Prediction of strength parameters of sedimentary rocks using artificial neural networks and regression analysis

Accurate laboratory measurement of geo-engineering properties of intact rock including uniaxial compressive strength (UCS) and modulus of elasticity (E) involves high costs and a substantial amount of time. For this reason, it is of great necessity to develop some relationships and models for estimating these parameters in rock engineering. The present study was conducted to forecast UCS and E in the sedimentary rocks using artificial neural networks (ANNs) and multivariable regression analysis (MLR). For this purpose, a total of 196 rock samples from four rock types (i.e., sandstone, conglomerate, limestone, and marl) were cored and subjected to comprehensive laboratory tests. To develop the predictive models, physical properties of studied rocks such as P wave velocity (V_p), dry density (γ_d), porosity, and water absorption (A_b) were considered as model inputs, while UCS and E were the output parameters. We evaluated the performance of MLR and ANN models by calculating correlation coefficient (R), mean absolute error (MAE), and root-mean-square error (RMSE) indices. The comparison of the obtained results revealed that ANN outperforms MLR when predicting the UCS and E.     

Multivariate Adaptive Regression Spline (Mars) for Prediction of Elastic Modulus of Jointed Rock Mass

This article presents multivariate adaptive regression spline (MARS) for determination of elastic modulus (E_j) of jointed rock mass. MARS is a technique to estimate general functions of high-dimensional arguments given sparse data. It is a nonlinear and non-parametric regression methodology. The input variables of model are joint frequency (J_n), joint inclination parameter (n), joint roughness parameter (r), confining pressure (σ₃) and elastic modulus (E_i) of intact rock. The developed MARS gives an equation for determination of E_j of jointed rock mass. The results from the developed MARS model have been compared with those of artificial neural networks (ANNs) using average absolute error. The developed MARS gives a robust model for determination of E_j of jointed rock mass.     

Effect of heat treatment on dynamic properties of selected rock types taken from the Salt Range in Pakistan

Temperature is one of the variables that influence the elasto-plastic behavior and integrity of rock outcrops. Fluctuations in temperature can trigger alteration of some of the mineral properties and impact the brittle-plastic transition. Initiation and propagation of thermally induced tension cracks tend to weaken most rock types. The principal goal of this study was to anticipate impacts of thermal stress-strain cycles on the dynamic response of representative rock units exposed in the Khewra Gorge of the Salt Range Punjab of Pakistan. Ten types of sedimentary rock units were sampled, including marl, dolomite, three types of limestone, and five different sandstones exhibiting varying characteristics in outcrop. Boulder specimens were collected from the field and transported to the laboratory to prepare 50 drill cores that could be subjected to thermal cycling between 50 and 200 °C in increments of 50 °C. Room temperature core samples were tested using an Erudite resonance frequency meter to measure their Q-factors and the resonance frequency (F_r) at an applied loading frequency of 7 KHz with 0.01 V output voltage. Results suggest that thermal cycling tends to reduce the dynamic Young’s modulus (E_d) and Q-factor. Other parameters, such as damping ratio (ξ), specific damping capacity (Ψ), and loss factor (?) appeared to increase with increasing temperature cycles, likely as a result of developing thermally induced tensile fractures. The resultant values of the null hypothesis (t-critical and t-stats) suggests that the null hypothesis can be discarded because there was no observable difference between the measured and expected values for the cores tested. The observations and data emanating from this study might be useful in designing low-level radioactive waste landfills, nuclear waste repositories, and deep underground excavations where the increased temperature could alter the mechanical behavior of the parent rock mass.     

Correlation of in situ modulus of deformation with degree of weathering, RMR and Q-system

Geotechnical investigation projects in Korea produced data on the in situ modulus of deformation of rock masses (E _M) measured with the borehole test, rock mass rating (RMR), and Q-system. The modulus of deformation of rock masses was correlated with the degree of weathering, RMR, and Q values. Determination of E _M for each degree of weathering allows for the results to be used to classify the degree of weathering or to predict E _M. The relation between E _M and RMR is represented by $ E_{\text{M}} = 10^{{\frac{{{\text{RMR}} - 16}}{50}}} $ , which returns values 2–3 times lower than those reported in previous studies. Despite scatter in the values, due to larger dataset used in this study, the proposed equation may be used to predict the in situ modulus of deformation from RMR values. In addition, the relation between modulus of deformation and Q values is $ E_{\text{M}} = 10^{{0.32{ \log }Q + 0.585}} $ .     

Artificial neural network model for prediction of rock properties from sound level produced during drilling

In many rock engineering applications such as foundations, slopes and tunnels, the intact rock properties are not actually determined by laboratory tests, due to the requirements of high quality core samples and sophisticated test equipments. Thus, predicting the rock properties by using empirical equations has been an attractive research topic relating to rock engineering practice for many years. Soft computing techniques are now being used as alternative statistical tools. In this study, artificial neural network models were developed to predict the rock properties of the intact rock, by using sound level produced during rock drilling. A database of 832 datasets, including drill bit diameter, drill bit speed, penetration rate of the drill bit and equivalent sound level (L_eq) produced during drilling for input parameters, and uniaxial compressive strength (UCS), Schmidt rebound number (SRN), dry density (ρ), P-wave velocity (V_p), tensile strength (TS), modulus of elasticity (E) and percentage porosity (n) of intact rock for output, was established. The constructed models were checked using various prediction performance indices. Goodness of the fit measures revealed that recommended ANN model fitted the data as accurately as experimental results, indicating the usefulness of artificial neural networks in predicting rock properties.     

Experimental Anelastic Strain Recovery Compliance of Three Typical Rocks

The experimental determination of anelastic strain recovery (ASR) compliances for three types of rocks (granite, marble, and sandstone) was performed in the laboratory. Preloading of specimens for uniaxial compression creep tests was at 50 % of the uniaxial compressive strength (UCS) for each rock type. We obtained the shear mode Jas(t) and volumetric mode Jav(t) ASR compliances and calculated the ratio of Jas(t) to Jav(t). The Kelvin model for rock rheology was then applied in numerical simulations and the results were in good agreement with the measured data for Jas(t) and Jav(t). These results showed that both the magnitude and rate of increase of the ASR compliances are strongly dependent on the rock type, and the values of the Jas(t)/Jav(t) ratio for a loading of 50 % of the UCS showed a trend leading to different constants for each of the three rock types. Further experimental and numerical analyses showed approximate power-law relationships between the ASR compliances at 50 % of UCS, and both the UCS and the tangential Young’s modulus at 50 % of UCS (E _t50). These relationships may be useful for the preliminary estimation of ASR compliances.     

Prediction of pressuremeter modulus and limit pressure of clayey soils by simple and non-linear multiple regression techniques: a case study from Mersin, Turkey

The Standard Penetration Test (SPT) is one of the most frequently applied tests during the geotechnical investigation of soils. Due to its usefulness, the development of empirical equations to predict mechanical and compressibility of soil parameters from the SPT blow count has been an attractive subject for geotechnical engineers and engineering geologists. The purpose of this study is to perform regression analyses between the SPT blow counts and the pressuremeter test parameters obtained from a geotechnical investigation performed in a Mersin (Turkey) city sewerage project. In accordance with this purpose, new empirical equations between pressuremeter modulus (E _M) and corrected SPT blow counts (N ₆₀) and between limit pressure (P _L) and corrected SPT blow counts (N ₆₀) are developed in the study. When developing the empirical equations, in addition to the SPT blow counts, the role of moisture content and the plasticity index of soils on the pressuremeter parameters are also assessed. A series of simple and nonlinear multiple regression analyses are performed. As a result of the analyses, several empirical equations are developed. It is shown that the empirical equations between N ₆₀ and E _M, and N ₆₀ and P _L developed in this study are statistically acceptable. An assessment of the prediction performances of some existing empirical equations, depending on the new data, is also performed in the study. However, the prediction equations proposed in this study and the previous studies are developed using a limited number of data. For this reason, a cross-check should be applied before using these empirical equations for design purposes.     

Pressuremeter test parameters of a compacted illitic soil under thermal cycling

The incorporation of heat exchangers in geostructures changes the temperature of the adjacent soil, raising important issues concerning the effect of temperature variations on hydro-mechanical soil behaviour. The objective of this paper is to improve the understanding and quantification of the impact of temperature variation on the bearing capacity of thermo-active piles. Currently, the design of deep foundations is based on the results of in situ penetrometer or pressuremeter tests. However, there are no published data on the effect of temperature on in situ soil parameters, preventing the specific assessment of the behaviour of thermo-active piles. In this study, an experimental device is developed to perform mini-pressuremeter tests under controlled laboratory conditions. Mini-pressuremeter tests are performed on an illitic soil in a thermo-regulated metre-scale container subjected to temperatures from 1 to 40 °C. The results reveal a slight decrease in the pressuremeter modulus (E _p) and a significant decrease in the creep pressure (p _f) and limit pressure (p _l) with increasing temperature. The results also reveal the reversibility of this effect during a heating–cooling cycle throughout the investigated temperature range, whereas the effect of a cooling–heating cycle was only partially reversible. In the case of several thermal cycles, the effect of the first cycle on the soil parameters is decisive.     

Artificial neural networks and nonlinear regression techniques to assess the influence of slake durability cycles on the prediction of uniaxial compressive strength and modulus of elasticity for carbonate rocks

Understanding rock material characterizations and solving relevant problems are quite difficult tasks because of their complex behavior, which sometimes cannot be identified without intelligent, numerical, and analytical approaches. Because of that, some prediction techniques, like artificial neural networks (ANN) and nonlinear regression techniques, can be utilized to solve those problems. The purpose of this study is to examine the effects of the cycling integer of slake durability index test on intact rock behavior and estimate some rock properties, such as uniaxial compressive strength (UCS) and modulus of elasticity (E) from known rock index parameters using ANN and various regression techniques. Further, new performance index (PI) and degree of consistency (Cd) are introduced to examine the accuracy of generated models. For these purposes, intact rock dataset is established by performing rock tests including uniaxial compressive strength, modulus of elasticity, Schmidt hammer, effective porosity, dry unit weight, p‐wave velocity, and slake durability index tests on selected carbonate rocks. Afterward, the models are developed using ANN and nonlinear regression techniques. The concluding remark given is that four‐cycle slake durability index (I_d4) provides more accurate results to evaluate material characterization of carbonate rocks, and it is one of the reliable input variables to estimate UCS and E of carbonate rocks; introduced performance indices, both PI and Cd, may be accepted as good indicators to assess the accuracy of the complex models, and further, the ANN models have more prediction capability than the regression techniques to estimate relevant rock properties. Copyright © 2011 John Wiley & Sons, Ltd.     

Modelling the effects of porous media deformation on the propagation of water-table waves in a sandy unconfined aquifer

This paper examines the influence of porous media deformation on water-table wave dispersion in an unconfined aquifer using a numerical model which couples Richards’ equation to the poro-elastic model. The study was motivated by the findings of Shoushtari et al. (J Hydrol 533:412–440, 2016) who were unable to reproduce the observed wave dispersion in their sand flume data with either numerical Richards’ equation models (assuming rigid porous media) or existing analytic solutions. The water-table wave dispersion is quantified via the complex wave number extracted from the predicted amplitude and phase profiles. A sensitivity analysis was performed to establish the influence of the main parameters in the poro-elastic model, namely Young’s modulus (E) and Poisson’s ratio (ν). For a short oscillation period (T?=?16.4 s), the phase lag increase rate (k _i) is sensitive to the chosen values of E and ν, demonstrating an inverse relationship with both parameters. Changes in the amplitude decay rate (k _r), however, were negligible. For a longer oscillation period (T?=?908.6 s), variations in the values of E and ν resulted in only small changes in both k _r and k _i. In both the short and long period cases, the poro-elastic model is unable to reproduce the observed wave dispersion in the existing laboratory data. Hence porous media deformation cannot explain the additional energy dissipation in the laboratory data. Shoushtari SMH, Cartwright N, Perrochet P, Nielsen P (2016) The effects of oscillation period on groundwater wave dispersion in a sandy unconfined aquifer: sand flume experiments and modelling. J Hydrol 533:412–440.     

Piles shaft capacity from CPT and CPTu data by polynomial neural networks and genetic algorithms

Cone penetration test (CPT) is one of the most common in situ tests which is used for pile design because it can be realized as a model pile. The measured cone resistance (q_c) and sleeve friction (f_s) usually are employed for estimation of pile unit toe and shaft resistances, respectively. Thirty three pile case histories have been compiled including static loading tests performed in uplift, or in push with separation of shaft and toe resistances at sites which comprise CPT or CPTu sounding. Group method of data handling (GMDH) type neural networks optimized using genetic algorithms (GAs) are used to model the effects of effective cone point resistance (q_E) and cone sleeve friction (f_s) as input parameters on pile unit shaft resistance, applying some experimentally obtained training and test data. Sensitivity analysis of the obtained model has been carried out to study the influence of input parameters on model output. Some graphs have been derived from sensitivity analysis to estimate pile unit shaft resistance based on q_E and f_s. The performance of the proposed method has been compared with the other CPT and CPTu direct methods and referenced to measured piles shaft capacity. The results demonstrate that appreciable improvement in prediction of pile shaft capacity has been achieved.     

Empirical modeling of plate load test moduli of soil via gene expression programming

New empirical models were developed to predict the soil deformation moduli using gene expression programming (GEP). The principal soil deformation parameters formulated were secant (E_s) and reloading (E_r) moduli. The proposed models relate E_s and E_r obtained from plate load-settlement curves to the basic soil physical properties. The best GEP models were selected after developing and controlling several models with different combinations of the influencing parameters. The experimental database used for developing the models was established upon a series of plate load tests conducted on different soil types at depths of 1–24 m. To verify the applicability of the derived models, they were employed to estimate the soil moduli of a part of test results that were not included in the analysis. The external validation of the models was further verified using several statistical criteria recommended by researchers. A sensitivity analysis was carried out to determine the contributions of the parameters affecting E_s and E_r. The proposed models give precise estimates of the soil deformation moduli. The E_s prediction model provides considerably better results in comparison with the model developed for E_r. The simplified formulation for E_s significantly outperforms the empirical equations found in the literature. The derived models can reliably be employed for pre-design purposes.     

Correlation of production rate of ornamental stone with rock brittleness indexes

The prediction of production rate in ornamental stones sawing is very important in cost estimation and process planning of the rock sawing plants. The main aim of this paper is finding a mathematical correlation between production rate and rock brittleness indexes. The utilized data have been collected from several stone factories in Iran. Seventeen different granite and carbonate rocks have been experienced sawing conditions with large-diameter circular saws. The laboratory tests such as uniaxial compressive strength and tensile strength, were carried out on the rock samples which were collected from these factories. The ratio of compressive strength to tensile strength (B ₁), the ratio of compressive strength minus tensile strength to compressive strength plus tensile strength (B ₂), and the area under the line of compressive strength and the line of tensile strength (B ₃) were used as brittleness indexes in this study. Then, the relationships between these brittleness indexes and production rate were investigated by using a regression analysis. The results show that, there is not a good relationship between the production rate and B ₁ and B ₂. However, the production rate correlates with B ₃ very well. Therefore, there is a reliable prediction for ornamental stones production based on B ₃ as a brittleness indexes.     

Application of statistical methods for predicting uniaxial compressive strength of limestone rocks using nondestructive tests

Uniaxial compressive strength (UCS) of an intact rock is an important geotechnical parameter for engineering applications. Using standard laboratory tests to determine UCS is a difficult, expensive and time-consuming task. The main purpose of this study is to develop a general model for predicting UCS of limestone samples and to investigate the relationships among UCS, Schmidt hammer rebound and P-wave velocity (V _P). For this reason, some samples of limestone rocks were collected from the southwestern Iran. In order to evaluate a correlation, the measured and predicted values were examined utilizing simple and multivariate regression techniques. In order to check the performance of the proposed equation, coefficient of determination (R ²), root-mean-square error, mean absolute percentage error, variance accounts for (VAF %), Akaike Information Criterion and performance index were determined. The results showed that the proposed equation by multivariate regression could be applied effectively to predict UCS from its combinations, i.e., ultrasonic pulse velocity and Schmidt hammer hardness. The results also showed that considering high prediction performance of the models developed, they can be used to perform preliminary stages of rock engineering assessments. It was evident that such prediction studies not only provide some practical tools but also contribute to better understanding of the main controlling index parameters of UCS of rocks.     

Studying the effect of non‐spherical micro‐particles on Hoek–Brown strength parameter mi using numerical true triaxial compressive tests

The strength parameter m_i in the Hoek–Brown strength criterion is empirical and was developed by trial and error. To better understand the fundamental relationship between m_i and the physical characteristics of intact rock, this paper presents a systematic study of m_i by representing intact rock as a densely packed cemented particle material and simulating its mechanical behavior using particle flow modeling. Specifically, the three‐dimensional particle flow code (PFC3D) was used to conduct numerical true triaxial compression tests on intact rock and to investigate the effect of non‐spherical micro‐particle parameters on m_i. To generate numerical intact rock specimens containing non‐spherical micro‐particles, a new genesis process was proposed, and a specific loop algorithm was used based on the efficiency of the process and the acceptability of generated specimens. Four main parameters—number, aspect ratio, size, and shape—of non‐spherical micro‐particles were studied, and the results indicated that they all have great effect on m_i. The strength parameter m_i increases when the number, aspect ratio, or size is larger or the shape becomes more irregular, mainly as a result of the higher level of interlocking between particles. This confirms the observations from engineering experience and laboratory experiments. To simulate the right strength parameter m_i, it is important to use appropriate non‐spherical micro‐particles by controlling these four parameters. This is further demonstrated by the simulation of two widely studied rocks, Lac du Bonnet granite and Carrara marble. Copyright © 2014 John Wiley & Sons, Ltd.     

High-resolution saturated hydraulic conductivity logging of borehole cores using air permeability measurements

Saturated hydraulic conductivity (K _s) is one of the most important parameters determining groundwater flow and contaminant transport in both unsaturated and saturated porous media. The hand-held air permeameter technique was investigated for high-resolution hydraulic conductivity determination on borehole cores using a spatial resolution of ～0.05 m. The suitability of such air permeameter measurements on friable to poorly indurated sediments was tested to improve the spatial prediction of classical laboratory-based K _s measurements obtained at a much lower spatial resolution (～2 m). In total, 368 K _s measurements were made on ～350 m of borehole cores originating from the Campine basin, northern Belgium, while ～5,230 air permeability measurements were performed on the same cores, resulting in a K _s range of seven orders of magnitude. Cross-validation demonstrated that, using air permeameter data as the secondary variable for laboratory based K _s measurements, the performance increased from R ²?=?0.35 for ordinary kriging (laboratory K _s only) to R ²?=?0.61 for co-kriging. The separate treatment of horizontal and vertical hydraulic conductivity revealed considerable anisotropy in certain lithostratigraphical units, while others were clearly isotropic at the sample scale. Air permeameter measurements on borehole cores provide a cost-effective way to improve spatial predictions of traditional laboratory based K _s.     

Application of the Dynamic Cone Penetrometer (DCP) for determination of the engineering parameters of sandy soils

Determination of the in situ engineering properties of foundation materials has always been a challenge for geotechnical engineers and, thus, several methods have been developed so far. Dynamic Cone Penetration (DCP) test is one of the most versatile amongst them. In the present research, a light weight simple DCP device was developed and used for evaluation of the engineering properties of sandy soils in laboratory conditions. The device consisted of an 8-kg hammer that drops over a height of 575 mm, and drives a 60° cone tip with 20 mm base diameter into the ground. To control the validation of the results, laboratory direct shear and plate load tests were used as reference tests. The soil sample was a poorly graded sandy soil (SP) taken from alluvial deposits of the Tehran plain. All DCP tests and PLTs were undertaken on compacted soil in a mould with 700 mm diameter and 700 mm height. Based on the results of the experiments, the relationships between Dynamic Penetration Index (DPI), relative density (D_r), modulus of elasticity (E), shear modulus (G), modulus of subgrade reaction (K_S), and the friction angle of the soil were obtained with a high coefficient of determination (> 90%). The repeatability of the test results was also evaluated by calculating the coefficient of variations (C_v), which was less than 30% for all tests.     

Is there Link between the Type of the Volumetric Strain Curve and Elastic Constants, Porosity, Stress and Strain Characteristics ?

The stress [crack damage stress (σ _cd) and uniaxial compressive strength (σ _c)] and strain characteristics [maximum total volumetric strain (ε _cd), axial failure strain (ε _af)], porosity (n) and elastic constants [elastic modulus (E) and Poisson’s ratio (ν)] and their ratios were coordinated with the existence of two different types (type 1 and type 2) of volumetric strain curve. Type 1 volumetric strain curve has a reversal point and, therefore, σ _cd is less than the uniaxial compressive strength (σ _c). Type 2 has no reversal point, and the bulk volume of rock decreases until its failure occurs (i.e., σ _cd = σ _c). It is confirmed that the ratio between the elastic modulus (E) and the parameter λ = n/ε _cd strongly affects the crack damage stress (σ _cd) for both type 1 and type 2 volumetric strain curves. It is revealed that heterogeneous carbonate rock samples exhibit different types of the volumetric strain curve even within the same rock formation, and the range of σ _cd/σ _c = 0.54–1 for carbonate rocks is wider than the range (0.71 < σ _cd/σ _c < 0.84) obtained by other researchers for granites, sandstones and quartzite. It is established that there is no connection between the type of the volumetric strain curve and values of n, E, σ _cd, ν, E/(1 ? 2ν), M _R = E/σ _c and E/λ. On the other hand, the type of volumetric strain curve is connected with the values of λ and the ratio between the axial failure strain (ε _af) and the maximum total volumetric strain (ε _cd). It is argued that in case of small ε _af/ε _cd–small λ, volumetric strain curve follows the type 2.     

Effect of rock discontinuities on certain rock strength and fracture energy parameters under uniaxial compression

Summary Five series of test blocks of Pendeli marble with artificially created discontinuities of different crack densities (simulating three mutually orthogonal joint sets) were tested in uniaxial compression in order to study the effect of discontinuities on: (a) the compressive strength and the modulus of elasticity, and (b) certain fracture energy parameters expressed by the ratio W _A/W _V, where W _A is the surface energy and W _V the volume elastic strain energy. Mathematical relationships are derived similar to those suggested by other authors relating strength parameters to crack densities. Such relationships clearly show a reduction in strength with increased crack density. The experimental results obtained permit the extension of Persson's relation (which refers to ideal intact rock) to the more realistic case of discontinuous rock mass by introducing the appropriate term that takes into consideration the effect of rock mass discontinuities on the energy ratio W _A/W _V. A comparison between laboratory results and field observations was subsequently carried out assuming the rock mass to behave as a linearly elastic material, obeying the Hoek and Brown failure criterion. This comparison showed that laboratory results can be extended to larger scale. Furthermore, in order to predict the in situ strength and stability of a rock mass in uniaxial compression (which is of major importance in underground excavations) certain concepts are proposed based on laboratory tests, in situ investigations and first principles of linear elastic fracture mechanics.     

Load–settlement behavior of rock-socketed drilled shafts using Osterberg-Cell tests

Osterberg-Cell (O-Cell) tests are widely used to predict the load–settlement behavior of large-diameter drilled shafts socketed in rock. The loading direction of O-Cell tests for shaft resistance is opposite to that of conventional downward load tests, meaning that the equivalent top load–settlement curve determined by the summation of the mobilized shaft resistance and end bearing at the same deflection neglects the pile-toe settlement caused by the load transmitted along the pile shaft. The emphasis is on quantifying the effect of coupled shaft resistance, which is closely related to the ratios of pile diameter to soil modulus (D/E_s) and total shaft resistance to total applied load (R_s/Q) in rock-socketed drilled shafts, using the coupled load-transfer method. The proposed analytical method, which takes into account the effect of coupled shaft resistance, was developed using a modified Mindlin’s point load solution. Through comparisons with field case studies, it was found that the proposed method reasonably estimated the load-transfer behavior of piles and coupling effects due to the transfer of shaft shear loading. These results represent a significant improvement in the prediction of load–settlement behaviors of drilled shafts subjected to bi-directional loading from the O-Cell test.