Geometric Bias Correction Of Discontinuity In Rockmass Caused By Line Sampling

Most engineering works involve soil and rock; in particular, the construction and operation of important structures such as roads, tunnels, dam foundations, port foundations, bridge foundations, and tall buildings are concerned with rockmass. Rockmass is not perfect a manmade material that is homogeneous and continuous; it is a complicated geological body formed over a long period, even millions of years of physical geological and engineering geological process. Such a long-term process results in its inhomogeneous and discontinuous structure; this structure is composed of two elements, i.e. intact rock and discontinuities. Discontinuities are defined as this whole range of discontinuities while at the same time avoiding any inference concerning their geological origins. They include bedding planes, faults, fissures, fractures, joints and other mechanical defects which, although formed from a wide range of geological processes, possess the common characteristics of low shear strength, negligible tensile strength and high fluid conductivity compared with the surrounding rock material. Numerous engineering practices recorded that the geometry of discontinuities greatly dominates the mechanical performance of engineering rockmass such as types of deformation and stability. The geometric features of discontinuities are typically observed in field by two common sampling technologies, i.e. window sampling and line sampling.This thesis only focuses on line sampling. Line sampling contains two strategies:borehole sampling and scanline sampling. This thesis considers scanline sampling. Such a sampling technology is widely used on outcrops. A clean, approximately planar rock face is selected that is large relative to the size and spacing of the discontinuities exposed. Such exposures can be found on beach cliffs, in gorges, road cuts, quarries, and open pit mines. Care should be exercised when selecting the face, to ensure that the rock material and discontinuities are representative of those across the site. Many natural and excavated rock faces are formed along major joints or faults; if such faces are selected for sampling it is important to set up additional scanlines on other faces, and at different orientations, to provide a three-dimensional sample of the discontinuity network.Many researchers have shown that, line sampling inevitably introduces bias to the probability distribution of orientations, that is. the scanline has a larger probability to intersect discontinuities that have a large intersection angle. The earliest English literature for correcting the bias appeared in Terzaghi1965. She obtained the corrected frequency by dividing the observed frequency by the sine of intersection angle. For the past50years this proposition has been widely applied by a number of specialists.Terzaghi’s procedure is invalid when applied to orientations that intersect the scanline at shallow angles of0to30°. This is the so-called blind zone.1. While a number of recommendations have been provided to avoid blind zone, little attention has been paid to the effectiveness outside blind zone. To the authors’knowledge, the source for low effectiveness outside the blind zone is unknown and the methods to improve the effectiveness are not available.2. As one of the internal processes within Terzaghi’s procedure, meshing the projection diagram was undertaken and is likely to be regarded as a necessary process by previous people. However, it remains unclear whether the meshing is a necessary process and whether the meshing contributes error to correction result.3. The recent work of Tang (2013) found that Terzaghi’s method results in large errors, even if the optimizing countermeasures are exerted. An alternative correction method of higher-effectiveness is unavailable.The purpose of this thesis includes three aspects:1. It clues the low effectiveness source of Terzaghi’s method when applying outside blind zone, and presents suggestions for improving effectiveness.2. For Terzaghi’s method, necessity of meshing is to be evaluated. If meshing is unnecessary, feasibility of the replacement of no meshing is to be tested and accuracy comparison between the correction results respectively from the two cases is to be conducted.3. A more effective correction method than Terzaghi’s method is proposed, based on a solution of the orientation probability distribution in three-dimensional space.The study methods of these three aspects respectively are:1. We suspect the low effectiveness arises from Terzaghi’s Equation. Hence, to clue the source, Terzaghi’s Equation is derived. Despite Terzaghi (1965) gave such a simple derivation, we are to provide a more detailed and reasonable derivation, by using analytic geometry, probability theory and integral. Then in the derivation, we attempt to find the step that is likely to introduce error. For effectiveness improvement, suggested values of two parameters (grid size and sample density) are presented. Grid size is a set parameter within Terzaghi’s correction procedures. Sample density is the division of sample size by the product of the two variables (the difference between the upper limit and lower limit of dip direction, and that of dip angle).(1) Firstly, the effectiveness under the two parameters with a series of values is tested by experiment.(2) Then, the two parameters with the highest effectiveness are determined as the suggested values.(3) Finally, the suggested values of grid size and sample density are verified by using a real discontinuity orientation dataset observed on a lithic arkose exposure in Wenchuan, Sichuan, China. First, the observed orientations are obtained by scanline sampling technique. Second, the observed orientations are corrected by Terzaghi’s method. Third, using the corrected orientations, the three-dimensional network of discontinuities is modeled by means of stochastic simulation. In the model, the orientations intersected by virtual scanline are observed. The orientation of virtual scanline is the same of the actual scanline used in field. To distinguish from the discontinuity orientations observed in field, the observed discontinuity orientations from model are named "model orientations". Fourth, difference of observed orientations and model orientations is tested by Kolmogorov-Smirnov Two-sample Test, under different grid sizes.2. Magnitude of error due to meshing process is to be evaluated, by comparing effectiveness with the case of no meshing.(1) At first, it is necessary to test the feasibility of no meshing.(2) Secondly, the effectiveness comparison between meshing and no meshing is to be conducted using the observed discontinuity orientation mentioned previously.55sampled orientations were simulated from the correction result. The distribution difference between raw orientations and simulated ones was tested with Two Independent Samples Kolmogorov-Smirnov Test. In other hand, the effectiveness test is conducted by discontinuity stochastic simulation that has more combinations of scanline orientations and sample numbers than case study. First, discontinuities, outcrop and scanline were modeled, and the orientations of the discontinuities intersecting the scanline were observed. This process was accomplished by Discontinuities Stochastic Simulation. To distinguish from these observed orientations, the orientations distributed in rockmass are named the "original" here. Next, the frequencies of these observed orientations were respectively corrected by Terzaghi’s method in meshing case or no meshing case. Afterwards, the difference between the corrected orientations distribution and the original one was tested by Two Independent Samples Kolmogorov-Smirnov Test.(3) Last, the differences corresponding to meshing and no meshing were compared. If the case of no meshing has lower difference, its replacing meshing is to be discussed.3. A more effective correction method towards the bias is proposed.(1) First, the solution of the probability density function is derived by means of calculus.(2) Then, on the basis of such a solution, the numerical solution method is proposed which includes two assumptions and five procedures.The assumptions are:(a) The orientation distribution should be independent of diameter distribution in each cluster.(b) The distributions of the two elements of orientation (i.e. dip direction and angle) in each cluster in three-dimensional space of rockmass should be in dependent of each other. And the same assumption should apply to those discontinuities that are intersected by scanline or borehole.The procedures are: (a) Cluster orientations. It can be implemented by the software Dips. It should be minded that all the analyses and the correction should be executed in each cluster.(b) Test the independence between dip direction and angle. As assumed, dip direction and angle should be independent of each other. So it is necessary to test whether the observed orientations meet this assumption. There are lots of independence testing methods; one of them is Pearson’s chi-squared (χ2) test.(c) Correct the discrete cumulative probabilities of dip direction and angle according to the numerical solution that is derived.(d) Judge the distribution type of the corrected orientations.(e) Calculate the distribution parameters of the corrected orientations. The parameter form is related to the distribution type, which has been determined by Step4. For example, for normal distribution, the parameters are mean and standard deviation (or variance); for lognormal distribution, that are means and standard deviation (or variance) of logarithm; for uniform distribution, that are lower limit and upper limit; and for exponential distribution, that is mean. Having determining the parameter form, the parameter value is obtained by approaching via Least Squares Method which can be executed by Matlab or SPSS.For procedure (d), in common statistical analysis, there are two alternatives for judging distribution type:from the shape of the probability density curve or from the shape of the cumulative distribution curve. In order to select the most effective way to judge between the probability density curve and the cumulative distribution curve, they are tested against each other on the effectiveness, under4orientation distribution types. The procedures of this test are:(a) First, the true distribution of orientation is hypothesized, as well as other parameters that are necessary for the following modeling (e.g. size, density, and aperture). The hypothesized orientation distributions include4types, as normal, lognormal, uniform and exponential; for each type, the sample number is set at7values, such as50,100,150,200,300,500and1,000, respectively.(b) Second, inputting these parameters, a three-dimensional network of discontinuities is modeled by means of stochastic simulation. Due to4different groups of distributions,4different models of discontinuity network are constructed.(c) Third, in the models, the orientations of discontinuities intersected by the scanline are observed. In each model, the orientation observation is conducted under7different sample numbers. Thus for4models,28series of observed orientations are obtained.(d) Fourth, because the proposed method is based on the assumption of independence between observed dip direction and dip angle, before correction, the observed orientations were tested on the independence by Pearson’s chi-squared (χ2) test.(e) Lastly, to obtain the probability density curve, the28observation series were corrected; to get the cumulative distribution curve, these series were corrected. Their effectiveness is compared here. (3) The effectiveness of proposed method is evaluated against Terzaghi’s method by using the real orientation dataset mentioned previously.The findings of the thesis are:1. A more detailed and reasonable derivation is provided for Terzaghi’s correction equation. The derivation reveals that an approximate substitution gives rise to the theoretical error of Terzaghi’s method when applied outside blind zone. This would cause the reduction of bias correction effectiveness. Then, to improve the effectiveness, two suggested values (grid size in the meshing and sample density in observation) were presented through experiments and verification of a study case. Results show the highest effectiveness is achieved at the grid size of2°×2°and at the sample density of0.05°-2, respectively.2. As one of the internal processes within Terzaghi’s orientation bias correction method, meshing the projection diagram tends to introduce a considerable error to the correction result of Terzaghi’s method. Then it is certified that no meshing is feasible. At last, the accuracy comparison reveals that the correction result would be more accurate in the case of no meshing than that in the case of meshing.3. In the derivation process of the probability distribution of orientation in three-dimensional space, it is difficult to obtain an analytic solution. Instead the numerical approximate solution is derived, which solves the orientation distribution from the observed orientations samples by line sampling. Based on such solution, a numerical solution method for correcting line-sampling bias of orientation distribution is proposed which contains two assumptions and five procedures.4. In order to select the most effective judgment way between probability density curve and cumulative distribution curve, they are tested against each other on the effectiveness. Test result shows the distribution type can hardly be judged clearly from probability density curve, whereas can be easily seen from cumulative distribution curve. It suggests, on the distribution type judgment, the cumulative distribution curve performs more effectively than the probability density curve. So cumulative distribution curve and its function (cumulative distribution function) are preferred as the optimal form judgment way for proposed method, rather than probability density curve and its function (probability density function).5. To find the optimal sample number for the numerical solution method, comparison experiments of various sample numbers on accuracy are conducted. The result reveals:(1) when the dip direction or angle follows exponential distribution, the proposed method is inaccurate. It suggests the proposed method does not suit the orientation of exponential distribution.(2) when the dip direction or angle follows uniform distribution, the proposed method is quite accurate. And increasing sample number hardly affects its accuracy. It suggests the proposed method suits the orientation of uniform distribution well.(3) when the orientation follows normal distribution and sample number is over500, increasing the number can hardly improve the accuracy. So the sample number should be limited below500.(4) when the dip direction or angle follows lognormal distribution and the sample number is over150, increasing the number cannot significantly improve the accuracy. So150would be the best sample number for lognormal distribution. Since the distribution type of orientation is unknown during the sampling till the correction, the optimal sample number could be one of the four results above. Thus, after combining the four results above, the optimal sample number is around150. And it is recommended to sample the discontinuities around the number of150in each cluster.6. The correction effectiveness comparison with Terzaghi’s method suggests the proposed method is more effective. Terzaghi’s method makes the assumption that all the discontinuities in each counting circle are parallel. However, actually, all the discontinuities are not parallel. Lower effectiveness of Terzaghi’s method might result from the discrepancy between this assumption and the reality, while, the proposed method does not require such assumption and so is more effectiveness.