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Development of a multivariate empirical model for predicting weak rock mass modulus
International Journal of Mining Science and Technology xxx (2015) xxx–xxx
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Development of a multivariate empirical model for predicting weak rock mass modulus Kallu Raj R. a,⇑, Keffeler Evan R. b, Watters Robert J. c, Agharazi Alireza d a
Department of Mining Engineering, University of Nevada, Reno, NV 89557, USA RESPEC Consulting and Services, Rapid City, SD 57709-0725, USA c University of Nevada, Reno, NV 89557, USA d Itasca Houston Inc., Stafford, TX 77477, USA b
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 18 January 2015 Accepted 16 February 2015 Available online xxxx Keywords: In-situ modulus Weak rock mass Preliminary design Elastic deformation
a b s t r a c t Estimating weak rock mass modulus has historically proven difficult although this mechanical property is an important input to many types of geotechnical analyses. An empirical database of weak rock mass modulus with associated detailed geotechnical parameters was assembled from plate loading tests performed at underground mines in Nevada, the Bakhtiary Dam project, and Portugues Dam project. The database was used to assess the accuracy of published single-variate models and to develop a multivariate model for predicting in-situ weak rock mass modulus when limited geotechnical data are available. Only two of the published models were adequate for predicting modulus of weak rock masses over limited ranges of alteration intensities, and none of the models provided good estimates of modulus over a range of geotechnical properties. In light of this shortcoming, a multivariate model was developed from the weak rock mass modulus dataset, and the new model is exponential in form and has the following independent variables: (1) average block size or joint spacing, (2) field estimated rock strength, (3) discontinuity roughness, and (4) discontinuity infilling hardness. The multivariate model provided better estimates of modulus for both hard-blocky rock masses and intensely-altered rock masses. Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Underground mines are increasingly being excavated in relatively deep and weak mineralized zones that are typically intensely-fractured and highly-altered with geotechnical properties resembling those of stiff soils, and civil engineering projects such as dams and tunnels are more frequently being constructed in geological domains that contain zones of highly-fractured rock. For example, most of the modern underground mines in Nevada, United States, are produced from Carlin-type deposits where disseminated gold and silver are hosted by intensely-fractured and moderately- to highly-altered sedimentary sequences. The host rocks are predominantly an assemblage of limestone, shale, volcanics, and breccias which have been fractured, folded, and locally hydrothermally altered resulting in a wide range of mechanical properties. Given the cost and complexity of in-situ deformability tests, direct measurements of rock mass moduli are typically not performed at active mines, which make the use of empirical models ⇑ Corresponding author. Tel.: +1 7756826448. E-mail address: [email protected] (R.R. Kallu).
attractive. However, the majority of published empirical equations are largely based on data sets with few measurements in weak and very weak rock masses. Additionally, common rock mass classification systems (i.e. Rock Mass Rating, RMR; Rock Tunneling Quality Index, Q) tend to be insensitive in these types of rocks, and geotechnical engineers at mines typically have relatively little geotechnical data to work with compared to civil engineering projects. Consequently, using predictive correlations in a mining environment can be difficult because of a lack of reliable input data. For many civil engineering projects such as dams and tunnels, available locations or alignments may be sited in geologic domains that contain zones of highly-fractured or altered rock masses. While large civil engineering projects typically include in-situ rock mass modulus testing programs, these field tests are expensive and time consuming, and interpreting the resulting data can be ambiguous [1]. At the feasibility and preliminary (high-level) design stages, field testing of rock mass modulus typically cannot be justified, and as a result predictive empirical models are employed [2]. During these early stages, geotechnical data are likely limited to basic rock mass parameters that have been quantified by field mapping, a limited number of geotechnical core holes, or literature reviews. Consequently, the civil engineering
http://dx.doi.org/10.1016/j.ijmst.2015.05.005 2095-2686/Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
Please cite this article in press as: Kallu RR et al. Development of a multivariate empirical model for predicting weak rock mass modulus. Int J Min Sci Technol (2015), http://dx.doi.org/10.1016/j.ijmst.2015.05.005
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sector also has a need for a reasonably accurate weak rock mass modulus correlation that requires only basic geotechnical parameters as inputs for feasibility analyses and preliminary designs. Test data and geotechnical characterizations were obtained for stiff plate loading tests performed on heavily fractured rock masses at two underground mines in Nevada (United States), the Portugues Dam project in Puerto Rico (United States Territory), and the Bakhtiary Dam project in Iran. The weak rock mass moduli measured at civil and mining projects were combined to create a weak rock mass modulus database with associated detailed geotechnical characterizations. The moduli in the database were compared to several published rock mass modulus models to determine which, if any, are suitable for estimating weak rock mass deformation properties when limited geotechnical data are available. Based on the limitations identified in the published equations, an empirical multivariate model was developed, and the model only requires geotechnical parameters that could be easily quantified by field mapping or core logging. 2. Review of predictive models for weak rock mass modulus An abundance of predictive empirical in-situ modulus models can be found in the literature, and one of the objectives of this study was to determine which, if any, of the existing models are adequate for predicting the moduli of weak rock masses when geotechnical data are limited. The mining environment and preliminary design stages of civil projects have unique limitations on available geotechnical data, and typically only lithological, rock mass classification (RMR, Q; or Geological Strength Index, GSI), and point load strength data may be available. Laboratory strength or deformability data for intact rock may be limited or nonexistent. Eight rock mass modulus models were identified in the literature that are suitable for use when only basic geotechnical data are available (Table 1); these models: (1) were based on data sets that included at least some weak rock mass modulus measurements and (2) only require geomechanical data that can be quantified by field mapping or core logging. The Simplified Hoek and Diederichs [1] equation is based on a dataset of in-situ modulus tests performed in China and Taiwan. The dataset has a population of 494 data points with GSI values between 10 and 95, and excludes moduli measured with downhole deformation jacks and dilatometers, and in many cases GSI was calculated directly from RMR. The model requires a disturbance factor, D, to be estimated, which varies from zero to one. A value of zero represents a tightly interlocked rock mass, while one corresponds to a fully disturbed and loosened rock mass. The disturbance factor has proven difficult to estimate and at best is selected subjectively [1,3]. Table 1 Rock mass modulus models that are applicable to weak rock masses and can be used when only limited geotechnical data are available.
Palstrom and Singh [2] assembled a database of in-situ modulus from 42 tests performed at hydroelectric projects in India, Bhutan, and Nepal. Rock types included gneiss, granite, mica schist, sandstone, mudstone, siltstone, and dolerite, and RMR varied between 46 and 75 (1.1 6 Q 6 30). Correction factors of up to 7.5 were applied to moduli to account for effects of blast damage and volume of rock tested. Even with these corrections, previously published models over-predicted moduli of the lower quality rock masses in their dataset. Palstrom and Singh [2] suggested a new Q-based correlation that has some applicability for rock masses with Q-values near 1. Galera et al. [4] assembled a database of in-situ moduli values from downhole dilatometer and pressure meter tests. Tests with moduli less than 0.5 GPa or that were performed on highly-weathered rock masses were censored from the database to eliminate ‘‘soil behavior’’, which resulted in 427 data points that included rock mass modulus, RMR, and Rock Quality Designation (RQD). Their analyses indicated RMR was a better predictor of rock mass modulus than RQD and that lithology was not a significant parameter. Based on the censored dataset, they developed linear and quadratic models. Serafim and Pereira created a relationship based on a combination of their own data and the dataset presented by Bieniawski [5,6]. The field test methods were not specified. The dataset overall has RMR values from 23 to 85, and the resulting model is exponential in form. The relationship by Read et al. was a re-fitting of the Serafim and Pereira and Bieniawski datasets with rock mass modulus constrained to equal 100 GPa when RMR was 100 [5– 7]. This constraint was imposed to prevent the correlation from predicting unrealistically high modulus values when RMR approached 100. Gokceoglu et al. created a new rock mass modulus database by combining the results of 58 situ deformability tests with 57 moduli originally published by Kayabasi et al. [3,8]. The data were generated from dilatometer and plate loading tests were conducted at the Deriner Dam site and Ermenek Dam site in Turkey. The tests were performed on quartz diorite, limestone, and heavily-jointed marl, and RMR varied between 20 and 85 with 17 tests in weak rock masses. Each data point includes RMR, GSI, RQD, uniaxial compressive strength of the intact rock, elastic modulus of the intact rock, and discontinuity properties. Gokceoglu et al. [3] developed new single variable and multi-variable models. For the single input equations, they evaluated linear, logarithmic, power, and exponential models; for both GSI and RMR the exponential correlations had the highest coefficients of regression. Multivariate regression analyses indicated intact rock modulus, uniaxial compressive strength, RQD, and discontinuity weathering degree were the most significant variables for predicting rock mass modulus. However, this multivariate model can be difficult to use when geotechnical data are limited because reliable measurements of intact rock properties may not be available, and in weak rock masses RQD is typically zero or near zero.
Model
Dependent variable
Applicability
Equation (GPa)
Hoek and Diederichs Simplified [1] Palstrom and Singh [2] Galera et al. [4] (Linear) Galera et al. [4] (Exponential) Serafim and Pereira [5] Read et al. [7]
GSI
20
1D=2 E ¼ 100 1þe½ð75þ25DGSIÞ=11
3. Database of weak rock mass modulus
Q
1 < Q < 30
E ¼ 8Q 0:4
3.1. Overview of data sources
RMR
10 < RMR < 50
E ¼ 0:0876RMR
RMR
10 < RMR < 80
E¼e
RMR
30 < RMR < 50
E ¼ 10ðRMR10Þ=40
RMR
20 < RMR < 80
Gokceoglu et al. [3] Gokceoglu et al. [3]
RMR GSI
20 < RMR < 85 15 < GSI < 77
E ¼ 0:1ðRMR=10Þ3 E ¼ 0:0736e0:0755RMR E ¼ 0:1451e0:0654GSI
A database of weak rock mass modulus was assembled for the results of stiff plate loading tests performed at two underground mines in Nevada, the Bakhtiary Dam and Hydroelectric Project in Iran, and the Portugues Dam in Puerto Rico. In-situ modulus was calculated from the reload portions of the force–displacement plots, and geotechnical parameters were determined from core logs following the guidelines developed in Section 3 or by window mapping.
ðRMR10Þ=18
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3.2. Underground mines of Nevada Plate loading tests were performed at two underground gold mines in Nevada yielding 14 modulus measurements. The field methods are described in detail along with the geotechnical conditions of each test site [9]. In both mines, the deposits are disseminated gold and silver hosted by altered sedimentary sequences, and the country rock is predominantly limestone and marl transected by altered porphyry dikes. Tests were performed at depths between 510 and 730 m, and the lithology of test sites included fault breccia, argillized porphyry dikes, and decalcified, oxidized, and/or argillized limestone. 3.3. Bakhtiary dam site The Bakhtiary Dam and Hydroelectric Project is located in the Zagros Mountains of southwest Iran and when finished will include a 315 m tall double-curvature concrete arch dam. The Bakhtiary Dam project is sited in the marly and siliceous limestone beds of the Cretaceous Sarvak formation. Kink bands associated with the Siah Kuh Anticline created zones of intensely folded and fractured rock in the abutments and foundation. An extensive foundation study included plate loading tests performed in exploration adits excavated into the abutments. Full details on the site geology and field test methods are provided in Agharazi et al. and Agharazi [10,11]. Agharazi et al. [10] analyzed the results of plate loading tests at the Bakhtiary Dam site and determined that for the intensely fractured rock, in-situ deformation behavior was a combination of both normal closure and shear displacement of joints, which ultimately results in compaction of the rock mass. With successive load–unload cycles, the fractured rock stiffened indicating the pieces became more-tightly interlocked after each cycle, and the stress–displacement plots were mainly non-linear with high total and permanent displacements. Based on force–displacement data from plate loading tests, eight rock mass moduli were calculated for intensely fractured but minimally weathered rock masses at depths between 20 and 210 m. Representative geotechnical properties were determined from core logs, and the geotechnical properties were similar for each site. Discontinuities were moderately rough to very rough, and infillings consisted of calcite and iron oxide up to 2 mm thick. Intact rock was slightly to faintly weathered. Laboratory strength testing of core samples indicates intact saturated uniaxial compressive strengths of 110 ± 30 MPa and intact deformation moduli of 69 ± 10 GPa [10].
performed a foundation stability investigation for the project from 1983 to 1988. The foundation study is detailed by USACE, and rock mass moduli are summarized by Stephens and Banks. The dam is sited in upper Cretaceous to Eocene siltstones, sandstones, and conglomerates that were metamorphosed to low-grade metasediments by an Eocene diorite stock. Numerous intrusive dikes radiate from the stock and transect the dam area. The rock has been highly altered by the humid-subtropical climate, but degree and depth of weathering vary greatly. Three adits were excavated into the abutments at depths up to 45 m below the ground surface in order to establish access for plate loading tests. The design memorandum by USACE contains detailed core logs for each plate loading test location, which include the following geotechnical parameters: lithology, recovery, fracture frequency, modified RQD, and intact rock soundness; waviness, roughness, aperture, weathering, and infilling were described for individual joints as well. Moderate to high-angle (relative to the core axis) discontinuities were ignored when calculating the modified RQD because ‘‘the in-situ quality of the rock is much better than the standard RQD indicated’’, and as such the reported RQD is biased high. Several samples were selected from each core hole for laboratory uniaxial compression strength and elastic deformation testing. Representative geotechnical properties for each plate loading tests site were determined from the core and discontinuity logs.
3.5. Description of database An empirical weak rock mass modulus database with 26 data points was assembled, and representative geotechnical properties were determined by window mapping for the Nevada test sites, and from geotechnical core logs for the Bakhtiary and Portugues Dam sites. Each data point includes the following attributes:
Field estimated rock strength (R-strength) Aperture (mm)/clast support Infilling type Infilling hardness Infilling thickness (mm) Weathering/alteration Block size/average joint spacing (cm) Roughness RMR-76 and parameter ratings [12] RMR-89 and parameter ratings [13] Reload (in-situ) modulus.
3.4. Portugues Dam site The Portugues Dam site is located near Ponce, Puerto Rico, and the Jacksonville District of the US Army Corp of Engineers (USACE)
Insufficient geotechnical data were available to confidently determine Q-values or GSI for many of the data points. RQD was excluded from analyses because more the half of the test sites
Table 2 Summary of geotechnical conditions for in-situ modulus measurements. Parameter
Nevada mines
Bakhtiary dam
Portugues dam
Number of in-situ measurements Field estimated rock strength
13 Stiff cohesive Strong to very strong rock 0.1 3 0 0 Slightly to moderately Decomposed Stiff cohesive Very rough 15 47 0.031 4.2
8 Strong to very strong rock Very strong rock 2 7 3 25 Slightly Faintly Slightly Rough to very rough 43 45 3.2 19.5
5 Medium strong rock Very strong rock 4 8 19 47 Slightly Slightly to moderately Slightly Moderately 39 55 2.1 18
Joint spacing (cm) RQD Alteration/weathering Discontinuity roughness RMR-76 In-situ modulus (GPa)
Min Max Min Max Min Max Min Max Min Max Min Max Min Max
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had RQD equal to zero and RQD for the Portugues Dam site is unrealiable. The database includes rock masses with a wide range of geotechnical properties with RMR-76 varying from 15 to 55 - the transition between ‘‘fair rock’’ and ‘‘good rock’’ is RMR-76 equal to 60 [12] - and degree of alteration variations from faintly weathered to decomposed (Table 2). 4. Analytical methods 4.1. Evaluation of models The performance of predictive rock mass modulus models can be assessed by several methods; however the dataset compiled for this dissertation is relatively small with 26 data points and is not particularly robust. Linear correlation coefficient, root mean square error (RMSE), F-statistic, and error ratio were used as metrics to quantify the predictive performance of existing and proposed empirical models. The linear correlation coefficient, r, is a measure of the linearity between bi-variate data and is calculated via
r¼
N 0 1 X Y i Y 0a Yi Ya N 1 i¼1 SY 0 SY
ð1Þ
where N = number of data points, Yi0 = predicted dependent variable, Ya0 = average of predicted dependent variable, SY0 = standard deviation of predicted dependent variable, Yi = measured (true) dependent variable, Ya = average of measured dependent variable, and SY = standard deviation of measured dependent variable. The linear correlation coefficient varies between 1 and 1; an absolute value of unity indicates a perfect linear correlation between predicted and measured moduli, while zero indicates no correlation. Interpreting the linear correlation coefficient is subjective, but in general a good fit can be inferred if the absolute value of the correlation coefficient is greater than 0.7. However, the correlation coefficient does not address lack of precision in the data. RMSE measures the departure of predicted values from measured values and indicates both bias and lack of precision with a low RMSE denoting high predictive power. RMSE is calculated via
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN 2 RMSE ¼ ðY 0i Y i Þ i¼1 N
ð2Þ
The F-test is a hypothesis test used to compare the variances of two data sets, and the F-statistic is commonly used to evaluate the quality of and make comparisons between regressions. For the purposes of evaluating the quality of regression, the F-statistic compares the variance of predicted values against the normalized variance of error. For single-variate models, the F-statistic is calculated via
F¼
varðY 0 Þ Y 0Þ
1 varðY N2
ð3Þ
where F = F-statistic, N = number of data points, and var(i) = variance of dataset i. The F-statistic for multivariate regressions is calculated by Scheaffer and McClave [14] as follows:
F¼
R2 =k ð1 R2 Þ=½N ðk þ 1Þ
ð4Þ
where R2 = multiple coefficient of determination, and k = number of independent variables. The multiple coefficient of determination is calculated via Scheaffer and McClave [14]:
P 2 ðY i YÞ R2 ¼ 1 P 0 2 ðY i Y i Þ
ð5Þ
= mean of true (measured) dependent variable. where Y A high F-statistic signifies the error is low relative to the predicted value, and as such regressions with higher F-statistics can be considered to have better predictive performance for a given dataset. The rock mass moduli assembled for this study vary over three orders of magnitude, and as a result RMSE and F-statistic are most strongly influenced by errors associated with the largest modulus values. However, a logarithmic transformation on modulus can be used to address this deficiency. The linear correlation coefficient, F-statistic, and RMSE are overall metrics of predictive performance and as such do not highlight the specific strengths or weaknesses of the models - for example, specific ranges over which the correlations have the best and worst predictive performances. Calculating error ratios between predicted and measured values is a versatile method of quantifying predictive performance; error ratio, ER, is calculated by
ER ¼
E0 E E
ð6Þ
Error ratio can be plotted against a rock mass classification or modulus to determine ranges over which models are most accurate and any obvious outliers may influence the linear correlation coefficient, RMSE, or F-statistic. In addition, plotting error ratio against measured modulus or rock mass quality may indicate biases not readily apparent from other metrics. The average of the absolute values, minimum, and maximum error ratios can be used as overall metrics of a model’s predictive performance. 4.2. Multivariate linear regression The multivariate linear regression technique is an extension of the commonly used least-squares linear regression method and was used to develop an empirical multivariate weak rock mass modulus model. Multivariate linear regression utilizes the following mathematical model:
y0 ¼ c0 þ c1 x1 þ c2 x2 þ þ ci xi
ð7Þ
where ci = constants or coefficients, xi = independent variables, and y0 = dependent variable. As the regression method is strictly based on a linear model, the first step was to select independent variables, and then based on the field data linearize those variables to the dependent variable or a transformation of the dependent variable (i.e. exponential or logarithmic, among others). Variables were linearized by applying transformation to numerical data or using ‘‘look-up’’ transformations (coding) to convert verbal descriptors such as roughness and alteration intensity into linearized numeric data. Linearization was accomplished entirely by direct manipulation. Once a set of suitable linearized independent variables was identified, multivariate linear regressions were performed using the software program Visual_Data 12.6. Ideally, the empirical database would be divided into two subgroups—one for performing regressions and another for evaluating regression performance. However, the weak rock mass database did not contain a sufficient quantity of data for these two separate tasks; consequently, the complete database was used for both regression development and testing. An important indicator of the validity of the regression is the residuals between predicted and measured values which should have a mean of zero, appear random when plotted against measured modulus, and reasonably follow the normal distribution. If the residuals strongly deviate from these conditions, validity of the resulting correlation is suspect, and transformations may need to be re-evaluated. Normality of the distribution of residuals from the multivariate regression was checked using the Probability Plot Correlation Coefficient (PPCC) test of normality. The PPCC test is a simple
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Modulus (GPa)
R.R. Kallu et al. / International Journal of Mining Science and Technology xxx (2015) xxx–xxx
20 18 16 14 12 10 8 6 4 2
^ = value of dependent variable predicted by multivariate where y model, t = value of t-distribution with N (k + 1) degrees of freedom for a/2, s2 = variance of residuals, x0 = column vector of inde^, and X = matrix of explanatory pendent variables used to predict y observations from multivariate regression.
Nevada data Serafim & Pereira Portugues data Bakhtiary dam data
Read et al. Galera et al. exponential
5. Performance of published models
Galera et al. linear
0
10
20
30
Gokceoglu et al. 50 60
40
The predictive performance of published rock mass modulus models (Table 1) was evaluated against the weak rock mass database. Because Q and GSI could not be reliably estimated for the Portugues and Bakhtiary Dam datasets, only models based on RMR were evaluated. For each model RMR-76 was used as the input and was assumed to be without error. The moduli and published models were plotted against RMR-76 as shown in Fig. 1, and error ratio plots and one-to-one plots (predicted modulus versus measured modulus) are provided in Fig. 2. Overall performance metrics are summarized in Table 3. The linear correlation coefficient is similar for all the models. The minimum, maximum, and average absolute value of error ratio provide the best insight into relative predictive performance, and based on these metrics, models by Gokceoglu et al. and Read et al. provide the best overall predictive performance [3,7]. The equation by Gokceoglu et al. provides good approximations of the Nevada data but underestimates modulus for data from the Portugues and Bakhtiary Dam site data, while the model by Read et al. overpredicts modulus for the Nevada dataset but has the best predictive performance for the dam data [3,7]. Models by Galera et al. and Serafim and Pereira provided overall poor performance for the weak rock mass modulus dataset [4,5].
RMR-76 Fig. 1. Weak rock mass moduli and published models plotted against RMR-76.
method for verifying the adequacy of using the normal distribution to describe a dataset. The fundamental premise of the PPCC test is a normally distributed dataset which will plot as a straight line on a normal probability plot. A probability plot is created by plotting the measured values versus normal quantiles (as calculated from the plotting positions). The linear correlation coefficient is then calculated between the measured values and the normal quantiles and compared against a table of critical values. If the coefficient of linear correlation is lower than the critical value then the null hypothesis of ‘‘data are normally distributed’’ is rejected, and the alternative hypothesis of ‘‘data are not normally distributed’’ is adopted. Probabilistic prediction intervals were calculated for the multivariate model and confidence intervals for predicted modulus at some significance level, a. Prediction intervals incorporate the unexplained variability of the dependent variable as well as the uncertainties associated with the estimates of coefficients for the independent variables. If the residuals are normally distributed, the prediction intervals should contain approximately (1 a) ⁄ 100% of the field data, and a/2 ⁄ 100% of the data should fall beyond each side of the error bounds. For a multivariate regression, the prediction intervals, YPI, are calculated via
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s2 h1 þ xT0 ðX T XÞ x0 i
Serafim & Pereira Read et al. Galera et al. (exponential) Galera et al. (linear) Gokceoglu et al. 1-to-1
Error ratio
40 30 20 10
50 40 30 20 10 0
0 -10
One of the goals of this study was to create a new empirical model that could be used to estimate weak rock mass modulus when only basic field geotechnical data are available;
Estimated modulus (GPa)
50
6.1. Multivariate model
ð8Þ
Error ratio
^t Y PI ¼ y
6. Development of multivariate weak rock mass model
0
10
20
30
40
50
-10 0.01
60
0.1
1
10
100
(a) Error ratio versus RMR-76
15 10 5
5
0
10
15
20
Measured modulus (GPa)
Rock mass modulus (GPa)
RMR-76
20
(b) Error ratio versus measured rock mass modulus
(c) Predicted versus measured modulus
Fig. 2. Error ratio plots and one-to-one plots (predicted modulus versus measured modulus).
Table 3 Performance metrics of published models for the weak rock mass modulus dataset. Model
Linear correlation coefficient
RMSE of ln(modulus) [ln(GPa)]
F-statistic for ln(modulus)
Average of absolute value of error ratio
Minimum error ratio
Maximum error ratio
Serafim and Pereira [5] Read et al. [7] Galera et al. [4] (Exponential) Galera et al. [4] (Linear) Gokceoglu et al. [3] (RMR)
0.49 0.50 0.49 0.48 0.47
1.30 1.10 1.30 1.20 1.10
8.10 33.00 7.40 2.20 16.00
3.50 2.30 3.40 2.70 0.80
0.62 0.53 0.64 0.64 0.89
45.00 14.00 44.00 44.00 6.90
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Table 4 Coding for transforming field estimate of rock strength to SF.
Table 7 Performance metrics for proposed multivariate model.
Estimate of field strength
Factor
Metric
Value
S3 firm clay Dense granular R0 extremely weak rock R0–R1 R1 very weak rock R1–R2 R2 weak rock R2–R3 R3 medium strong rock R3–R4 R4 strong rock R4–R5 R5 very strong rock
3 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
RMSE (GPa) RMSE of natural log of modulus [ln(GPa)] Multiple coefficient of determination Minimum error ratio Average of absolute value of error ratio Maximum error ratio F-statistic F-statistic for natural log of modulus
3.60 0.69 0.62 0.73 0.71 4.00 8.60 17.00
linear correlation coefficients for the natural logarithm of modulus were approximately two times higher than those for the untransformed modulus. The linearizations for the natural logarithm of modulus were then refined. Multiple regressions were performed to identify the combination of inputs and linearizations that produced a model with minimum error. The first criterion for culling regressions was that coefficients on dependent variables must be positive because the linearizations had positive slopes. For example, a negative coefficient for weathering would imply that highly altered rock would have a higher modulus than fresh rock. RMSE and error ratio were used to determine which of the remaining regressions provided the best overall predictive performance. Secondary criteria for selecting the best overall multivariate model were: (1) reducing the number of independent variables and (2) using independent variables that are less ambiguous to define in the field. The best overall multivariate linear regression was
Table 5 Coding for transforming discontinuity roughness to RF. Discontinuity roughness
Factor
Soil-like Smooth Smooth to slightly rough Slightly rough Slightly rough to moderately rough Moderately rough Moderately rough to rough Rough Rough to very rough Very rough
0 1 2 3 4 5 6 7 8 9
E ¼ exp½0:731 þ 0:0846SF þ 0:382lnðBSÞ þ 0:134RF þ 0:157IF
Table 6 Coding for transforming discontinuity infilling hardness to IF. Discontinuity infillings
Factor
Soft infillings Mix of soft and hard infillings Hard infillings
0 1 7
consequently, the new relationship should only use routinely collected geotechnical parameters such as joint spacing or block size, alteration, field estimated rock strength, infilling, and roughness as inputs. Discontinuity aperture was excluded as an input to regressions because it is difficult to accurately measure in core and the definition is not consistent between practitioners. Field estimated rock strength, weathering or alteration, roughness, infilling hardness, and infilling description (hardness and thickness) data were preliminarily linearized against both modulus and the natural logarithm of modulus, and correlation matrices were created. The
15
where E = estimated modulus in GPa, SF = transformed strength factor, BS = average block size or joint spacing in cm, RF = transformed roughness factor, and IF = transformed infilling hardness. The independent variable transformations (codings) are presented in Tables 4–6. Error ratio and one-to-one plots for the multivariate correlation are provided in Fig. 3, and performance metrics are summarized in Table 7. The multivariate model provides good estimates of modulus for the Nevada dataset and 7 of 12 moduli from the Portugues and Bakhtiary Dam sites. The multivariate model under-predicts moduli for the five stiffest rock masses, but the error ratios for these data points are less than an absolute value of 0.6 (60% relative error). With the exception of three moduli, error ratios are within the range of plus-or-minus one. The mean of the residuals of the natural logarithm of modulus was 0.0 ln(GPa), and the normal distribution was adequate for describing the residuals at a significance level of 0.05.
5
Nevada Portugues dam Bakhtiary dam 1-to-1
4
Error ratio
Predicted modulus (GPa)
20
ð9Þ
10
3 2 1
5 0 0
5
10
15
20
-1
0
5
10
15
Measured modulus (GPa)
Measured modulus (GPa)
(a) Predicted versus measured multivariate model
(b) Error ratio plot for proposed multivariate mod
20
Fig. 3. Error ratio and one-to-one plots for the multivariate correlation.
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7
20
Error ratio
Error ratio
20
10
0
-10
0
10
20
30
40
50
Read et al. Gokceoglu et al. Proposed model 1-to-1
10
0
-10 0.01
60
RMR-76
(a) Error ratio versus RMR-76
0.1
1
10
100
Estimated modulus (GPa)
R.R. Kallu et al. / International Journal of Mining Science and Technology xxx (2015) xxx–xxx
20 15 10 5 0
5
10
15
20
Measured rock mass modulus (GPa)
Measured modulus (GPa)
(b) Error ratio versus measured modulus
(c) Predicted versus measured modulus
Fig. 4. Error ratio versus RMR-76 and measured modulus, predicted versus measured modulus for proposed multivariate model and published models by Gokceoglu et al. and Read et al. [3,7].
Consequently, probabilistic prediction intervals, EPI, can be calculated for the multivariate model and are defined as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ^ t 0:4960h1 þ xT0 ðX T XÞ x0 i EPI ¼ exp y
ð10Þ
where
3 2 1 a 7 6 6 SF 7 6 6g 7 6 6 1 T 7 6 x0 ¼ 6 lnðBSÞ 7 ðX XÞ ¼ 6 6h 7 6 6 5 4 RF 4i IF j 2
g
h i
b
k
j
3
7 m7 7 k c n p 7 7 7 l n d q 5 m p q f l
where a = 0.28257; b = 0.04143; c = 0.08523; d = 8.2242 103; f = 6.1440 103; g = 0.06058; h = 0.09877; i = 0.01973; j = 2.0912 103; k = 0.03880; l = 6.4285 103; m = 7.2426 103; n = 3.0643 103; p = 9.5047 104; q = 1.2286 103. To simplify use of the new multivariate correlation in practice, a spreadsheet was created to automate the calculations and is provided as a supplemental file. The spreadsheet requires the average block size or joint spacings and verbal descriptions of field estimated rock strength, discontinuity roughness, and infilling hardness. Block size is keyed-in, and the spreadsheet automatically validates that the value is between 0 and 8 cm. The spreadsheet provides the mean estimated modulus as well as probabilistic prediction intervals at a user defined confidence level, and the architecture of the spreadsheet prevents the user from entering parameters outside those in the empirical database. 6.2. Comparison to published single variate models Comparisons were made between the proposed model and single-variate models by Gokceoglu et al. and Read et al. to determine whether the new multivariate model provides better estimates of rock mass modulus for this dataset [3,7]. The proposed model provides better performance in terms of overall metrics and more accurately predicts modulus over a wider range of rock mass quality than the previous simple-to-use models. While the previous models produced reasonable estimates of modulus over finite ranges of rock mass quality, the new correlation provides good predictions over the range of RMR-76 values 15–55 (Fig. 4a). Similarly, the new correlation can predict very low and moderately high moduli without significant bias (Fig. 4). 7. Conclusions and discussion The data analyses presented in this study depend on assembling a reliable database of weak rock mass moduli and associated geotechnical parameters; implicit assumptions of this database
are that geotechnical parameters and rock mass moduli are isotropic and homogeneous. Variability in logging geotechnical parameters can be relatively large between individual geologists and engineers, and the empirical database captures some of this variability by using data from civil and mining sources. The weak rock mass modulus database was used to test the predictive performance of several published rock mass modulus models that had applicability to weak rock masses and that could be used when geotechnical data are limited such as in the mining industry or in the preliminary stages of civil engineering projects. These RMR-based models proved to be accurate over narrow ranges of rock mass quality for this database, and none provided uncertainty or error bounds. Given the limitations of the existing relationships, a new empirical multivariate modulus model was developed based on the weak rock mass database. The geotechnical inputs were assumed to be discrete values without any associated error, and as such the proposed model provides high-level estimates of modulus. The multivariate model requires block size or average joint spacing, roughness, field estimated strength, and infilling hardness as inputs and is exponential in the form matching many models in the literature. For this dataset, infilling hardness proved to be acceptable as an input, but this likely does not hold for all rock masses. The new model has the additional benefit of providing reasonable estimates of modulus for a wide range of geotechnical conditions from ‘‘soil-like’’ to ‘‘good-quality’’ rock masses. Because the multivariate model is empirical, the same limitations of the database are imposed on the results of the correlation; namely, the predicted moduli only apply to rock masses that are isotropic, homogeneous, and reasonably approximated as linear elastic. Additionally, predicted moduli have a precision of two digits at best. While the new model requires less geotechnical data than is needed to calculate RMR, the structure of the correlation is intricate and can be time consuming to use. The error bounds can be large because of substantial variability in the small empirical database. This variability could have been reduced by further culling the data, but data points that produced large residuals were included in the analyses to preserve the inherent uncertainty associated with predicting rock mass mechanical properties. While the new multivariate model seems to provide better estimates of weak rock mass modulus over a wider range of geotechnical conditions, it is not without limitations. The empirical database does not contain a robust number of data points, and the performance of the model could not be independently tested. As with all empirical methods, the multivariate correlation should not be used to estimate moduli for conditions not encompassed by the weak rock mass database. The major limitations for using the new correlation are joint spacings, which must be less than 8 cm,
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discontinuities contain infillings, and depths should range between 45 and 730 m. Even with these limitations and deficiencies, the multivariate model may prove useful for preliminary or high-level designs of excavations in intensely-fractured and altered rock masses when only basic geotechnical data are available. Once more weak rock mass modulus data become available, the multivariate model can be independently tested and the probabilistic error bounds can be refined. Acknowledgments This research was funded by the National Institute of Occupational Safety and Health through research contract 200-2011-39965 (Principal Investigator Dr. Kallu) and University of Nevada, Reno, NV. References [1] Hoek E, Diederichs MS. Empirical estimation of rock mass modulus. Int J Rock Mech Min Sci 2006;43:203–15. [2] Palstrom A, Singh R. The deformation modulus of rock masses-comparisons between in-situ tests and indirect estimates. Tunn Undergr Space Technol 2001;16:115–31. [3] Gokceoglu C, Somnez H, Kayabasi A. Predicting the deformation moduli of rock masses. Int J Rock Mech Min Sci 2003;40:701–10.
[4] Galera JM, Alvarez M, Bieniawski ZT. Evaluation of the deformation modulus of rock masses using RMR: comparison with dilatometer tests. In: ISRM, Lisbon; 2007. p. 71–7. [5] Serafim JL, Pereira JP. Considerations on the geomechanical classification of Bieniawski. In: International symposium on engineering geology and underground construction. Lisbon; 1983. p. 33–44. [6] Bieniawski ZT. Determining rock mass deformability: experience from case histories. Int J Rock Mech Min Sci Geomech Abstr 1978;15:237–47. [7] Read SAL, Perrin ND, Richards LR. Applicability of the Hoek-Brown failure criterion to New Zealand greywacke rocks. In: 90th International congress on rock mechanics, Paris; 1999. p. 655–60. [8] Kayabasi A, Gokceoglu C, Ercanoglu M. Estimating the deformation modulus of rock masses: a comparative study. Int J Rock Mech Min Sci 2003;40:55–63. [9] Keffeler ER. Measurement and prediction of in-situ weak rock mass modulus: case studies from Nevada, Puerto Rico, and Iran. Reno: University of NevadaReno; 2014. [10] Agharazi A, Tannant DD, Martin CD. Characterizing rock mass deformation mechanisms during plate loading tests at the Bakhtiary dam project. Int J Rock Mech Min Sci 2012;49:1–11. [11] Agharazi A. Development of a 3D equivalent continuum model for deformation analysis of systematically jointed rock masses. Edmonton: University of Alberta; 2013. [12] Bieniawski ZT. Rock mass classification in rock engineering. In: Proceedings of the symposium on exploration for rock engineering. Johannesburg; 1976. p. 97–106. [13] Bieniawski ZT. Engineering rock mass classifications. New York: John Wiley & Sons; 1989. [14] Scheaffer RL, McClave JT. Probability and statistics for engineers. Belmont: Wadsworth Publishing Company; 1995.
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Contents lists available at ScienceDirect
International Journal of Mining Science and Technology journal homepage: www.elsevier.com/locate/ijmst
Development of a multivariate empirical model for predicting weak rock mass modulus Kallu Raj R. a,⇑, Keffeler Evan R. b, Watters Robert J. c, Agharazi Alireza d a
Department of Mining Engineering, University of Nevada, Reno, NV 89557, USA RESPEC Consulting and Services, Rapid City, SD 57709-0725, USA c University of Nevada, Reno, NV 89557, USA d Itasca Houston Inc., Stafford, TX 77477, USA b
a r t i c l e
i n f o
Article history: Received 15 December 2014 Received in revised form 18 January 2015 Accepted 16 February 2015 Available online xxxx Keywords: In-situ modulus Weak rock mass Preliminary design Elastic deformation
a b s t r a c t Estimating weak rock mass modulus has historically proven difficult although this mechanical property is an important input to many types of geotechnical analyses. An empirical database of weak rock mass modulus with associated detailed geotechnical parameters was assembled from plate loading tests performed at underground mines in Nevada, the Bakhtiary Dam project, and Portugues Dam project. The database was used to assess the accuracy of published single-variate models and to develop a multivariate model for predicting in-situ weak rock mass modulus when limited geotechnical data are available. Only two of the published models were adequate for predicting modulus of weak rock masses over limited ranges of alteration intensities, and none of the models provided good estimates of modulus over a range of geotechnical properties. In light of this shortcoming, a multivariate model was developed from the weak rock mass modulus dataset, and the new model is exponential in form and has the following independent variables: (1) average block size or joint spacing, (2) field estimated rock strength, (3) discontinuity roughness, and (4) discontinuity infilling hardness. The multivariate model provided better estimates of modulus for both hard-blocky rock masses and intensely-altered rock masses. Ó 2015 Published by Elsevier B.V. on behalf of China University of Mining & Technology.
1. Introduction Underground mines are increasingly being excavated in relatively deep and weak mineralized zones that are typically intensely-fractured and highly-altered with geotechnical properties resembling those of stiff soils, and civil engineering projects such as dams and tunnels are more frequently being constructed in geological domains that contain zones of highly-fractured rock. For example, most of the modern underground mines in Nevada, United States, are produced from Carlin-type deposits where disseminated gold and silver are hosted by intensely-fractured and moderately- to highly-altered sedimentary sequences. The host rocks are predominantly an assemblage of limestone, shale, volcanics, and breccias which have been fractured, folded, and locally hydrothermally altered resulting in a wide range of mechanical properties. Given the cost and complexity of in-situ deformability tests, direct measurements of rock mass moduli are typically not performed at active mines, which make the use of empirical models ⇑ Corresponding author. Tel.: +1 7756826448. E-mail address: [email protected] (R.R. Kallu).
attractive. However, the majority of published empirical equations are largely based on data sets with few measurements in weak and very weak rock masses. Additionally, common rock mass classification systems (i.e. Rock Mass Rating, RMR; Rock Tunneling Quality Index, Q) tend to be insensitive in these types of rocks, and geotechnical engineers at mines typically have relatively little geotechnical data to work with compared to civil engineering projects. Consequently, using predictive correlations in a mining environment can be difficult because of a lack of reliable input data. For many civil engineering projects such as dams and tunnels, available locations or alignments may be sited in geologic domains that contain zones of highly-fractured or altered rock masses. While large civil engineering projects typically include in-situ rock mass modulus testing programs, these field tests are expensive and time consuming, and interpreting the resulting data can be ambiguous [1]. At the feasibility and preliminary (high-level) design stages, field testing of rock mass modulus typically cannot be justified, and as a result predictive empirical models are employed [2]. During these early stages, geotechnical data are likely limited to basic rock mass parameters that have been quantified by field mapping, a limited number of geotechnical core holes, or literature reviews. Consequently, the civil engineering
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sector also has a need for a reasonably accurate weak rock mass modulus correlation that requires only basic geotechnical parameters as inputs for feasibility analyses and preliminary designs. Test data and geotechnical characterizations were obtained for stiff plate loading tests performed on heavily fractured rock masses at two underground mines in Nevada (United States), the Portugues Dam project in Puerto Rico (United States Territory), and the Bakhtiary Dam project in Iran. The weak rock mass moduli measured at civil and mining projects were combined to create a weak rock mass modulus database with associated detailed geotechnical characterizations. The moduli in the database were compared to several published rock mass modulus models to determine which, if any, are suitable for estimating weak rock mass deformation properties when limited geotechnical data are available. Based on the limitations identified in the published equations, an empirical multivariate model was developed, and the model only requires geotechnical parameters that could be easily quantified by field mapping or core logging. 2. Review of predictive models for weak rock mass modulus An abundance of predictive empirical in-situ modulus models can be found in the literature, and one of the objectives of this study was to determine which, if any, of the existing models are adequate for predicting the moduli of weak rock masses when geotechnical data are limited. The mining environment and preliminary design stages of civil projects have unique limitations on available geotechnical data, and typically only lithological, rock mass classification (RMR, Q; or Geological Strength Index, GSI), and point load strength data may be available. Laboratory strength or deformability data for intact rock may be limited or nonexistent. Eight rock mass modulus models were identified in the literature that are suitable for use when only basic geotechnical data are available (Table 1); these models: (1) were based on data sets that included at least some weak rock mass modulus measurements and (2) only require geomechanical data that can be quantified by field mapping or core logging. The Simplified Hoek and Diederichs [1] equation is based on a dataset of in-situ modulus tests performed in China and Taiwan. The dataset has a population of 494 data points with GSI values between 10 and 95, and excludes moduli measured with downhole deformation jacks and dilatometers, and in many cases GSI was calculated directly from RMR. The model requires a disturbance factor, D, to be estimated, which varies from zero to one. A value of zero represents a tightly interlocked rock mass, while one corresponds to a fully disturbed and loosened rock mass. The disturbance factor has proven difficult to estimate and at best is selected subjectively [1,3]. Table 1 Rock mass modulus models that are applicable to weak rock masses and can be used when only limited geotechnical data are available.
Palstrom and Singh [2] assembled a database of in-situ modulus from 42 tests performed at hydroelectric projects in India, Bhutan, and Nepal. Rock types included gneiss, granite, mica schist, sandstone, mudstone, siltstone, and dolerite, and RMR varied between 46 and 75 (1.1 6 Q 6 30). Correction factors of up to 7.5 were applied to moduli to account for effects of blast damage and volume of rock tested. Even with these corrections, previously published models over-predicted moduli of the lower quality rock masses in their dataset. Palstrom and Singh [2] suggested a new Q-based correlation that has some applicability for rock masses with Q-values near 1. Galera et al. [4] assembled a database of in-situ moduli values from downhole dilatometer and pressure meter tests. Tests with moduli less than 0.5 GPa or that were performed on highly-weathered rock masses were censored from the database to eliminate ‘‘soil behavior’’, which resulted in 427 data points that included rock mass modulus, RMR, and Rock Quality Designation (RQD). Their analyses indicated RMR was a better predictor of rock mass modulus than RQD and that lithology was not a significant parameter. Based on the censored dataset, they developed linear and quadratic models. Serafim and Pereira created a relationship based on a combination of their own data and the dataset presented by Bieniawski [5,6]. The field test methods were not specified. The dataset overall has RMR values from 23 to 85, and the resulting model is exponential in form. The relationship by Read et al. was a re-fitting of the Serafim and Pereira and Bieniawski datasets with rock mass modulus constrained to equal 100 GPa when RMR was 100 [5– 7]. This constraint was imposed to prevent the correlation from predicting unrealistically high modulus values when RMR approached 100. Gokceoglu et al. created a new rock mass modulus database by combining the results of 58 situ deformability tests with 57 moduli originally published by Kayabasi et al. [3,8]. The data were generated from dilatometer and plate loading tests were conducted at the Deriner Dam site and Ermenek Dam site in Turkey. The tests were performed on quartz diorite, limestone, and heavily-jointed marl, and RMR varied between 20 and 85 with 17 tests in weak rock masses. Each data point includes RMR, GSI, RQD, uniaxial compressive strength of the intact rock, elastic modulus of the intact rock, and discontinuity properties. Gokceoglu et al. [3] developed new single variable and multi-variable models. For the single input equations, they evaluated linear, logarithmic, power, and exponential models; for both GSI and RMR the exponential correlations had the highest coefficients of regression. Multivariate regression analyses indicated intact rock modulus, uniaxial compressive strength, RQD, and discontinuity weathering degree were the most significant variables for predicting rock mass modulus. However, this multivariate model can be difficult to use when geotechnical data are limited because reliable measurements of intact rock properties may not be available, and in weak rock masses RQD is typically zero or near zero.
Model
Dependent variable
Applicability
Equation (GPa)
Hoek and Diederichs Simplified [1] Palstrom and Singh [2] Galera et al. [4] (Linear) Galera et al. [4] (Exponential) Serafim and Pereira [5] Read et al. [7]
GSI
20
1D=2 E ¼ 100 1þe½ð75þ25DGSIÞ=11
3. Database of weak rock mass modulus
Q
1 < Q < 30
E ¼ 8Q 0:4
3.1. Overview of data sources
RMR
10 < RMR < 50
E ¼ 0:0876RMR
RMR
10 < RMR < 80
E¼e
RMR
30 < RMR < 50
E ¼ 10ðRMR10Þ=40
RMR
20 < RMR < 80
Gokceoglu et al. [3] Gokceoglu et al. [3]
RMR GSI
20 < RMR < 85 15 < GSI < 77
E ¼ 0:1ðRMR=10Þ3 E ¼ 0:0736e0:0755RMR E ¼ 0:1451e0:0654GSI
A database of weak rock mass modulus was assembled for the results of stiff plate loading tests performed at two underground mines in Nevada, the Bakhtiary Dam and Hydroelectric Project in Iran, and the Portugues Dam in Puerto Rico. In-situ modulus was calculated from the reload portions of the force–displacement plots, and geotechnical parameters were determined from core logs following the guidelines developed in Section 3 or by window mapping.
ðRMR10Þ=18
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3.2. Underground mines of Nevada Plate loading tests were performed at two underground gold mines in Nevada yielding 14 modulus measurements. The field methods are described in detail along with the geotechnical conditions of each test site [9]. In both mines, the deposits are disseminated gold and silver hosted by altered sedimentary sequences, and the country rock is predominantly limestone and marl transected by altered porphyry dikes. Tests were performed at depths between 510 and 730 m, and the lithology of test sites included fault breccia, argillized porphyry dikes, and decalcified, oxidized, and/or argillized limestone. 3.3. Bakhtiary dam site The Bakhtiary Dam and Hydroelectric Project is located in the Zagros Mountains of southwest Iran and when finished will include a 315 m tall double-curvature concrete arch dam. The Bakhtiary Dam project is sited in the marly and siliceous limestone beds of the Cretaceous Sarvak formation. Kink bands associated with the Siah Kuh Anticline created zones of intensely folded and fractured rock in the abutments and foundation. An extensive foundation study included plate loading tests performed in exploration adits excavated into the abutments. Full details on the site geology and field test methods are provided in Agharazi et al. and Agharazi [10,11]. Agharazi et al. [10] analyzed the results of plate loading tests at the Bakhtiary Dam site and determined that for the intensely fractured rock, in-situ deformation behavior was a combination of both normal closure and shear displacement of joints, which ultimately results in compaction of the rock mass. With successive load–unload cycles, the fractured rock stiffened indicating the pieces became more-tightly interlocked after each cycle, and the stress–displacement plots were mainly non-linear with high total and permanent displacements. Based on force–displacement data from plate loading tests, eight rock mass moduli were calculated for intensely fractured but minimally weathered rock masses at depths between 20 and 210 m. Representative geotechnical properties were determined from core logs, and the geotechnical properties were similar for each site. Discontinuities were moderately rough to very rough, and infillings consisted of calcite and iron oxide up to 2 mm thick. Intact rock was slightly to faintly weathered. Laboratory strength testing of core samples indicates intact saturated uniaxial compressive strengths of 110 ± 30 MPa and intact deformation moduli of 69 ± 10 GPa [10].
performed a foundation stability investigation for the project from 1983 to 1988. The foundation study is detailed by USACE, and rock mass moduli are summarized by Stephens and Banks. The dam is sited in upper Cretaceous to Eocene siltstones, sandstones, and conglomerates that were metamorphosed to low-grade metasediments by an Eocene diorite stock. Numerous intrusive dikes radiate from the stock and transect the dam area. The rock has been highly altered by the humid-subtropical climate, but degree and depth of weathering vary greatly. Three adits were excavated into the abutments at depths up to 45 m below the ground surface in order to establish access for plate loading tests. The design memorandum by USACE contains detailed core logs for each plate loading test location, which include the following geotechnical parameters: lithology, recovery, fracture frequency, modified RQD, and intact rock soundness; waviness, roughness, aperture, weathering, and infilling were described for individual joints as well. Moderate to high-angle (relative to the core axis) discontinuities were ignored when calculating the modified RQD because ‘‘the in-situ quality of the rock is much better than the standard RQD indicated’’, and as such the reported RQD is biased high. Several samples were selected from each core hole for laboratory uniaxial compression strength and elastic deformation testing. Representative geotechnical properties for each plate loading tests site were determined from the core and discontinuity logs.
3.5. Description of database An empirical weak rock mass modulus database with 26 data points was assembled, and representative geotechnical properties were determined by window mapping for the Nevada test sites, and from geotechnical core logs for the Bakhtiary and Portugues Dam sites. Each data point includes the following attributes:
Field estimated rock strength (R-strength) Aperture (mm)/clast support Infilling type Infilling hardness Infilling thickness (mm) Weathering/alteration Block size/average joint spacing (cm) Roughness RMR-76 and parameter ratings [12] RMR-89 and parameter ratings [13] Reload (in-situ) modulus.
3.4. Portugues Dam site The Portugues Dam site is located near Ponce, Puerto Rico, and the Jacksonville District of the US Army Corp of Engineers (USACE)
Insufficient geotechnical data were available to confidently determine Q-values or GSI for many of the data points. RQD was excluded from analyses because more the half of the test sites
Table 2 Summary of geotechnical conditions for in-situ modulus measurements. Parameter
Nevada mines
Bakhtiary dam
Portugues dam
Number of in-situ measurements Field estimated rock strength
13 Stiff cohesive Strong to very strong rock 0.1 3 0 0 Slightly to moderately Decomposed Stiff cohesive Very rough 15 47 0.031 4.2
8 Strong to very strong rock Very strong rock 2 7 3 25 Slightly Faintly Slightly Rough to very rough 43 45 3.2 19.5
5 Medium strong rock Very strong rock 4 8 19 47 Slightly Slightly to moderately Slightly Moderately 39 55 2.1 18
Joint spacing (cm) RQD Alteration/weathering Discontinuity roughness RMR-76 In-situ modulus (GPa)
Min Max Min Max Min Max Min Max Min Max Min Max Min Max
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had RQD equal to zero and RQD for the Portugues Dam site is unrealiable. The database includes rock masses with a wide range of geotechnical properties with RMR-76 varying from 15 to 55 - the transition between ‘‘fair rock’’ and ‘‘good rock’’ is RMR-76 equal to 60 [12] - and degree of alteration variations from faintly weathered to decomposed (Table 2). 4. Analytical methods 4.1. Evaluation of models The performance of predictive rock mass modulus models can be assessed by several methods; however the dataset compiled for this dissertation is relatively small with 26 data points and is not particularly robust. Linear correlation coefficient, root mean square error (RMSE), F-statistic, and error ratio were used as metrics to quantify the predictive performance of existing and proposed empirical models. The linear correlation coefficient, r, is a measure of the linearity between bi-variate data and is calculated via
r¼
N 0 1 X Y i Y 0a Yi Ya N 1 i¼1 SY 0 SY
ð1Þ
where N = number of data points, Yi0 = predicted dependent variable, Ya0 = average of predicted dependent variable, SY0 = standard deviation of predicted dependent variable, Yi = measured (true) dependent variable, Ya = average of measured dependent variable, and SY = standard deviation of measured dependent variable. The linear correlation coefficient varies between 1 and 1; an absolute value of unity indicates a perfect linear correlation between predicted and measured moduli, while zero indicates no correlation. Interpreting the linear correlation coefficient is subjective, but in general a good fit can be inferred if the absolute value of the correlation coefficient is greater than 0.7. However, the correlation coefficient does not address lack of precision in the data. RMSE measures the departure of predicted values from measured values and indicates both bias and lack of precision with a low RMSE denoting high predictive power. RMSE is calculated via
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 XN 2 RMSE ¼ ðY 0i Y i Þ i¼1 N
ð2Þ
The F-test is a hypothesis test used to compare the variances of two data sets, and the F-statistic is commonly used to evaluate the quality of and make comparisons between regressions. For the purposes of evaluating the quality of regression, the F-statistic compares the variance of predicted values against the normalized variance of error. For single-variate models, the F-statistic is calculated via
F¼
varðY 0 Þ Y 0Þ
1 varðY N2
ð3Þ
where F = F-statistic, N = number of data points, and var(i) = variance of dataset i. The F-statistic for multivariate regressions is calculated by Scheaffer and McClave [14] as follows:
F¼
R2 =k ð1 R2 Þ=½N ðk þ 1Þ
ð4Þ
where R2 = multiple coefficient of determination, and k = number of independent variables. The multiple coefficient of determination is calculated via Scheaffer and McClave [14]:
P 2 ðY i YÞ R2 ¼ 1 P 0 2 ðY i Y i Þ
ð5Þ
= mean of true (measured) dependent variable. where Y A high F-statistic signifies the error is low relative to the predicted value, and as such regressions with higher F-statistics can be considered to have better predictive performance for a given dataset. The rock mass moduli assembled for this study vary over three orders of magnitude, and as a result RMSE and F-statistic are most strongly influenced by errors associated with the largest modulus values. However, a logarithmic transformation on modulus can be used to address this deficiency. The linear correlation coefficient, F-statistic, and RMSE are overall metrics of predictive performance and as such do not highlight the specific strengths or weaknesses of the models - for example, specific ranges over which the correlations have the best and worst predictive performances. Calculating error ratios between predicted and measured values is a versatile method of quantifying predictive performance; error ratio, ER, is calculated by
ER ¼
E0 E E
ð6Þ
Error ratio can be plotted against a rock mass classification or modulus to determine ranges over which models are most accurate and any obvious outliers may influence the linear correlation coefficient, RMSE, or F-statistic. In addition, plotting error ratio against measured modulus or rock mass quality may indicate biases not readily apparent from other metrics. The average of the absolute values, minimum, and maximum error ratios can be used as overall metrics of a model’s predictive performance. 4.2. Multivariate linear regression The multivariate linear regression technique is an extension of the commonly used least-squares linear regression method and was used to develop an empirical multivariate weak rock mass modulus model. Multivariate linear regression utilizes the following mathematical model:
y0 ¼ c0 þ c1 x1 þ c2 x2 þ þ ci xi
ð7Þ
where ci = constants or coefficients, xi = independent variables, and y0 = dependent variable. As the regression method is strictly based on a linear model, the first step was to select independent variables, and then based on the field data linearize those variables to the dependent variable or a transformation of the dependent variable (i.e. exponential or logarithmic, among others). Variables were linearized by applying transformation to numerical data or using ‘‘look-up’’ transformations (coding) to convert verbal descriptors such as roughness and alteration intensity into linearized numeric data. Linearization was accomplished entirely by direct manipulation. Once a set of suitable linearized independent variables was identified, multivariate linear regressions were performed using the software program Visual_Data 12.6. Ideally, the empirical database would be divided into two subgroups—one for performing regressions and another for evaluating regression performance. However, the weak rock mass database did not contain a sufficient quantity of data for these two separate tasks; consequently, the complete database was used for both regression development and testing. An important indicator of the validity of the regression is the residuals between predicted and measured values which should have a mean of zero, appear random when plotted against measured modulus, and reasonably follow the normal distribution. If the residuals strongly deviate from these conditions, validity of the resulting correlation is suspect, and transformations may need to be re-evaluated. Normality of the distribution of residuals from the multivariate regression was checked using the Probability Plot Correlation Coefficient (PPCC) test of normality. The PPCC test is a simple
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Modulus (GPa)
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20 18 16 14 12 10 8 6 4 2
^ = value of dependent variable predicted by multivariate where y model, t = value of t-distribution with N (k + 1) degrees of freedom for a/2, s2 = variance of residuals, x0 = column vector of inde^, and X = matrix of explanatory pendent variables used to predict y observations from multivariate regression.
Nevada data Serafim & Pereira Portugues data Bakhtiary dam data
Read et al. Galera et al. exponential
5. Performance of published models
Galera et al. linear
0
10
20
30
Gokceoglu et al. 50 60
40
The predictive performance of published rock mass modulus models (Table 1) was evaluated against the weak rock mass database. Because Q and GSI could not be reliably estimated for the Portugues and Bakhtiary Dam datasets, only models based on RMR were evaluated. For each model RMR-76 was used as the input and was assumed to be without error. The moduli and published models were plotted against RMR-76 as shown in Fig. 1, and error ratio plots and one-to-one plots (predicted modulus versus measured modulus) are provided in Fig. 2. Overall performance metrics are summarized in Table 3. The linear correlation coefficient is similar for all the models. The minimum, maximum, and average absolute value of error ratio provide the best insight into relative predictive performance, and based on these metrics, models by Gokceoglu et al. and Read et al. provide the best overall predictive performance [3,7]. The equation by Gokceoglu et al. provides good approximations of the Nevada data but underestimates modulus for data from the Portugues and Bakhtiary Dam site data, while the model by Read et al. overpredicts modulus for the Nevada dataset but has the best predictive performance for the dam data [3,7]. Models by Galera et al. and Serafim and Pereira provided overall poor performance for the weak rock mass modulus dataset [4,5].
RMR-76 Fig. 1. Weak rock mass moduli and published models plotted against RMR-76.
method for verifying the adequacy of using the normal distribution to describe a dataset. The fundamental premise of the PPCC test is a normally distributed dataset which will plot as a straight line on a normal probability plot. A probability plot is created by plotting the measured values versus normal quantiles (as calculated from the plotting positions). The linear correlation coefficient is then calculated between the measured values and the normal quantiles and compared against a table of critical values. If the coefficient of linear correlation is lower than the critical value then the null hypothesis of ‘‘data are normally distributed’’ is rejected, and the alternative hypothesis of ‘‘data are not normally distributed’’ is adopted. Probabilistic prediction intervals were calculated for the multivariate model and confidence intervals for predicted modulus at some significance level, a. Prediction intervals incorporate the unexplained variability of the dependent variable as well as the uncertainties associated with the estimates of coefficients for the independent variables. If the residuals are normally distributed, the prediction intervals should contain approximately (1 a) ⁄ 100% of the field data, and a/2 ⁄ 100% of the data should fall beyond each side of the error bounds. For a multivariate regression, the prediction intervals, YPI, are calculated via
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 s2 h1 þ xT0 ðX T XÞ x0 i
Serafim & Pereira Read et al. Galera et al. (exponential) Galera et al. (linear) Gokceoglu et al. 1-to-1
Error ratio
40 30 20 10
50 40 30 20 10 0
0 -10
One of the goals of this study was to create a new empirical model that could be used to estimate weak rock mass modulus when only basic field geotechnical data are available;
Estimated modulus (GPa)
50
6.1. Multivariate model
ð8Þ
Error ratio
^t Y PI ¼ y
6. Development of multivariate weak rock mass model
0
10
20
30
40
50
-10 0.01
60
0.1
1
10
100
(a) Error ratio versus RMR-76
15 10 5
5
0
10
15
20
Measured modulus (GPa)
Rock mass modulus (GPa)
RMR-76
20
(b) Error ratio versus measured rock mass modulus
(c) Predicted versus measured modulus
Fig. 2. Error ratio plots and one-to-one plots (predicted modulus versus measured modulus).
Table 3 Performance metrics of published models for the weak rock mass modulus dataset. Model
Linear correlation coefficient
RMSE of ln(modulus) [ln(GPa)]
F-statistic for ln(modulus)
Average of absolute value of error ratio
Minimum error ratio
Maximum error ratio
Serafim and Pereira [5] Read et al. [7] Galera et al. [4] (Exponential) Galera et al. [4] (Linear) Gokceoglu et al. [3] (RMR)
0.49 0.50 0.49 0.48 0.47
1.30 1.10 1.30 1.20 1.10
8.10 33.00 7.40 2.20 16.00
3.50 2.30 3.40 2.70 0.80
0.62 0.53 0.64 0.64 0.89
45.00 14.00 44.00 44.00 6.90
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Table 4 Coding for transforming field estimate of rock strength to SF.
Table 7 Performance metrics for proposed multivariate model.
Estimate of field strength
Factor
Metric
Value
S3 firm clay Dense granular R0 extremely weak rock R0–R1 R1 very weak rock R1–R2 R2 weak rock R2–R3 R3 medium strong rock R3–R4 R4 strong rock R4–R5 R5 very strong rock
3 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
RMSE (GPa) RMSE of natural log of modulus [ln(GPa)] Multiple coefficient of determination Minimum error ratio Average of absolute value of error ratio Maximum error ratio F-statistic F-statistic for natural log of modulus
3.60 0.69 0.62 0.73 0.71 4.00 8.60 17.00
linear correlation coefficients for the natural logarithm of modulus were approximately two times higher than those for the untransformed modulus. The linearizations for the natural logarithm of modulus were then refined. Multiple regressions were performed to identify the combination of inputs and linearizations that produced a model with minimum error. The first criterion for culling regressions was that coefficients on dependent variables must be positive because the linearizations had positive slopes. For example, a negative coefficient for weathering would imply that highly altered rock would have a higher modulus than fresh rock. RMSE and error ratio were used to determine which of the remaining regressions provided the best overall predictive performance. Secondary criteria for selecting the best overall multivariate model were: (1) reducing the number of independent variables and (2) using independent variables that are less ambiguous to define in the field. The best overall multivariate linear regression was
Table 5 Coding for transforming discontinuity roughness to RF. Discontinuity roughness
Factor
Soil-like Smooth Smooth to slightly rough Slightly rough Slightly rough to moderately rough Moderately rough Moderately rough to rough Rough Rough to very rough Very rough
0 1 2 3 4 5 6 7 8 9
E ¼ exp½0:731 þ 0:0846SF þ 0:382lnðBSÞ þ 0:134RF þ 0:157IF
Table 6 Coding for transforming discontinuity infilling hardness to IF. Discontinuity infillings
Factor
Soft infillings Mix of soft and hard infillings Hard infillings
0 1 7
consequently, the new relationship should only use routinely collected geotechnical parameters such as joint spacing or block size, alteration, field estimated rock strength, infilling, and roughness as inputs. Discontinuity aperture was excluded as an input to regressions because it is difficult to accurately measure in core and the definition is not consistent between practitioners. Field estimated rock strength, weathering or alteration, roughness, infilling hardness, and infilling description (hardness and thickness) data were preliminarily linearized against both modulus and the natural logarithm of modulus, and correlation matrices were created. The
15
where E = estimated modulus in GPa, SF = transformed strength factor, BS = average block size or joint spacing in cm, RF = transformed roughness factor, and IF = transformed infilling hardness. The independent variable transformations (codings) are presented in Tables 4–6. Error ratio and one-to-one plots for the multivariate correlation are provided in Fig. 3, and performance metrics are summarized in Table 7. The multivariate model provides good estimates of modulus for the Nevada dataset and 7 of 12 moduli from the Portugues and Bakhtiary Dam sites. The multivariate model under-predicts moduli for the five stiffest rock masses, but the error ratios for these data points are less than an absolute value of 0.6 (60% relative error). With the exception of three moduli, error ratios are within the range of plus-or-minus one. The mean of the residuals of the natural logarithm of modulus was 0.0 ln(GPa), and the normal distribution was adequate for describing the residuals at a significance level of 0.05.
5
Nevada Portugues dam Bakhtiary dam 1-to-1
4
Error ratio
Predicted modulus (GPa)
20
ð9Þ
10
3 2 1
5 0 0
5
10
15
20
-1
0
5
10
15
Measured modulus (GPa)
Measured modulus (GPa)
(a) Predicted versus measured multivariate model
(b) Error ratio plot for proposed multivariate mod
20
Fig. 3. Error ratio and one-to-one plots for the multivariate correlation.
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20
Error ratio
Error ratio
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-10
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Read et al. Gokceoglu et al. Proposed model 1-to-1
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-10 0.01
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RMR-76
(a) Error ratio versus RMR-76
0.1
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Estimated modulus (GPa)
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20 15 10 5 0
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Measured rock mass modulus (GPa)
Measured modulus (GPa)
(b) Error ratio versus measured modulus
(c) Predicted versus measured modulus
Fig. 4. Error ratio versus RMR-76 and measured modulus, predicted versus measured modulus for proposed multivariate model and published models by Gokceoglu et al. and Read et al. [3,7].
Consequently, probabilistic prediction intervals, EPI, can be calculated for the multivariate model and are defined as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ^ t 0:4960h1 þ xT0 ðX T XÞ x0 i EPI ¼ exp y
ð10Þ
where
3 2 1 a 7 6 6 SF 7 6 6g 7 6 6 1 T 7 6 x0 ¼ 6 lnðBSÞ 7 ðX XÞ ¼ 6 6h 7 6 6 5 4 RF 4i IF j 2
g
h i
b
k
j
3
7 m7 7 k c n p 7 7 7 l n d q 5 m p q f l
where a = 0.28257; b = 0.04143; c = 0.08523; d = 8.2242 103; f = 6.1440 103; g = 0.06058; h = 0.09877; i = 0.01973; j = 2.0912 103; k = 0.03880; l = 6.4285 103; m = 7.2426 103; n = 3.0643 103; p = 9.5047 104; q = 1.2286 103. To simplify use of the new multivariate correlation in practice, a spreadsheet was created to automate the calculations and is provided as a supplemental file. The spreadsheet requires the average block size or joint spacings and verbal descriptions of field estimated rock strength, discontinuity roughness, and infilling hardness. Block size is keyed-in, and the spreadsheet automatically validates that the value is between 0 and 8 cm. The spreadsheet provides the mean estimated modulus as well as probabilistic prediction intervals at a user defined confidence level, and the architecture of the spreadsheet prevents the user from entering parameters outside those in the empirical database. 6.2. Comparison to published single variate models Comparisons were made between the proposed model and single-variate models by Gokceoglu et al. and Read et al. to determine whether the new multivariate model provides better estimates of rock mass modulus for this dataset [3,7]. The proposed model provides better performance in terms of overall metrics and more accurately predicts modulus over a wider range of rock mass quality than the previous simple-to-use models. While the previous models produced reasonable estimates of modulus over finite ranges of rock mass quality, the new correlation provides good predictions over the range of RMR-76 values 15–55 (Fig. 4a). Similarly, the new correlation can predict very low and moderately high moduli without significant bias (Fig. 4). 7. Conclusions and discussion The data analyses presented in this study depend on assembling a reliable database of weak rock mass moduli and associated geotechnical parameters; implicit assumptions of this database
are that geotechnical parameters and rock mass moduli are isotropic and homogeneous. Variability in logging geotechnical parameters can be relatively large between individual geologists and engineers, and the empirical database captures some of this variability by using data from civil and mining sources. The weak rock mass modulus database was used to test the predictive performance of several published rock mass modulus models that had applicability to weak rock masses and that could be used when geotechnical data are limited such as in the mining industry or in the preliminary stages of civil engineering projects. These RMR-based models proved to be accurate over narrow ranges of rock mass quality for this database, and none provided uncertainty or error bounds. Given the limitations of the existing relationships, a new empirical multivariate modulus model was developed based on the weak rock mass database. The geotechnical inputs were assumed to be discrete values without any associated error, and as such the proposed model provides high-level estimates of modulus. The multivariate model requires block size or average joint spacing, roughness, field estimated strength, and infilling hardness as inputs and is exponential in the form matching many models in the literature. For this dataset, infilling hardness proved to be acceptable as an input, but this likely does not hold for all rock masses. The new model has the additional benefit of providing reasonable estimates of modulus for a wide range of geotechnical conditions from ‘‘soil-like’’ to ‘‘good-quality’’ rock masses. Because the multivariate model is empirical, the same limitations of the database are imposed on the results of the correlation; namely, the predicted moduli only apply to rock masses that are isotropic, homogeneous, and reasonably approximated as linear elastic. Additionally, predicted moduli have a precision of two digits at best. While the new model requires less geotechnical data than is needed to calculate RMR, the structure of the correlation is intricate and can be time consuming to use. The error bounds can be large because of substantial variability in the small empirical database. This variability could have been reduced by further culling the data, but data points that produced large residuals were included in the analyses to preserve the inherent uncertainty associated with predicting rock mass mechanical properties. While the new multivariate model seems to provide better estimates of weak rock mass modulus over a wider range of geotechnical conditions, it is not without limitations. The empirical database does not contain a robust number of data points, and the performance of the model could not be independently tested. As with all empirical methods, the multivariate correlation should not be used to estimate moduli for conditions not encompassed by the weak rock mass database. The major limitations for using the new correlation are joint spacings, which must be less than 8 cm,
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discontinuities contain infillings, and depths should range between 45 and 730 m. Even with these limitations and deficiencies, the multivariate model may prove useful for preliminary or high-level designs of excavations in intensely-fractured and altered rock masses when only basic geotechnical data are available. Once more weak rock mass modulus data become available, the multivariate model can be independently tested and the probabilistic error bounds can be refined. Acknowledgments This research was funded by the National Institute of Occupational Safety and Health through research contract 200-2011-39965 (Principal Investigator Dr. Kallu) and University of Nevada, Reno, NV. References [1] Hoek E, Diederichs MS. Empirical estimation of rock mass modulus. Int J Rock Mech Min Sci 2006;43:203–15. [2] Palstrom A, Singh R. The deformation modulus of rock masses-comparisons between in-situ tests and indirect estimates. Tunn Undergr Space Technol 2001;16:115–31. [3] Gokceoglu C, Somnez H, Kayabasi A. Predicting the deformation moduli of rock masses. Int J Rock Mech Min Sci 2003;40:701–10.
[4] Galera JM, Alvarez M, Bieniawski ZT. Evaluation of the deformation modulus of rock masses using RMR: comparison with dilatometer tests. In: ISRM, Lisbon; 2007. p. 71–7. [5] Serafim JL, Pereira JP. Considerations on the geomechanical classification of Bieniawski. In: International symposium on engineering geology and underground construction. Lisbon; 1983. p. 33–44. [6] Bieniawski ZT. Determining rock mass deformability: experience from case histories. Int J Rock Mech Min Sci Geomech Abstr 1978;15:237–47. [7] Read SAL, Perrin ND, Richards LR. Applicability of the Hoek-Brown failure criterion to New Zealand greywacke rocks. In: 90th International congress on rock mechanics, Paris; 1999. p. 655–60. [8] Kayabasi A, Gokceoglu C, Ercanoglu M. Estimating the deformation modulus of rock masses: a comparative study. Int J Rock Mech Min Sci 2003;40:55–63. [9] Keffeler ER. Measurement and prediction of in-situ weak rock mass modulus: case studies from Nevada, Puerto Rico, and Iran. Reno: University of NevadaReno; 2014. [10] Agharazi A, Tannant DD, Martin CD. Characterizing rock mass deformation mechanisms during plate loading tests at the Bakhtiary dam project. Int J Rock Mech Min Sci 2012;49:1–11. [11] Agharazi A. Development of a 3D equivalent continuum model for deformation analysis of systematically jointed rock masses. Edmonton: University of Alberta; 2013. [12] Bieniawski ZT. Rock mass classification in rock engineering. In: Proceedings of the symposium on exploration for rock engineering. Johannesburg; 1976. p. 97–106. [13] Bieniawski ZT. Engineering rock mass classifications. New York: John Wiley & Sons; 1989. [14] Scheaffer RL, McClave JT. Probability and statistics for engineers. Belmont: Wadsworth Publishing Company; 1995.
Please cite this article in press as: Kallu RR et al. Development of a multivariate empirical model for predicting weak rock mass modulus. Int J Min Sci Technol (2015), http://dx.doi.org/10.1016/j.ijmst.2015.05.005