Prediction of void ratio and shear wave velocity for soil in quaternary alluvium using cone penetration tests

Abstract

Several correlations are available to determine the shear wave velocity using cone penetration test (CPT) data. Available correlations are applied for the studied region, which shows the requirement for developing a new correlation for the study area. This study uses CPT, standard penetration test (SPT), and multichannel analysis of surface wave (MASW) data to formulate correlations for predicting void ratio (e) and shear wave velocity (V_S). The estimated void ratio at various depths was taken from the SPT bore log available for the site. A regression model has been formulated for predicting e from normalized cone tip resistance (Q_tn). In developing the shear wave velocity prediction model, two types of cone data are used: mechanical and electrical. In the prediction model of V_S, various parameters, such as cone tip resistance (q_c), soil behavior type index (I_C), effective stress (${{\sigma }_{0}}{\prime}$), e, and depth (z), are considered. The correlation regarding shear wave velocity gives a good prediction with both CPT cones. A cone factor (K_C) is introduced in the developed correlation for predicting ‘e’. The proposed correlations allow design soil parameters to be easily obtained from cone penetration test data.

Introduction

The shear wave velocity (V_S) is an essential property used in dynamic analysis and is related to the stiffness of the soil. Site characterization (determination of site condition), liquefaction hazard assessment, seismic hazard analyses and ground response analyses use shear wave velocity (V_S) as input parameters. Site classification and liquefaction hazard assessment can be performed for a city (Chakrabortty et al. 2018) or a region (Wang et al. 2017). The expected ground motion estimated from probabilistic or deterministic seismic hazard assessment also required knowledge of shear wave velocity (Al-Ajamee et al. 2022). Therefore, it is imperative to accurately measure or predict shear wave velocity (V_S) for seismic design purposes. Undisturbed samples need to be collected to accurately estimate the soil properties in the laboratory. However, collecting undisturbed samples is often not possible or very difficult. A disturbed soil sample will not give the actual value of V_S, as the soil structure will change, and particles will be oriented in a different configuration. The behavior of altered soil particles will ultimately differ from that of soil deposits, and the properties determined will not depict the actual information. Direct field measurement of velocity should be taken for determining stiffness parameters such as shear modulus or Young's modulus, as it provides convenient and reliable results (Jardine et al. 1986). Various laboratory methods, such as resonant column and bender element tests, as well as field methods such as geophysical techniques, are used to determine shear wave velocity (Gu et al. 2015). In a study by Nilay et al. (2022), three different in situ tests, CPT, SPT, and MASW, were considered for liquefaction hazard mapping. The conclusion drawn was that a CPT-based assessment tends to yield conservative liquefaction potential results for sites within the studied region. Reflecting on the previous discussion, it becomes evident that numerous direct and indirect methods exist for determining V_S. The choice of the appropriate method depends on the specific requirements of the task at hand. Therefore, developing a multivariable nonlinear regression prediction model (Das and Chakrabortty 2022) between V_S and cone test parameters is beneficial, as it gives a reasonably accurate measurement of V_S for both region-wide and site-specific ground response assessments (McGann et al. 2015). Shear wave velocity is affected by soil type, aging conditions, cementation properties, and effective stress (Andrus et al. 2007). The prediction model can consider aging conditions and cementation properties using effective stress and void ratio terms. The value of V_s in soil deposits of the Pleistocene age is greater than that of the Holocene age. This difference influences the researchers to introduce an age scaling factor (SF) in the correlation. Many CPT-V_S correlations are available worldwide for different sites with different cone parameters. Initially, correlation model was proposed using only two parameters by Baldi et al. (1990). Later on, three parameters were considered by Hegazy and Mayne (1995) and Andrus et al. (2007). Subsequently, models involving four parameters were introduced by Hegazy and Mayne (2006) and Robertson (2009). Some of these correlations applicable to the study area (given in Table 1) have been used to predict Vs for the studied soil. Hegazy and Mayne (1995) formulated three correlations (considering sand, clay, and all soil) for Vs determination. Hegazy and Mayne (2006) selected a site with relatively complex stratigraphy and proposed a global correlation for V_S determination.

Table 1 Applicable correlations between CPT and Vs used for the study area

The soil behavior index (I_C) was considered in this correlation. Robertson (2009) gave a global relation for V_S as a function of cone tip q_c, soil behavior index I_C, and effective vertical stress ${{\sigma }{\prime}}_{0}$. Mousa and Hussein (2022) most recently, provided seven (7) different correlations for shear wave determination using CPT. From the literature, it has been inferred that the Vs-CPT correlation improves considering e. The void ratio articulates the denseness of strata. It is closely related to soil compressibility, permeability, and shear strength and depends upon the particle size and distribution of particle size. As an important parameter, a direct method for estimating ‘e’ at a desired depth is not available. Therefore, CPT data can be used as an effective way to estimate ‘e’.

A correlation model between CPT and ‘e’ was developed in this study with available CPT and SPT data to materialize this concept. A power regression model was fitted with the available data between the factored void ratio (FVR) and normalized cone tip resistance. The factored void ratio is defined here as e^0.5 multiplied by (I_C)ⁿ. A correction factor has been proposed to consider the effect of cone type. This proposed model is one of the novelties of the present study. In the next part of this study, two site-specific prediction models have been proposed for estimating Vs from CPT data. The first Vs-CPT model has been presented with four parameters (q_c, I_C, ${{\sigma }{\prime}}_{0}$ and z). The second model has been proposed with five parameters (q_c, I_C, ${{\sigma }{\prime}}_{0}$, z and e) based on regression analyses. The need for the proposed site-specific models for estimating shear wave velocity from CPTs is explained in Sect. "Shear wave velocity (VS) prediction model".

Study area

Geology

The data used in this study were collected from different soil reports available for the IIT Patna campus (Fig. 1). The study area lies in the alluvium plain of Ganga and its tributaries with the most recent geologic age termed Quaternary alluvium (Sahu et al. 2015). This Quaternary alluvium refers to sedimentary deposits formed in the most recent geological period through the action of flowing water from the Ganga River and its tributaries, such as the Sone, Gandak, and Koshi Rivers. The entire region lies in the Middle Ganga Plain (MGP), which has an almost flat topography. The geology of the studied region is influenced by fine sand particles deposited by the Sone River. This is the only river flowing in this area is dynamic in nature as mentioned by Sahu et al. (2010). The sediment type found near Sone is locally called Sone sand, which contains fine to medium fine-grained sand and gravel with a size range of 0.15 to 1.18 mm. Generally, these sediments can be found in various settings, including floodplains, deltas, alluvial fans, and terraces. The tectonics of the study area lie in an alluvial plain, an active tectonic region underlain by transverse and oblique faults. Two significant faults, namely, the East Patna and West Patna faults, are considered the most active because of the continuous subsidence of the Indian plate into the Eurasian plate. Hence, to thoroughly understand the region's seismicity, proper estimation of dynamic soil properties is essential.

Fig. 1

Locations of CPT, SPT, and MASW tests marked by various symbols on the study area (IIT Patna campus) map prepared by modifying the google map

Database for formulating prediction models

The database is formed by collecting data from three different types of testing, namely, i) CPT, ii) SPT, and iii) MASW testing. The collected CPT data have two different types of cones, namely, mechanical and electrical cones. From CPT, two essential readings are obtained: q_c and sleeve friction (f_s). The collected CPT data using a mechanical cone were obtained from existing soil reports containing data at 15 locations on the campus with a penetration depth of 30 m. The collected ECPT data were obtained from soil reports conducted at 27 sites with a maximum penetration depth of approximately 20 m. Continuous readings are available in electrical cone penetration testing (ECPT). As velocity measurements taken from MASW are available at every 1-m depth until 30 m depth, CPT readings are also selected at the same level with 1-m intervals from both the ECPT and mechanical cone. Data from SPT testing near that of CPT and MASW testing are considered to determine properties such as unit weight, e, etc., along the depth. A comparison of q_c, shear wave velocity (V_S), and SPT N-value (N) along depth is shown in Fig. 2. The test results shown in Fig. 2 for a particular location (e.g., C1-M2-S5) are close to each other, with a maximum of 150 m apart. In SPT profiling, some distinct markers are shown in red, and these red marker values are those with an N-value equal to or greater than 100.

Fig. 2

Comparison between recorded cone tip resistance (q_c), SPT N-value (N), and shear wave velocity (V_S) that are close to each other along with depth at seven locations (e.g., C8, M6 and S17 are tested at nearby locations as shown in Fig. 1)

Soil classification

Cone penetration testing has been in use for nearly 40 years. It has a sound theoretical aspect and a simplified testing procedure. In this testing, a cone penetrates into the soil. The resistance offered by the soil to the cone gives essential results, which are called cone parameters. With much advancement in the past years using CPT, information regarding soil properties such as soil type, behavior, and strength can be obtained very quickly. Soil stratigraphy and soil type classification are significant applications of CPT. Using CPT, early soil identification charts were given by Douglas and Olsen (1981). Later, normalized and nonnormalized charts provided by Robertson (1990) and his coworkers gained much popularity. Robertson (1990) proposed the concept of normalization for the cone tip and friction ratio, which is shown in Eq. (1) and (2) as follows:

$${\mathrm{Q}}_{\mathrm{t1}}\mathrm{ = [}\frac{{\mathrm{q}}_{\mathrm{t}}-{\upsigma }_{0}}{{{\upsigma }{\prime}}_{0}}\mathrm{]}$$

(1)

where, q_t is the cone tip resistance, ${\sigma }_{0}$ is the total stress and ${{\sigma }_{0}}{\prime}$ is the effective stress at the tested depth.

$${\mathrm{F}}_{\mathrm{r}}\mathrm{=}\left[\frac{{\mathrm{f}}_{\mathrm{s}}}{{\mathrm{q}}_{\mathrm{t}}-{\sigma }_{0}}\right]\times \mathrm{100 (\%)}$$

(2)

where, f_s is the sleeve friction, and F_r is the friction ratio. Robertson and co-workers (Robertson and Wride 1998; Zhang et al. 2002) proposed an modified version of Eq. (1), which introduces a normalized cone tip resistance, expressed as:

$${\mathrm{Q}}_{\mathrm{tn}}\mathrm{ = [}\frac{{\mathrm{q}}_{\mathrm{t}} \, -{\upsigma }_{0}}{{\mathrm{p}}_{\mathrm{a}}}] \, \times \, {\mathrm{(}\frac{{\mathrm{p}}_{\mathrm{a}}}{{{\upsigma }{\prime}}_{0}}\mathrm{)}}^{\mathrm{n}}$$

(3)

where, Q_tn is the normalized cone tip resistance; n is the stress exponent; and P_a is atmospheric pressure. Jefferies and Davies (1993) introduced I_c (soil behavior type index) to characterize the soil zone in Qt₁—F_r charts, defining it as the boundary in terms of the radius of concentric circles. Robertson and Wride (1998) provided an equation, Eq. (4) For these concentric circles and the updated Robertson's 1990 chart. These charts are plotted between the normalized cone tip resistance (Q_tn) and friction ratio (F_r), and the entire graph is divided into nine different soil zones. Each soil zone provides information about the soil type in that stratum corresponding to a range of I_C values.

$${\mathrm{I}}_{\mathrm{C }}= \mathrm{ } {\mathrm{[}{\mathrm{(3.47-log}{\mathrm{Q}}_{\mathrm{t1}}\mathrm{)}}^{2}\mathrm{ } + {\mathrm{(log}{\mathrm{F}}_{\mathrm{r}}\mathrm{ + 1.22)}}^{2}\mathrm{]}}^{0.5}$$

(4)

The stress exponent (n) in Eq. 3 is a function of I_C and overburden pressure and is given by the following equation:

$$\mathrm{n = 0.381}{\mathrm{I}}_{\mathrm{C}}\mathrm{ + 0.05(}\frac{\upsigma {^{\prime}}_{0}}{{\mathrm{p}}_{\mathrm{a}}}\mathrm{) - 0.15}\text{; }\text{Where n }\le {1}\mathrm{.}$$

(5)

For soil classification using CPT, I_C is estimated using the abovementioned equations. It is an iterative process that starts with steps from Eq. 1 to Eq. 5. This iteration will begin by assuming an initial stress exponent 'n' equal to 1. It will stop when the change in two consecutive 'n' values is less than 0.01 (Δn < 0.01). The Δn is the change observed in 'n' from two successive observations. When the difference in n is below or equal to 0.01, Q_tn and I_C at that stage will be termed final values.

Identification of soil strata is completed based on I_C values, in which soil ranges from Zone-3 to Zone-6. Zone 3 belongs to the clayey soil type with an I_C value between 2.95 and 3.6, and Zone 6 belongs to the purely clean sandy type soil (Table 2). From Fig. 3, it can be observed that several datasets lie in the zone of silt mixtures to sand mixtures. The I_C value at all the locations is calculated and plotted along the depth (Fig. 4). It consists of a silt mixture in the first four meters of depth and a sandy mixture in the next eight to ten meters. A sharp change in the I_C value shows a sudden shift in stratigraphy. This abrupt change indicates a layer of different material present at that depth. The measured q_c value is affected by the presence of these thin layers. At the interface, the cone senses these thin layers before entering them from a certain distance. The transition effect induced variation in the q_c value. At the boundary, q_c is affected by both layers, i.e., layer ahead and layer behind. This variation continues up to a certain depth in the next layer. This effect is termed the "thin layer effect".

Table 2 Soil classification based on soil behavior type index (Robertson and Wride 1998)

Fig. 3

Soil classification using normalized cone tip resistance (Q_tn) and friction ratio (F_r) based on Robertson and Wride (1998)

Fig. 4

Variation of soil behavior type index (I_c) along with the depth in the studied area

Void ratio (e) Prediction model

Prediction using mechanical cone

The void ratio is usually estimated from laboratory tests of collected soil samples from SPT. To eliminate the dependency on SPT, a prediction model for ‘e’ from CPT data was proposed in this study. A rigorous statistical analysis was conducted using the available data, and it was found that a power relation exists between e^0.5I_Cⁿ and Q_tn. The term plotted on the Y-axis in Fig. 5b, i.e., e^0.5I_Cⁿ, is called the Factored Void Ratio (FVR) here. The actual measured void ratio needed for correlation formulation is obtained from the results of SPT testing available for the site. It has been assumed that there is no/little change in soil properties within small distances between SPT and nearby CPT locations (within a distance of 150 m.). From that location, cone tip parameters are chosen at a depth of known void ratios. A total of 194 CPT data points are gathered from the mechanical cone, and regression analysis is carried out. The datasets used are shown as bar charts with individual counts and their respective CPT locations in Fig. 5a. The trend between the FVR and I_C is shown in Fig. 5b. The functional form of the equation obtained between the FVR and Q_tn is also shown in the figure. The proposed CPT-e correlation with an R² value of 0.92 is given below:

Fig. 5

(a) Dataset distribution (total count: 194) at various locations (e.g., C1 is CPT test location 1 and S5 is SPT test location as shown in Fig. 1) used for formulating void ratio prediction model, (b) Trend between factored void ratio (FVR) and normalized cone tip resistance (Q_tn)

$${\mathrm{e}}^{0.5}{{\mathrm{I}}_{\mathrm{c}}}^{\mathrm{n}}\mathrm{ = 5.2976}\times {{\mathrm{Q}}_{\mathrm{tn}}}^{-0.274}$$

(6)

Validation of the prediction model for mechanical cone data

In this section, the formulated CPT-e correlation given in Eq. 6 is validated with available CPT data from different locations on campus, which was not used in developing the relationship. Figure 6a compares the measured and predicted FVR against depth; both values were almost identical. Figure 6b corresponds to the predicted FVR and FVR estimated (based on the estimated ‘e’). Most of the data points are on a 45° degree line or nearby. The data points chosen for validation using mechanical cones agree well with the proposed model. However, electrical cone penetration test (ECPT) data, when validated using the proposed model, show a downward shift in the FVR compared to the proposed correlation (as shown in Fig. 7). Cone tip resistance (q_c) values from both cones at nearly the same site are plotted against depth. The values of q_c are similar, and not much noticeable change is detected. Additionally, the same vertical soil profile was obtained from the I_C value calculated from two different types of CPT testing parameters. After checking all the relevant parameters, a downward shift in the plotted value of FVR and Q_tn with ECPT was present. All other parameters are nearly identical; the only change was that an electrical cone is used in ECPT, which is entirely different from the mechanical cone. This observed downward shift may be because of this changed cone type. This necessitates introducing a particular factor to the formulated correlation that will take care of this cone effect, known as the cone factor (K_C). The trend line is plotted across the data between FVR and Q_tn from ECPT to calculate the cone factor. The ECPT data, collected from nearly the exact location of mechanical testing, are used, and FVR-Q_tn is plotted, which gives the following equation:

Fig. 6

Validation of void ratio prediction model developed using mechanical cone data (a) variation of factored void ratio (FVR) along with depth, (b) predicted and estimated FVR plotted along with 45° line

Fig. 7

Estimation of cone type effect on power relation between factored void ratio (FVR) and normalized cone tip resistance (Q_tn)

$${e}^{0.5}{{I}_{c}}^{n} = 4.2903 \times {{Q}_{tn}}^{-0.274}$$

(7)

By comparing Eq. 6 and Eq. 7, the cone factor for the electrical cone is proposed as follows:

$${\mathrm{K}}_{\mathrm{c}}\mathrm{=}\frac{\mathrm{4.2903}\times {{\mathrm{Q}}_{\mathrm{tn}}}^{-\mathrm{0.274}}}{\mathrm{5.2976}\times {{\mathrm{Q}}_{\mathrm{tn}}}^{-\mathrm{0.274}}} \, \mathrm{=} \, \mathrm{0.809}$$

(8)

Therefore, the proposed mechanical and electrical cone factors are 1 and 0.809, respectively. This factor can be multiplied by Eq. 6 to predict ‘e’ from various cones. The modified predicted model takes the following form:

$${\mathrm{e}}^{0.5}{{\mathrm{I}}_{\mathrm{c}}}^{\mathrm{n}}\mathrm{=5.2976}\times {{\mathrm{Q}}_{\mathrm{tn}}}^{-\mathrm{0.274}}\times \mathrm{(}{\mathrm{K}}_{\mathrm{C}}\mathrm{)}$$

(9)

Validation of the prediction model for ECPT data

To validate the proposed model, values obtained from electrical cones at different testing locations are used in Eq. 9 considering the cone factor (K_C). The predicted and estimated FVR from ECPT data is shown in Fig. 8. In Fig. 9a, FVR predicted and measured values lie on or near the 45-degree line, with a comparison of FVR along depth in Fig. 9b. Both figures show the excellent predictability of e from the proposed model.

Fig. 8

Validation of void ratio prediction model using the ECPT data with consideration of cone factor

Fig. 9

(a) Efficiency of model showing predicted and measured FVR, (b) Validation of FVR along depth using electrical Cone

Shear wave velocity (VS) prediction model

The correlation between cone penetration testing (CPT) and shear wave velocity (V_S) is commonly represented in literature through various forms: linear, as demonstrated by Sykora and Stokoe (1983), nonlinear with a single parameter proposed by Jaime and Romo (1988), or nonlinear with multiple parameters as explored by Robertson (2009). Some existing relations, such as the one by Andrus et al. (2007), adopt a power law equation, while others, like those presented by Hegazy and Mayne (1995, 2006) rely on logarithmic relationships. However, this simple power law equation with a single parameter does not work well, as previous studies show that V_S depends upon many factors other than soil type and testing conditions. For this reason, a nonlinear model with multiple variables gives better efficiency in prediction. These multiple variables are direct (e.g., q_c) or indirect cone parameters (e.g., I_c) or in situ soil properties (e.g., total or effective stresses). A series of correlation models available for predicting Vs based on q_c data for the Quaternary alluvial deposit available in the studied region. Some of these applicable models, as presented in Table 1, have been used to predict Vs. The predicted Vs (V_pre) and measured Vs (V_mea) from MASW for locations C2-M5 are shown in Fig. 10. The result indicates that none works well and cannot accurately provide the V_S. The difference between V_pre and V_mea along with the depth shows the requirement of developing a site-specific model for the studied region. Using the functional form of the correlation, a simple CPT-V_S correlation has been proposed by Mishra et al. (2023) that is applicable to the same study region. However, in that study, a limited number (90 datasets from 3 different locations) of available datasets was used. Therefore, to improve the prediction model, large datasets (453 pairs) from both testing types (mechanical and electrical) are combined, forming an updated CPT-V_S correlation.

Fig. 10

Variation of shear wave velocity along depth estimated using different available CPT-V_S models and experimentally measured for the studied site

Existing models have been considered for the selection of the functional form of the model and parameters to be incorporated in the model. The CPT and Vs relationship has been investigated by various researchers since the 1980s (Robertson and Campanella 1983; Robertson et al. 1986; Hegazy and Mayne 1995; Mayne and Rix 1995). One of the limitations of these earlier-developed models is that most of these relations are valid for either sand or clays. Later, these have been addressed by including parameters such as Ic and e, which relate the soil type with predicted Vs values (Piratheepan 2002; Andrus et al. 2007; Robertson 2009; Long and Donohue 2010; Gadeikis et al. 2013; Cai et al. 2014; Sara 2014; Ahmad et al. 2015; Mola-Abasi et al. 2015; McGann et al. 2015; Abbaszadeh Shahri and Naderi 2016; Mohamed Ahmed and Ahmed 2017; Zhang and Tong 2017; Tun and Ayday 2018; Fayed and Mousa 2020; Yang et al. 2022; Mousa and Hussein 2022; Khan et al. 2022; Mishra et al. 2023). Two models have been proposed in the following subsections, one without considering e (correlation model 1) and another considering e (correlation model 2).

Correlation model 1

First, a multiparameter regression model was formulated to predict Vs based on the CPT data. Both mechanical and electrical cone test data have been used to formulate CPT-V_S correlation for the study area. The shear wave velocity (V_S) has been estimated from MASW tests conducted at different locations and reported in the literature (Nilay et al. 2022). A total of 453 data pairs from 33 CPT and 16 MASW sites were considered. The site locations and the number of datasets at each location are shown in Fig. 11. The data pairs are very close, having a maximum distance of 100 m. Statistical regression analysis has been performed on these datasets, and a nonlinear multivariable equation has been proposed to predict V_S. The proposed equation is as follows:

$${\mathrm{V}}_{\mathrm{s}}\mathrm{ = }\mathrm{156.885}{{\mathrm{ q}}_{\mathrm{c}}}^{\mathrm{0.0}{33}}{{\mathrm{I}}_{\mathrm{c}}}^{0.120}\upsigma {^{\prime} }^{-\mathrm{0.169}}{\mathrm{z}}^{\mathrm{0.}{366}}$$

(10)

where, q_c is cone tip resistance, I_C is soil behavior type index, ${{\sigma }_{0}}{\prime}$ is effective stress at particular depth, and z is the depth. The predictive equation for shear wave velocity (V_S) is derived through non-linear regression incorporating multiple variables. This choice is informed by an evaluation of existing models such as Hegazy and Mayne (1995), and Andrus et al. (2007). Additionally, insights from prior research highlight the significance of cone tip resistance (q_c) as a pivotal parameter associated with the undisturbed shear strength of the soil, as demonstrated by Hegazy and Mayne (2006). Their findings indicate that cone tip resistance (q_c) exhibits superior variability in predicting V_S compared to sleeve friction (f_s). In the present study, q_c (containing a correlation coefficient (r) of 0.34 with Vs) is considered, alongside the soil behavior type index (I_C) in the correlation equation. As soil samples are not extracted during CPT, the inclusion of the I_C (with a correlation coefficient of 0.26 with Vs) incorporated valuable insights into soil type and its behavior. The correlation equation also considers other parameters, namely effective stress (${{\sigma }_{0}}{\prime}$) and depth (z). The inclusion of these parameters is justified by the high correlation observed between shear wave velocity (Vs) with depth (r = 0.95), as well as effective stress (r = 0.90). The selection of these variables was made based on correlation coefficient analyses independently. The observed lower r-values for (q_c) and (I_C) indicate a non-linear relationship with Vs, while effective stress and depth demonstrate a robust linear relationship with Vs. A visual examination of the relationship between Vs and depth confirms a positive correlation, indicating an increase in Vs along depth. A similar relationship is observed with ${{\sigma }_{0}}{\prime}$. Therefore, the inclusion of all these parameters in the CPT-VS correlation is deemed beneficial.

Fig. 11

Number of data pairs at various locations (e.g., C1 is CPT test location 1 and M2 is MASW test location as shown in Fig. 1) used for formulating CPT-Vs prediction model

The above correlation Eq. (10) has a coefficient of determination of 0.858. ANOVA was used to estimate the significance of the model. The proposed model has also been validated for the site using the datasets not included in the model formulation and discussed in subSect. "Validation of the proposed models".

Correlation model 2

As mentioned earlier, including e in the prediction model increases the model's efficiency. In some studies, ‘e’ is considered an input parameter for the computation of V_S, and the CPT-V_S correlation model is significantly improved. Therefore, another prediction model has been proposed in this section considering e in the regression model. The uniqueness of this proposed model is that all the parameters used in the regression model can be estimated from the CPT tests including void ratio ‘e’ presented in Sect. "Void ratio (e) Prediction Model". Therefore, the dependency of the model on other test methods has been eliminated in this proposed model. After incorporating e in the equation, the following CPT-V_S model has been proposed:

$${\mathrm{V}}_{\mathrm{s}}\mathrm{ = }\mathrm{151.85}{{\mathrm{9 q}}_{\mathrm{c}}}^{0.044}{{\mathrm{I}}_{\mathrm{c}}}^{0.165}\upsigma {^{\prime} }^{-\mathrm{0.202}}{\mathrm{z}}^{0.397}{\mathrm{e}}^{0.035}$$

(11)

The above correlation equation has a coefficient of determination of 0.849. ANOVA was used to estimate the significance of the model.

Validation of the proposed models

The validation of the CPT-V_S correlation is essential to ensure the accuracy and reliability of the predictions. Several methods are available to validate CPT-V_S correlations, including laboratory and field testing. One standard method is the comparison of V_s measurements obtained from different techniques or instruments. Field testing involves conducting CPT and geophysical tests at the same site to compare the Vs directly. The developed CPT-V_S correlations have been compared in this section with the MASW data to validate the correlations. In this section, the correlations have been validated using the data from some selected locations inside the campus, as shown in Figs. 12a and 13a for correlation model 1. From both types of cones used, the measured and predicted V_S profiles show good agreement. Not much discrepancy is observed in the predicted and measured data. The absolute percentage difference in the expected value is less than 18% of the calculated value. Another term defined by Zhang and Tong (2017) as the velocity ratio (K), which is the ratio of the predicted to measured velocity, is shown in Figs. 12b and 13b for the mechanical and electrical cones, respectively. The value of K shows the variation in predicted velocity and estimated velocity. The value of K for most data closer to 1 offers the best predictability power of the correlation model. The figure shows that at every level of depth, 'K' values are within approximately 25% of the measured data.

Fig. 12

Validation of prediction model for various locations (C3-M1, C14-M13, C15-M13) inside the studied site using mechanical cone (a) shear wave velocity profile, (b) velocity ratio (K-value)

Fig. 13

Validation of prediction model with ECPT data sites (ECPT2-M3, ECPT3-M3, ECPT4-M4, ECPT9-M1, ECPT26-M4, ECPT26(A)-M3) (a) in terms of shear wave velocity profile, (b) velocity ratio (K-value)

Comparing the two correlation models given in Eq. (10) and (11), no significant improvement is observed in the predicted V_S values with the introduction of e, as shown in Fig. 14. There may be two probable reasons for such a minor variation. The first is the consideration of I_C values in both correlations. The void ratio expresses the grain compactness. The I_C can also incorporate the effects of soil type and grain compactness. The second reason is the incorporation of ${{\sigma }_{0}}{\prime}$ and q_c in both models. These two parameters already consider the stiffness of the soil. Therefore, including e does not increase the accuracy of the model.

Fig. 14

Comparison of Vs Correlation Models 1 and 2 at different locations (a) C3-M1 (b) C15-M13 (c) ECPT2-M3 (d) ECPT4-M4

Additionally, ANOVA was used on the collected and predicted data to determine the significance of Eq. (10) and (11). The degrees of freedom for the numerator and denominator were suitably determined based on the number of groups and the sample size. The significance level was set at 0.05. The obtained F values in this study are 0.0064 & 0.00068; the F-critical values are 3.854 & 3.854; and the resulting p values are 0.98 & 0.98, respectively. ANOVA for both correlations gives nearly the same F value, F_cri value, and p values. Therefore, it is clear that the acquired F value is much smaller when compared with the critical F value (3.854). Therefore, it can be concluded that the observed differences in the means are not statistically significant, or it is likely that the variation in the data can be attributable to random chance or elements unrelated to the treatments under comparison. ANOVA with a p value of 0.98 indicated no statistically significant difference between the observed variation across groups. In other words, the null hypothesis, which states no significant differences between the groups being compared, is strongly supported.

Another tool for evaluating different regression equations is the computation of residuals for the fitted regression models. Therefore, for this reason, residuals (ε) are computed using the following equation:

$$\varepsilon =\frac{{\mathrm{ln}}\left({V}_{mea}\right)- {\mathrm{ln}}\left({V}_{pre}\right)}{{S}_{V|X}}$$

(12)

where S_V|X is an estimate of the conditional standard deviation (Ang and Tang 2007) and defined as:

$${{\mathrm{S}}}_{{\mathrm{V}}|{\mathrm{X}}}=\sqrt{\frac{\sum {({\mathrm{ln}}\left({{\mathrm{V}}}_{{\mathrm{mea}}}\right)-{\mathrm{ln}}\left({{\mathrm{V}}}_{{\mathrm{pre}}})\right)}^{2}}{{\mathrm{n}}-4}}$$

(13)

where n is the number of datasets included in the regression. In Fig. 15, ε calculated from Eq. (12) is shown along the depth for a few typical sites (blue circle markers) considered for validation. Figure 15-A (a, b, c… etc.) displays ε for the Correlation Model 1, i.e., without consideration of void ratio, while in the right-side Fig. 15-B (i, ii, iii…etc.) are the computed ε for Correlation Model 2. The continuous black line is the moving average with ± σ. At locations C15-M3, ECPT9-M1, and ECPT4-M4, nearly zero residuals are observed, which means that the developed correlation perfectly predicts the V_S value. Locations C3-M1 and C14-M13 show concentrated biases at depths greater than 20 m and 25 m for the latter case, which results in a slight underestimation of the V_S value at the mentioned level of depth. Collectively, observing all the residues for the formulated correlation is consistent with the considered validation sites. For correlation model 2 (Fig. 15-B), the same trend for ε is seen, i.e., residual estimates for the same selected areas are practically negligible. In Fig. 16, the X-axis represents the V_pre values obtained from Eq. 10 and 11. In contrast, the Y-axis represents the residuals given by Eq. 12. The figure shows that the ε value with a random scattering of residuals points around the zero line suggests that the regression model captures the actual measured value of V_S. Additionally, comparing correlation models 1 and 2 in Fig. 16b, no significant change has been observed in the residue plot.

Fig. 15

Residue plot along depth for different locations (a−f) shows residue for Correlation Model 1; (i-vi) shows residue for Correlation Model 2

Fig. 16

Residue plot as a function of V_pre for (a) Correlation Model 1 (b) Correlation Model 2

Validation of VS at a similar site worldwide

Two external sites are selected from the literature, similar to the study areas. Here, explicitly mentioning site similarity means the site of the same geologic age and soil type formation. The literature shows CPT parameters along with the measured shear wave velocity. The V_pre value from the developed correlation is compared to the V_mea for validation. Figure 17a and b show the validation of V_S data for sites in the Korean peninsula region (Sun et al. 2013) and Eskisehir, Turkey, soil deposits (Mola-Abasi et al. 2015), respectively. The results show decent agreement between the predicted and measured results.

Fig. 17

Validation of CPT-MASW relation for a similar site outside the campus worldwide (a) Korean peninsula, Korea (b) Eskisehir, Turkey (V_mea and V_pre are measured and predicted value of V_S)

Conclusions

This research underway with the collection of CPT test data from two different types of cones and V_mea from MASW tests from the IIT Patna campus. Previous studies available in the literature describe that many correlations are available to estimate shear wave velocity from CPT data. However, shear wave velocity predicted (V_pre) and measured (V_mea) show a significant difference, necessitating the development of a new correlation for the studied site. While selecting the parameters for correlation, previous researchers indicated that consideration of void ratio would be beneficial for the developed model. Whereas, very limited correlations are available for predicting void ratio from the CPT data. Therefore, a prediction model has been proposed for estimating the void ratio based on the CPT results, and then CPT-V_S correlations are formulated. From this study, the following conclusions are drawn:

• One of the novelties of this study is the proposed Eq. (6) and given in Table 3 for predicting the void ratio from the CPT data. The data analysis of formulated model shows a power relation between the factored void ratio (FVR) (e^0.5I_Cⁿ) and normalized cone tip resistance (Q_tn). Formulated FVR-Q_tn relation for void ratio computation when used with ECPT data, a downward shift has been observed, which necessitates the introduction of a cone correction factor (K_C) for the electrical cone. The value for Kc has been estimated and proposed as 0.809 for the electrical cone. A similar study can be performed using other cones to propose the value for Kc for other cones.

• The second novelty of this study is the models (shown in Table 3) for prediction of shear wave velocity. During the estimation of V_S, cone parameters such as q_c, I_C, ${{\sigma }{\prime}}_{0}$_, and z are considered in the first correlation model (Eq. 10). Moreover, the void ratio effect in V_S prediction is checked by adding the e term in the model. The correlation model (discussed in Sect. "Void ratio (e) Prediction Model") for e can be used to estimate it from CPT data.

• No significant improvement was observed after the introduction of e in correlation model 2 (Eq. 11). This is because I_C is already included in both models. The I_C can integrate the effects of soil type and grain compactness. Therefore, it is recommended to use this I_C index if a V_S prediction model is to be formed using CPT parameters.

• The proposed equations are validated inside and outside the study area with similar soil conditions. The site conditions, geologic similarity, etc., must be considered before using this correlation. If the direct measurement of V_S is possible, then preference should be given to all those methods.

Table 3 Proposed equations for void ratio (e) and shear wave velocity (V_s) estimation for soil in Quaternary alluvial