minimum sample size of 150discontinuities is required,and discontinuity poles corresponding to patches on the hemisphere need to meet Lancaster’s criterion to be able to use Miller’s method. Hence,each simulated discontinuity set contains[150 discontinuities,and discontinuity poles are fairly dispersed on the hemisphere projection plot.Table3shows the parameters used for random generation.The discontinuity orientations of each simulated data set are represented by the lower-hemisphere,equal-area projection for visualiza-tion(Fig.3).
Identi?cation of structural domain boundaries
between arti?cial data sets
The Wald–Wolfowitz runs test and the method proposed by Miller were used to determine whether four arti?cial data sets were from the same structural domain.The four discontinuity data sets had an insigni?cant number of near-vertical dip angles(Fig.3).Hence,the34-patch scheme was utilized to process the orientation data.To minimize the subjectivity of the patch directions,each pair of Schmidt plots was analyzed18times as the patch network was rotated from0°to180°in10°increments(Miller 1983),as shown in Fig.4.In this study,the average p value of the18patch directions was used to determine whether the structural regions being compared were similar.A signi?cance level of0.05was selected as the threshold to determine whether the null hypothesis H0was correct.The results of the Wald–Wolfowitz runs test and Miller’s method are listed in Table4.
Table4shows that the results of the Wald–Wolfowitz runs test are different from the results provided by Miller’s method.The results calculated by Miller’s method indicate that each pair of simulated data sets complies with the null hypothesis H0at a signi?cance level of0.05;i.e.,the four
Table2Example showing how the discontinuities of compared regions are ranked
t j=n1,j?n2,j,1B j B34.
T i=0or1,1B i B
(t1?t2?_?t33?t34)Patches Schmidt
plot1
Schmidt
plot2
Total Rank according
to trace length
Code Number
of runs 1n1,1n2,1t1Z1;Z2;...;Z t1T1,T2,…,
T(t1?t2?_?t33?t34)
U
2n1,2n2,2t2Z(t1?1),Z(t1?2),…,Z(t1?t2)
3n1,3n2,3t3Z(t1?t2?1),Z(t1?t2?2),…,
Z(t1?t2?t3)
:::::
34n1,34n2,34t34Z(t1?t2?_?t33?1),
Z(t1?t2?_?t33?2),…,
Z(t1?t2?_?t33?t34)
1410S.Song et al.
data sets are from the same structural domain.However, the results obtained from the Wald–Wolfowitz runs test signify that only data sets1and4are from the same structural domain.The reason for this difference in the results of the two analyses is that Miller’s method only considers discontinuity orientation,whereas the new method considers both orientation and trace length when evaluating the homogeneity of structural populations.
To allow a visual comparison of the distributions of discontinuity orientations,histograms for the pole frequency in each patch of the34-patch network(Fig.1) are presented in Fig.5.Figure5shows that the pole fre-quencies in corresponding patches on the hemispheres of discontinuity orientations selected from four simulated data sets are very similar.In addition,the parameters of the bivariate normal distributions of orientation in the four simulated data sets are the same(Table3).Hence,the results provided by Miller’s method are reliable because data sets1,2,3,and4are considered to refer to the same structural domain.
Table3Parameters of the simulated discontinuity data sets Data
set
Bivariate normal distribution Normal distribution Number of
discontinuities Mean of dip
direction(°)
Variance of
dip
direction
Mean of
dip angle
(°)
Variance
of dip
angle
Mean of
trace length
(m)
Variance of
trace length
11807,000551,00020.25250 21807,000551,00010.25250 31807,000551,00020.50250 41807,000551,00020.25250
A multivariate method for identifying structural domain boundaries1411
To allow a visual comparison of the distributions of discontinuity trace length,histograms of discontinuity trace length frequency for the four simulated data sets are pre-sented in Fig.6.Figure 6shows that distributions of trace lengths selected from data sets 1and 4are very similar.Moreover,the orientation distributions of data sets 1and 4are also similar.Therefore,data sets 1and 4are considered to belong to the same structural domain by the new
Table 4Results from the evaluation of the similarity between the simulated discontinuity data sets
Data sets Wald-Wolfowitz runs test Miller’s method
p value Results
p value Results
Min.
Max.Average Min.
Max.Average 1and 27.34910-57 2.05910-54 2.85910-55Rejected 0.23470.99030.5328Accepted 1and 3 3.34910-40.01000.0031Rejected 0.03080.39810.1508Accepted 1and 40.06400.8778
0.4088
Accepted
0.03000.92000.3880Accepted 2and 37.87910-38
4.54910-32
7.91910-33
Rejected 0.01890.80280.3374Accepted 2and 47.34910-57 3.28910-53 5.95
910-54
Rejected 0.28570.96790.6896Accepted 3and 4
2.40910
-4
0.0535
0.0166
Rejected
0.0220
0.9961
0.4240
Accepted
Fig.5Pole frequencies in the patches on the hemisphere of discontinuity orientations collected from each simulated data set (a –d correspond to sets 1–4,respectively)
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method.When the trace length distribution of any other pair of data sets was considered,there was a signi?cant difference between the data sets in the mean or the variance of the distribution.Hence,the results obtained by the proposed method are reasonable because data sets 2and 3are considered to differ from the structural domain con-stituted by data sets 1and 4.
As mentioned above,the results calculated using Miller’s method and the Wald–Wolfowitz runs test are reasonable.The difference between the two methods is that the new method considers discontinuity orien-tation and discontinuity trace length whereas Miller’s method only considers discontinuity orientation when evaluating the homogeneity of structural populations.Normally,the results provided by the new method are more rigorous because compared structural popula-tions with similar orientation and trace length distri-butions are considered to belong to the same structural domain.
Application to real data Study area
To verify the practicability of the proposed method,the Wald–Wolfowitz runs test was applied to evaluate the
statistical homogeneity of the dam site of the Songta Hydropower Station in China.
The Songta Hydropower Station,sited between Tibet and Yunnan Province in the upper reaches of the Nu River,will be one of the largest hydropower sta-tions in the world when completed.The ?ow direction of the Nu River at the dam site is approximately 188°SW (Fig.7).
The dam site exhibits an asymmetric ‘‘V’’shape,which is a typical mountain-canyon geomorphology (Fig.8a).A concrete double-curved arch dam is planned for the Songta hydropower station.The height of the arch dam will be approximately 318m,the elevation of the normal water level will be 1,700m,and the water storage level will be 1,925m.The total storage capacity of the station will be 4.55billion m 3and the hydro-electric generating capacity of the station will be 3,600MW.
Biotite monzonitic granite and plagioclase amphibolite from the late Yanshanian (Cretaceous)period are exposed at the dam site.Biotite monzonitic granite is the predom-inant lithology and has been slightly metamorphosed;it has a cataclastic texture characterized by the fragmentation of feldspar phenocrysts.Plagioclase amphibolite comprises dykes with widths varying from 0.05to 5m.It has expe-rienced slight argillization in the region connecting the two types of lithology (Fig.8
b).
Fig.6Histograms of discontinuity trace length frequency for the four simulated data sets (a –d correspond to sets 1–4,respectively)
A multivariate method for identifying structural domain boundaries 1413
To study the engineering geological properties of the rock mass,a large number of exploration adits were created at different elevations on both banks.The window sampling method was used to collect data on the discon-tinuities within these adits.Only discontinuities with trace lengths of [0.3m that intersected the left wall of each adit
Legend County National boundary River
Songta dam site
Province boundary
Songta dam
Flow direction
Nu River
97°
98°99°100°
29°
28°
27°Chayu
Gongsha n
Weixi
Derong
Songta Yunnan Sichuan Tibet Nu River
Dulong Lancang River
River
Province
Province
N
N
(c)
PD232PDS1
PD252
PD242
Nu River
(a)Dam site
Nu River
(b)
Joints
Plagioclase amphibolite
Biotite monzonitic granite
Geological conditions at the dam site
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S.Song et al.
were measured.The discontinuity orientation,trace length, aperture,?lling,roughness,and weathering were collected.
The new method considers discontinuity orientation and trace length when evaluating the homogeneity of structural populations.Hence,the discontinuity orientation and trace length data collected from four adits—PDS1,PD232, PD242,and PD252—were utilized as a test case.These adits are adjacent and occur at the same elevation of 1,732m on the right bank of the Nu River(Fig.8c).Each adit is200m long and has a cross-section2m wide and 2m high.
The hemisphere projections of the discontinuity orien-tations collected from each adit are presented in Fig.9. Figure9shows that the projections for adits PDS1,PD232, and PD252are similar.It is dif?cult to determine whether these discontinuity clusters were collected from the same structural domain based on visual comparisons.
Identi?cation of structural domain boundaries between adits
A quantitative method,the Wald–Wolfowitz runs test,was used to evaluate whether the structural populations(i.e.,PDS1,PD232,PD242,and PD252)are statistically https://www.wendangku.net/doc/673080374.html,ler’s method was also applied to the populations as a benchmark(Table5).
Table5shows that the results calculated in the Wald–Wolfowitz runs test are the same as the results from Miller’s method at a signi?cance level of0.05.The results of the two methods indicate that the structural populations PD232and PD252are statistically homogeneous.Although PD242lies between PD232and PD252,the rock mass around PD242should be treated independently.
The Wald–Wolfowitz runs test and the method proposed by Miller consider different properties of discontinuities when determining structural domain boundaries,but the results of the two methods are the same.This leads us to the question of whether the results of each method are reliable.The histograms for the distributions of disconti-nuity orientation and trace length are presented in Figs.10 and11,respectively.
Figure10shows that the pole frequencies in corre-sponding patches on the hemispheres of discontinuity ori-entations selected from PD232and PD252are very similar. For the discontinuity orientations collected from PDS1,the pole frequencies that occur in patches1–13are signi?-cantly different from those of the other adits.For the
A multivariate method for identifying structural domain boundaries1415
discontinuity orientations selected from PD242,the pole frequencies that occur in patches18–34are also different from those of the other adits.
As mentioned above,the results provided by Miller’s method are reliable because the discontinuity orientations from PD232and PD252have similar distributions.
Figure11shows that trace lengths distributions from PD232and PD252are very similar,and the frequency of discontinuities with trace lengths varying from1.6to1.8m is largest,approximately20%.For the trace lengths selected from PDS1,the frequency of discontinuities with trace lengths varying from0.4to0.6m is largest,approximately 15%,meaning that the number of discontinuities with trace lengths of\0.6m is larger than seen for the other adits.For the trace lengths collected from PD242,the frequency of trace lengths ranging from1.6to1.8m is only14%,and the number of discontinuities with trace lengths of[2.4m is larger than seen for the other adits.
In conclusion,the distributions of discontinuity orien-tations and trace lengths collected from PD232and PD252 are very similar.Hence,according to the new method,the structural populations collected from PD232and PD252 are statistically homogeneous.Although the new method and Miller’s method are not contradictory,the new method
Table5Evaluation of the similarity of the structural populations for the adits
Adits Wald–Wolfowitz runs test Miller’s method
p value Results p value Results
Min.Max.Average Min.Max.Average
PDS1and PD2328.13910-70.01600.0022Rejected 1.16910-10 2.51910-6 2.71910-7Rejected PDS1and PD242 1.11910-10 1.19910-7 2.79910-8Rejected000Rejected PDS1and PD252 3.60910-60.0045 4.61910-4Rejected9.35910-70.00070.0001Rejected PD232and PD2428.10910-80.03310.0051Rejected0 2.74910-9 2.45910-10Rejected PD232and PD2520.05850.19470.1289Accepted0.41050.99570.7891Accepted
PD242and PD2520.00240.14250.0363Rejected9.80910-12 1.02910-8 2.71910-9
Rejected
Fig.10Pole frequencies in the patches on the hemisphere of discontinuity orientations collected from each adit(a–d correspond to PDS1, PD232,PD242,and PD252,respectively)
1416S.Song et al.
is more rigorous than Miller’s method because the new method considers not only discontinuity orientation but also discontinuity trace length.
Conclusions
In this work,a new method based on the Wald–Wolfowitz runs test for identifying structural domain boundaries in a rock mass was proposed.In the new method,both dis-continuity orientation and discontinuity trace length are considered.First,Schmidt plots are divided into 34patches that have equal surface areas on the hemisphere.Second,the discontinuities occurring in each corresponding patch are ranked according to their trace lengths.Third,a sequence is coded according to the ranks of discontinuities and utilized for statistical analysis.The Wald–Wolfowitz runs test was adopted to determine the homogeneity of structural populations.
For four simulated data sets,the results provided by the new method differed from those obtained using Miller’s method.The results of Miller’s method indicated that the four discontinuity data sets derived from the same struc-tural domain.However,the results of the new method showed that only data sets 1and 4were from the same structural domain.The reason for this difference in the results of the two methods is that the new method considers not only discontinuity orientation but also discontinuity
trace length when evaluating the homogeneity of structural populations.Normally,the results provided by the new method are more rigorous because Miller’s method deter-mines the structural domain based on orientation only whereas the new method consideres both orientation and trace length.
For the discontinuity data collected from four adjacent adits at the dam site of the Songta Hydropower Station in China,the results of the new method were the same as those calculated using Miller’s method.The results of the two methods signi?ed that the structural populations collected from PD232and PD252were statistically homogeneous.The reason that both methods gave the same results in this case is that the discontinuity data collected from PD232and PD252have not only similar orientation distributions but also similar trace length distributions.
A major advantage of the new method is that it provides more rigorous results than the method proposed by Miller when identifying structural domain boundaries.Moreover,the new method can be used to evaluate the homogeneity of structural populations with small sample sizes.
Acknowledgments This work was supported by the State Key Program of National Natural Science of China (grant no.41330636),2010non-pro?t scienti?c special research funds of the Ministry of Water Resources (grant no.201001008),Jilin University’s 985pro-ject (grant no.450070021107),and the Graduate Innovation Fund of Jilin University (grant no.
2014062).
Fig.11Frequencies of discontinuity trace lengths collected from each adit (a –d correspond to PDS1,PD232,PD242,and PD252,respectively)
A multivariate method for identifying structural domain boundaries 1417
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