Slake durability study of shaly rock and its predictions

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More than 35% of the earth’s crust is comprised of clay-bearing rocks, characterized by a wide variation in engineering properties and their resistance to short term weathering by wetting and drying phenomenon. The resistance to short-term weathering
  Slake durability study of shaly rockand its predictions T.N. Singh  Æ  A.K. Verma  Æ  Vasudev Singh  Æ  A. Sahu Abstract  More than 35% of the earth’s crust iscomprised of clay-bearing rocks, characterized by awide variation in engineering properties and theirresistance to short term weathering by wetting anddrying phenomenon. The resistance to short-termweathering can be determined by slake durability index test. There are various methods to determinethe slake durability indices of weak rock. The effectof acidity of water (slaking fluid) on slake durability index of shale in the laboratory is investigated.These methods are cumbersome and timeconsuming but they can provide valuableinformation on lithology, durability and weatherability of rock. Fuzzy set theory, Fuzzy logic andArtificial Neural Networks (ANN) techniques seemvery well suited for typical complex geotechnicalproblems. In conjunction with statistics andconventional mathematical methods, a hybridmethod can be developed that may prove a stepforward in modeling geotechnical problems. Duringthis investigation a model was developed andcompared with two other models i.e., Neuro-fuzzy systems (combination of fuzzy and artificial neuralnetwork systems) and artificial neural networksystem, for the prediction of slake durability indexof shaly rock to evaluate the performance of itsprediction capability. Keywords  ANN  Æ  Neuro-Fuzzy   Æ  Slake durability index  Æ  Fuzzy set theory   Æ  Strength properties of rock Introduction A durable rock is usually required for the constructionindustry. Temporary access roads, however, such asthose for opencast mines must be constructed with thematerial, which does not disintegrate allow ‘boggingdown’ of vehicles. The presence of swelling and non-durable rocks makes tunnel stability dangerous, espe-cially in coal mines. Roofs consisting of fine grainedsandstone, shale and clay are more prone to instability when exposed to cycles of weathering by wetting anddrying processes.Rocks containing a substantial amount of clay are moreprone to swelling, cracking and disintegration when ex-posed to cycles of weathering by wetting and drying(Balakrishna and Ramana 1968). This may cause problemin coal mines, where shale or coarse grained sandstoneroofs are present (Volarovich and Fan 1962). The slakedurability test is performed to know the behavior of rockunder water to assess its strength (Franklin and Chandra1972). Index test represent a compromise between sim-plicity and precision and gives an approximate idea of mechanical behavior of rocks.Many factors other than the nature of rocks such asenvironmental stress conditions and slaking fluid alsoinfluence the weathering rate. Hence, an attempt is madeto study the slake durability characteristics under acidicwater environment. The rock sample was collected from acoal mine of shaly rock.This paper presents an alternative modeling approach toassist the prediction of slake durability characteristics of shale under different acidic conditions. In the presentinvestigation two different models were designed usingtwo different systems, Artificial Neural Network (ANN)and Neuro-Fuzzy, and a comparison was made to dem-onstrate the superiority of one method over the other. Thefocus is not only on how to construct the model but, also,on how to use this modeling framework to interpret theresults and assess the reliability of the model. Received: 18 May 2004 / Accepted: 26 July 2004Published online: 15 October 2004 ª  Springer-Verlag 2004 T.N. Singh ( & )Department of Earth Sciences,Indian Institute of Technology-Bombay-400 076,Mumbai, IndiaE-mail: +91-22-2576727Fax: +91-22-25767253A.K. Verma  Æ  V. Singh  Æ  A. SahuInstitute of Technology, Banaras Hindu University,221005 Varanasi, India 246  Environmental Geology (2005)  47:246–253 DOI 10.1007/s00254-004-1150-9Original article  Methodology The rock blocks used for tests were shaly rock. Shale andshaly sandstone are the most common coal measure rocks.It is collected from Jayant Mine, Singrauli, NorthernCoalfield limited, M.P, which belongs to the lowerGondwana system. The apparatus used for conductingslake durability study was fabricated in the workshop andis similar to the one used by Franklin and Chandra (1972).It was prepared according to standards given by Interna-tional Society for Rock Mechanics (ISRM 1972).A sample comprising of nine rock lumps of shale roughly spherical in shape, each weighing 50±10 g for a totalweight of 500±50 g had been taken and placed in a drumto dry until a constant weight was obtained in an oven at105ºC for a duration of 4 to 5 hours. For the slake dura-bility test the drum was mounted on the trough and wascoupled to the motor. The trough was then filled withwater to a level of 20 mm below the drum axis and tomaintain the temperature at 25ºC. The drum had beenrotated at 20 rpm for a period of 10 minutes and the drumwas removed from the trough and placed in an oven anddried out at a temperature of 105ºC for 4 hours to drainout the remaining moisture in the samples. The trough hadbeen thoroughly cleaned and filled with acidic water of pH6.0. As similar test was conducted for different pH values(6.0, 4.0, 3.0 and 2.0). The slake durability index can becalculated as given below:The NX size samples were also used for determination of different physico-mechanical properties of rockmass. TheISRM standard was used to conduct the test on the servo-control auto feedback stiff testing machine (MTS) (ISRM1981). The samples were placed in buckets having differentpH values for more than 76 hours. Then, a test of slakedurability index was conducted. Slake durability index=(C-E)/(A-E) 3 100% Where,A= Initial wt. of sample + drum (kg),C= Wt. of sample + drum after second cycle of rota-tion (kg), andE= Wt. of empty drum. Artificial neural network Neural networks may be used as a direct substitute forauto correlation, multivariable regression, linear regres-sion, trigonometric and other statistical analysis andtechniques (Singh and others 2003).When a data stream is analyzed using a neural network, itis possible to detect important predictive patterns that arenot previously apparent to a non-expert. Thus, the neuralnetwork can act as an expert. The particular network canbe defined using three fundamental components: transferfunction, network architecture and learning law (Simpson1990). It is essential to define these components, to solvethe problem satisfactorily.A network needs first to be trained before interpreting new information. Several different algorithms are available fortraining of neural networks but the back-propagationalgorithm is the most versatile and robust one, whichprovides the most efficient learning procedure for multi-layer neural networks. Also, the fact that back-propagationalgorithms are especially capable of solving or predictingproblems makes them popular (Maulenkamp and Grima1999; Singh and others 2003).During training of the net-work, data are processed through the network until itreaches the output layer (forward pass). In this layer, theoutput is compared to the measured values (the ‘true’output). The difference or error between both is processedback through the network (backward pass) updating theindividual weights of the connections and the biases of theindividual neurons. The input and output data are mostly represented as vectors called training pairs. The process asmentioned above is repeated for all the training pairs inthe data set, until the network error converged to athreshold minimum defined by a corresponding costfunction; usually the root mean squared error (RMS) orsummed squared error (SSE). Fuzzy backpropagation system Fuzzy system is currently being used in a wide field of industrial and scientific applications. As the design andespecially the optimization process of fuzzy system can bevery time consuming, it is convenient to have algorithms,which construct and optimize them automatically. Onepopular approach is to combine fuzzy systems withlearning techniques derived from neural networks. Suchapproaches are usually called Neuro-fuzzy systems (Alv-arez and others 2000). Fuzzy system is based on the ideathat sets are not crisp but some are fuzzy, and these can bemodeled in linguistic human terms such as large, smalland medium. In the fuzzy system, rules can be formulatedthat use these linguistic human expressions (Takagi andSugeno 1985). The combination of ANN’s and fuzzy setsoffers a powerful method to model human behavior. TheANN is used to define the clustering in the solution space,which results in creation of fuzzy sets. A further advantageis that the solution space rather than being representedpoint by point as some expert systems ‘‘clumps’’ results infewer rules and lower computer resources (Jang and others1997). (Fig. 1) Fuzzy system The classical fuzzy set is a set with a crisp boundary. Thevariable either belongs or doesn’t belong to the set. Incontrast to a classical crisp set, a fuzzy set is definedwithout a crisp boundary, where the transition ‘‘belong toa set’’ is gradual and is characterized by the membershipfunctions in the range that give fuzzy sets flexibility inmodeling (Bezdek 1981). Environmental Geology (2005)  47:246–253  247 Original article  The membership may be described either in a discreteform as a set of membership values or as a continuousfunction valid in some range of values of the variable x. Tothe most popular types of membership functions belongthe triangle, trapezoidal, Gaussian or bell functions. Thegeneralized description of Gaussian function is used hereand given below:  f   x ;  r ;  c ð Þ¼ e    x  c ð Þ 2 2 r 2  ð 1 Þ This is Gaussian function that depends on two parameters r  and c. The parameters for Gaussian represent theparameters  r  and c listed in order in the vector.The most popular solution of the fuzzy networks is basedon the so called fuzzy inference system, fuzzy   if—then rules and fuzzy reasoning. Such a fuzzy inference systemimplements a non-linear mapping from the input space tothe output space. This mapping is accomplished by anumber of fuzzy   if- then  rules, each of which describes thelocal behavior of the mapping, like it is done in radial basisfunction networks. The antecedent of the rule defines thefuzzy region in the input space, while the consequentspecifies the output of the fuzzy region.There are different solutions of fuzzy inference systems.The most known models belong to the Mamdani fuzzy model, Tsukamoto fuzzy model and Takagi-Sugeno-Kang(TSK) model. In the present work the TSK model is used.A typical fuzzy rule in this model has the form:If x 1  is A 1  and x 2  is A 2  . . . And x N  is AN then y   ¼ f x ð Þð 2 Þ crisp function in the consequent. The function y=f(x) is apolynomial in the input variables x 1 , x 2, . . .  x N . The linearform of this function is applied here. The aggregated val-ues of the membership function for the vector x may beassumed either in the form of   MIN   operator or in aproduct form. For M fuzzy rules of the form (2), M suchmembership functions  l 1 ,  l 2 , .....,  l M  are there. It isassumed that each antecedent is followed by the conse-quent of the linear formN ¼ p io þ  R  p ij x  j  j ¼ 1 ð 3 Þ Where p ij  are the adjustable coefficients, for I=1, 2,..., Mand j=1, 2, ...., N.Applying fuzzy singleton, the generalized Gaussian mem-bership function is described by Eq. (1) and algebraicproduct aggregation of the input variables, at the existenceof M rules the neuro – fuzzy TSK system output signal y(x)upon excitation by the vector x is described by the equa-tion.The adjusted parameters of the system are the non-linearparameters ( c  j(k) ,  r  j(k) ,b  j(k) ) for j=1, 2, ...., N and k=1, 2,...., M of the fuzzifier functions and the linear parameters(weights p  kj  ) of TSK functions. Contrary to the Mamdanifuzzy inference system, the TSK model generates the crispoutput value instead of a fuzzy one. The defuzzifier is notnecessary.My  ð x Þ  1M  RR  pl r  x  j     K=IJ ¼ 1r ¼ 1N  N  dl k  x  j     ð pko þ R p kj  x  j Þ J ¼ 1 J ¼ 1 0@1A ð 4 Þ 248  Environmental Geology (2005)  47:246–253Fig. 1 Backpropagation Neural Network Original article  Neuro fuzzy network system The TSK fuzzy inference systems described in Eq. (4) canbe easily implanted in the form of the so-called neurofuzzy network structure. Figure 2 illustrates the 5-layer structureof the neurofuzzy network, realizing the TSK model of thefuzzy system. It is assumed that the functions y  i , y  i  = f  i  (x)are linear in the form:f  i  x ð Þ¼ p io þ R  N  p ij  x  j :  j ¼ 1  ð 5 Þ The adaptable parameters of the networks are thevariables of the membership functions c  j(k) ,  r  j(k) ,b  j(k) for j=1, 2 ;  . . .  ; N, k = 1, 2 ;  . . .  ;  M and the coefficients (linearweights)p ij  for i=1, 2 ;  . . .  ;  M and j=0, 1, 2 ;  . . .  ;  N of thelinear Takagi-Sugeno functions.The first layer performs the fuzzification according to themembership function  l k  (x  j ), described by the relation (1).The second layer aggregates the fuzzified results of theindividual scalar functions of every variable anddetermines the membership function of the whole vectorx. This is the product type aggregation. Each node of thislayer represents the firing strength of a rule. The thirdlayer calculates the aggregated signal of the fuzzy inferencefor each inference rule. The output signal of each unit of this layer is the product of the firing strength of the ruleand the consequent membership value. The fourth layerdetermines the output membership function. Layer fivecalculates only the sum of the signal of the second and thethird layers of the network. The final sixth layer containsonly one neuron for output. In the case of multiple outputsas many output neurons as needed can be added in afashion similar to the case of one output. The outputneuron computes the overall output signal according tothe relation (4). Thus, a neurofuzzy network is constructedthat is functionally equivalent to Takagi-Sugeno fuzzy model. Only the first and the third layers are parametric.The parameters of the first layer are associated with thenon-linear functions and the weights p ij  of the third layerare linear. Hybrid learning algorithm Learning of the neurofuzzy network, that is adaptation of the parameters of the first (cj (k) ,  r  j(k) ,b  j(k) ) and third (p ij )layers of the network can be done either in supervised orself–organizing mode. The supervised mode is more effi-cient for the purpose of approximation.In practical implementation, a hybrid approach has beenused. In this method, it is taken and accounted that thenetwork is linear in the parameters p ij,  thus, these linearparameters can be identified by a linear least squaresmethod based on singular value decomposition (SVD). Atthis stage it is assumed that all non-linear parameters arefixed. This is the first run of the learning stage. In thesecond run fix the linear parameters of the network andapply the gradient steepest descent method applied for theestimation of the non-linear parameters of the member-ship functions.In hybrid learning, each iteration is composed of a forwardpass and a backward one. In the forward pass, after theinput vector is presented, the node outputs in the networklayers are calculated and on the basis of this the linearparameters p ij  are adjusted using pseudo inverse based onthe SVD technique. After linear parameters are identified Environmental Geology (2005)  47:246–253  249 Fig. 2 ANFIS structure for the PRmodel, Neuro-fuzzy, with fourinput parameters and five rules. Layer-1  represents inputs,  layer-2 input membership function, layer-3  rules,  layer-4  outputmembership function,  layer-5 weighted sum output and  layer-6  output Original article  the error for training data pairs can be computed. In thebackward pass, the error signals propagate from the out-put end toward the input nodes; the gradient vector iscalculated and the non-linear parameters c  j(k) ,  r  j(k) , b  j(k) updated by steepest descent method. The learning step of the non-linear parameters update is adjusted using adap-tive approach. This process is repeated many times untilthere is sufficient change of the values of the adaptedparameters of the network.The important advantage of the hybrid algorithm issplitting the learning process into two independent stages:the adaptation of linear weights and adaptation of parameters of the non-linear membership functions. Thisalgorithm involves less complexity and at the same timeincreased efficiency of learning also.One of the most important stages of the neurofuzzy TSKnetwork generation is the establishment of the inferencerules (Takagi and Sugeno1985). The so-called grid methodis often used in which the rules are defined as the com-binations of the membership functions for each inputvariable. If the input variable range is split into a limitednumber (say n i  for i=1, 2,.. N) of membership functions,the combinations lead to many different inference rules.For example, for 10 input systems, at 3 membershipfunctions each, the maximum possible number of rules isequal M=3 10 =59049. The problem is that thesecombinations correspond in many cases to the regions of no data, hence a lot many of them may be deleted.This problem can be solved by using fuzzy self – organi-zation algorithm. This algorithm splits the data space intoa specified number of overlapping clusters. Each clustermay be associated with the specific rule of the centercorresponding to the center of the appropriate cluster. Inthis way all rules correspond to the regions of the spacecontaining a majority of the data and the problem of theempty rules can be avoided (Chiu 1994).The ultimate goal of data clustering is to partition the datainto similar subgroups. This is accomplished by employing some similar measures (e.g. the Euclideandistance). In this paper, data clustering is used to derivemembership functions from measured data, which, inturn, determine the number of If-then rules in the model(i.e. rules indication). Several clustering methods havebeen proposed in the literature (Nauck and others 1997).The method employed in this paper is the subtractiveclustering method. Network architecturefor ANN model The feed-forward network is adopted here as this archi-tecture is reported to be suitable for the problem based onpattern identification. Pattern matching is basically aninput/output mapping problem. The closer is the mappingthe better the performance of the network.In the small sample, as the data are analyzed here, across-validation technique termed leaving-one-out is moreappropriate. Out of 48 data sets, 38 were taken to train thenetwork and test the remaining data set. The procedure isrepeated 15 times, leaving one observation randomly chosen at a time. In the experiment the first observationwas predicted, using the outcome of an analysis based onthe observations 2, 3..., 55, and 55 th observation was pre-dicted from the observations 1, 2, 3..., 54. This method isadvantageous as it uses nearly the entire data-sets fortraining the network. This cross-validation technique wasused in this ANN network which was used for prediction.The network has an input layer consisting of 4 neuronsand an output layer consisting of 1 neuron. The number of hidden layers was decided by training and predicting the‘training data’ and ‘testing data’ by varying the number of  250  Environmental Geology (2005)  47:246–253Fig. 3a–d a  Membership function plot for input 1 (pH);  b  Membership functionplot for input 2 (UCS);  c  Membership function plot for input 3 (shearstrength);  d  Membership function for input 4 (tensile strength) Original article
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