Introduction

Various methods and approaches are being performed when trying to determine the bearing capacity and settlement properties of soils where structures will be placed upon. The most widely used methods are the Menard pressuremeter test (MPT), standard penetration test (SPT), cone penetration test (CPT), and the plate loading test. Besides these, laboratory tests are also used for the same purpose. Various factors like potential disturbances in the sample specimens and samples not reflecting the properties of the soil accurately often influence the accuracy of the parameters used in calculations. In situ tests have the significant advantage of providing more reliable and realistic results as the soil is not being disturbed as such (ASTM 1994, ASTM D4318–00 2000, ASTM D1586/D1586M-18 2018). Besides, it is possible in in situ testing to obtain samples from any desired depth among the vertical soil profile. Many statistical relationships between in situ and laboratory tests have been established in the literature for cohesive and non-cohesive soils. The correlations between SPT and cohesion, internal friction angle, and MPT values are frequently in literature. However, no correlation was found in the literature between MPT data and consolidation data, and the studies investigating the relationships between in situ and laboratory findings regarding overconsolidated soils are few in numbers.

A limited number of researchers have performed research on the relationship between SPT and MPT values (Gonin et al. 1992; Yagiz et al. 2008; Bozbey and Togrol 2010; Kayabaşı 2012; Kayabaşı and Gökceoğlu 2012; Aladağ et al. 2013; Ağan 2014; Cheshomi and Ghodrati 2015; Anwar 2016; Özvan et al. 2018, 2019). Various empirical correlations have been obtained in the literature between SPT and MPT on sandy and clay soils (Table 1). The study by Chiang and Ho (1980) was performed in Hong Kong in 1980 and evaluated the linear relationship between the SPT-N and EPMT and PL values of weathered granite, while the study of Ohya et al. (1982) investigated the correlation between SPT-N and EPMT in clayey soils. Meanwhile, Yagiz et al. (2008) investigated the relationship between the corrected SPT blow count (Ncor) and EPMT and PL and revealed that a linear relationship existed between the corrected Ncor and EPMT and PL values for silty sand with clay. Bozbey and Togrol (2010) performed a study and investigated the relationship between SPT-N60, EPMT, and PL values with a total of 182 tests performed on sandy and clayey soil samples, and have obtained empirical equations with high regression coefficient (R2) for each soil type, separately. Gonin et al. (1992) have correlated the SPT results for a total of nine different soil types with EPMT and PL. In some of these studies, high determination coefficients (R2) were determined between SPT-N and net limit pressure (PL) and Menard deformation modulus (EM) for different soil types. The researchers suggest that the equations obtained as part of the study will yield valid results in case they are applied to similar soil types, and they could be taken into consideration during the initial stages of geological projects (Phoon and Kulhawy 1999; Yagiz et al. 2008; Bozbey and Togrol 2010; Kayabaşı 2012; Ching and Phoon 2012, 2013, 2014; Phoon and Ching 2013; Cheshomi and Ghodrati 2015; Shaban and Cosentino 2016; Özvan et al. 2019; Firuzi et al. 2019; Akkaya et al. 2019; Cheshomi et al. 2020; Cheshomi and Khalili 2021). Özvan et al. (2018) have found high determination coefficient between SPT and MPT results for clayey soils. In this research suggested multiple regression analyses be performed on SPT and MPT laboratory tests as future studies to determine the physical and mechanical properties of clayey soils.

Table 1 Empirical relationships between EM, PL, and SPT-N in the literature
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Regression analyses based on single variable are generally available in the literature (Table 1). Due to the different physical and mechanical properties of geological structures, it is usual that multivariate analyses give more accurate results. Therefore, the aim of this study is to more accurately describe the geological structure with multiple regression analyses between in situ and laboratory data.

In the present study, SPT and MPT tests were performed on consolidated clayey units that are well-distinguished from weathered clay and that have high (CH) or low plasticity (CL) properties, and on severely weathered claystone and other lithological units that could be classified as overconsolidated units. SPT-N60 value was obtained from the SPT test, while EM and PL values were obtained from the MPT test. The results of these tests and the data obtained from a series of physical and mechanical tests performed in the laboratory were evaluated using multiple regression analyses, which were then compared to findings obtained from similar soil types in the past.

Geological properties of the study area

The study area consists of Quaternary (Pleistocene) aged old lake and stream sediments that deposited as a result of water movements of the Lake Van (Fig. 1). With different thicknesses and engineering properties, these sediments are particularly present in the wide fields towards the east of Lake Van. Lake Van Basin is a region where rocks of different ages starting with the Paleozoic aged outcrop to the surface and the region has a complex stratigraphy, in particular due to the influence of tectonic activities in the area (Özvan et al. 2005; Akkaya et al. 2015, 2017, 2018; Akkaya and Özvan 2019). The total thickness of old lake sediments is approximately 150 m in the area (Acarlar et al. 1991). According to the previous studies, old and fresh stream sediments were encountered in the study area in addition to the old lake sediments (Acarlar et al. 1991; Selçuk 2003; Koçyiğit 2013). These units are intersected in the north of the study area by the Van thrust fault which ruptured in the destructive earthquake on October 23, 2011 (Mw = 7.1), and the units extend from northwest of the study area to the Lake Van (Akkaya et al. 2015, 2017, 2018; Akkaya and Özvan 2019; Sengul et al. 2019). The Van Fault that intersects these units is a thrust type fault inclining towards the north (Fig. 2).

Fig. 1
figure 1

Location map of the study area

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Fig. 2
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Geological map of the study area and geological cross section of NE-SW line

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Testing program or experimental testing methods

In addition to the previous data obtained from the study area, 6 additional boreholes were drilled on the hanging wall and footwall of the fault to investigate the influence range of the thrust fault (Fig. 2), and the clayey soils in the area were investigated using both the in situ and laboratory test data.

In situ tests

SPT and MPT represent the most commonly used in situ tests. SPT aims to measure the penetration resistance of the soil and was developed initially in the USA towards the end of the 1920s. Since the test setup is fairly simple and the testing takes a relatively short time, SPT is a widely preferred in situ testing method. In the SPT method performed as part of this study, the test tube was driven into the soil using an automatic pile driver, and the SPT-N blow count was obtained, with which SPT-N60 values were calculated (Bowles 1997; Aggour and Radding 2001; British Standards Institution 2007). The SPT was performed in line with the ASTM D1586/D1586M-18 (2018) standards. During the SPT, the blow counts are highly sensitive to the length of rods, hammer energy, sampler type, borehole diameter, and overburden stress (Idriss and Boulanger 2008, 2010). Thus, a corrected penetration resistance is obtained using raw SPT data and a number of correction factors as shown in the following equation:

$${\left(NI\right)}_{60}={C}_{N}{C}_{E}{C}_{R}{C}_{B}{C}_{S}{N}_{m}$$

where CN, CE, CR, CB, and CS are the correction parameters, whereas Nm is the SPT blow count obtained in situ (Idriss and Boulanger 2008, 2010).

MPT is a test performed using this device and is often performed in areas where the soil is too weak and weathered soils to obtain proper test specimens for laboratory tests. Furthermore, a self-boring pressuremeter device was also developed to reduce the drilling disturbance in loose soils. The MPT equipment consists of four main parts as the reading unit, probe, pressure air tube, and the pipe section (Fig. 3). The probe through the borehole is either 76 mm or 89 mm in diameter and is made up of three parts consisting of the main body, compressed air cell, and compressed water compartment. The probe diameter is 74 mm. The measuring cell volume (Vc) was taken as 790 cm3. When the probe reaches the test level within the well, it is inflated using compressed air, and pressure is applied to the well every 60 s in an attempt to deform the soil. If the applied pressure fails the soil, the well walls start to deform and additional water is sent to the compressed air compartment. The amount of water sent to the water chamber is recorded every 15, 30, and 60 s. Here, the pressure level applied corresponds to the soil deformation pressure, and the amount of water sent in corresponds to the amount of deformation under that particular pressure level.

Fig. 3
figure 3

a Simultaneous SPT and MPT measurements from two adjacent boreholes at the same depth: undisturbed sample collection (left panel), MPT measurement equipment (middle panel), and theorical pressure–volume curve (bottom panel). b Consolidation measurements: test equipment (top panel), and consolidation curve at 2 m in SK-1 borehole (bottom panel)

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As a result of this test, a pressuremeter curve can be plotted which shows the pressure and volume change, and it is possible to calculate net limit pressure (PL) and Menard deformation modulus (EM) values for each depth level tested (Menard 1957; Shields and Bauer 1975; Baguelin et al. 1978; Mair and Wood 1978; Clarke 1995; ASTM 1994). PL represents the difference between the lift-off pressure and limit pressure. PL is widely utilized to define soil strength for use in design and analysis procedures (Shaban and Cosentino 2016). EM, on the other hand, is calculated from the pseudo-elastic slope of the corrected pressure–volume curve. These tests were performed as per the standards outlined in ASTM D4719-87 (1994) and AFNOR NF 94–110-1 (2006).

The data from the in situ tests were obtained from a total of 6 boreholes with an approximately 15-m spacing between them. MPT was performed every 1.5 m in the first well, and concurrent SPT measurements were performed in a second well that was approximately 5 m away from the first at the same depths. Furthermore, disturbed and undisturbed (UD) soil samples were collected from the boreholes when possible. Specimens were coated with paraffin to prevent exposure to air, which were then further covered with stretch film. The physical (water content, specific weight, unit volume weight, grain size, and consistency limit tests) and mechanical properties (consolidation and triaxial pressure tests) were determined in the laboratory.

Laboratory tests

The behavior of soils under different water content levels is called “consistency” and it is of extreme importance when trying to determine the physical properties of fine-grained units. Soil consistency is the strength with which soil materials are held together or the resistance of soils to deformation and rupture. Soil consistency is measured for wet, moist, and dry soil samples. The liquid limit (LL), plastic limit (PL), and shrinkage limit values are collectively known as Atterberg limits and are determined based on the water content of the soils. Atterberg limit tests were performed on the disturbed specimens collected from the 6 boreholes at different locations of the study area, adhering to the standards set forth by ASTM D-4318 (2000).

Similar to Cetin (1997, 2000), consolidation tests with ASTM D-2435–2009 standard were performed to determine the consolidation characteristics (e.g., pre-consolidation pressure) of the fine-grained units in the study area including the Van thrust fault. In this test, the constant weight increase period was set to 24 h and its multiples. Each stress increase was sustained until the excessive water pressure in the pores was completely depleted. The minimum diameter was 50 mm and the minimum height was 20 mm for the samples used in this study. The deformation changes in height were measured using a comparator with 0.01 mm sensitivity. In cases where the test was applied to a fully saturated sample or a sample from beneath a groundwater table, water was introduced to the consolidation compartment after the settlement load. In cases where the sample was not covered by water shortly after applying the settlement load, the consolidation device was covered with moist cotton to prevent evaporation, so that the sample volume could be preserved. The sample was then subjected to constant stress increases. To achieve a compression curve with a distinct break in the slope, and in turn, to obtain the pre-consolidation pressures, the final loading pressure was selected as four times the expected pre-consolidation pressure. Loadings were usually initiated so that at least 2.5 kPa stress could be created on the samples. To minimize the heave after the test, the samples were returned to their settlement loads (2.5 kPa) during removal.

Various researchers have developed different methods to determine the pre-consolidation pressure (σpc) (Casagrande 1936). The most commonly used is the method suggested by Casagrande (1936) and was used in this study to determine σpc as well. With this method, the maximum effective stress value (σı) that influences a given soil and gives its final structure and fabric is defined as the pre-consolidation pressure. The pre-consolidation pressure (σpc) was determined using the void ratio (e) – log effective stress (σ′) curve and the Casagrande method.

A triaxial pressure test (UU) was also performed on the UD samples under laboratory conditions. This was done by drawing the Mohr circles corresponding to the primary tensions (σ1, σ3) of the moment of fracture due to low load impact, the c and ϕ values for the Coulomb’s shear equation. In this study, UU (undrained-unconsolidated) triaxial test was performed using the ASTM D(2850)–15 2015 standards.

The index properties of fine-grained soils

Disturbed and undisturbed soil samples were collected from the boreholes as part of the study, and the physical properties of the samples were determined (Table 2). When the grain size ratio of the samples was inspected, it was revealed that the ratio of fine-grained silt and clay amount to total grain size was higher than 80%. When the natural water moisture content of the samples was investigated, it was found out that the fine-grained soil samples usually were not fully saturated with water. Inspection of the water content of these samples has shown that the highest water content was 32%, while the lowest was 11.6%. A great majority of the inspected samples contained 20–24% water, while the average water content among all samples was determined as 21.9% (Table 2).

Table 2 Physical properties and soil classes of the investigated clayey levels
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When the specific gravity and densities of the samples were evaluated, it was found that the highest specific gravity was 2.87 while the lowest specific gravity was 2.60, and the highest density was 2.14 g/cm3 while the lowest density was 1.82 g/cm3 (Table 2). Inspection of the consistency curve has shown that the highest liquid limit for these units was 88%, while the lowest was 25%, and the highest plasticity limit was 32%, and the lowest was 15%. When these values are placed in the plasticity chart, the inspected clayey levels were classified either low (CL) or high (CH) clayey soils.

In the in situ tests, N30 values of the SPT blow counts were refusal, especially at regions closer to the fault (> 50 blow/30 cm) (Table 2). Differing from previous studies, the present study attempted to continue the penetration after 50 blows in the SPT measurement, so that SPT data could be compared to MPT data (ASTM D1586/D1586M-18, 2018).

In the study area, MPT tests were performed every 1.5 m to make the measurements coincide with that of the SPT tests. This in situ test can be influenced by various factors like in open borehole wall collapses, or groundwater presence. Due to situations like these, when evaluating values obtained as a result of tests, Menard pressuremeter (elastic) modulus (EM) and net limit pressure (PL) could not be calculated for some depth levels. The EM and PL values were calculated for a total of 33 different depth levels in the study area (Table 2). The calculations show that EM values change between 58.7 and 658.9 kg/cm2, while PL values change between 8.7 and 67.1 kg/cm2. When these results are compared with values provided for typical EM and PL value ranges, it becomes apparent that the soil is very solid—hard clay. When these values are compared to the physical properties of the inspected soils and SPT-N60 values, the results were found to be compatible.

The evaluation of the average EM values obtained as part of this study has shown that the highest EM values were recorded in areas closer to the Van Fault, while the lowest EM were recorded in the well that was nearest to the lake (southwest) (BL-2) (Table 2). Similar to the SPT test, it was found out that the higher the depth, the higher the EM value.

Simple regression analysis

Regression analysis explains a functional relationship between two variables. In such a relationship, if the independent variable is X with ei representing an additive error term and the dependent variable is Y, the functional relationship between the two variables can be written as:

$${Y}_{i}=f\left({X}_{i},\beta \right)+{e}_{i}$$
(1)
$$f\left({X}_{i},\beta \right)= {\beta }_{0}+{\beta }_{1}{X}_{i}$$
(2)

The aim of the analysis is to estimate β parameters with different regression analysis types, such as least squares method. The estimations of the parameters with the least squares method and the correlation coefficient formulas are as follows:

$${\beta }_{1}=\frac{\sum ({x}_{i}-\overline{x })({y}_{i}-\overline{y })}{\sum {({x}_{i}-x)}^{2}}$$
(3)
$${\beta }_{0}=\overline{y }-{\beta }_{1}\overline{x }$$
(4)
$$R=\frac{\sum xy-\left(\sum x\right)\left(\sum y\right)/n}{\sqrt{\left[\sum {x}^{2}-{\left(\sum x\right)}^{2}/n\right]\left[\sum {y}^{2}-{\left(\sum y\right)}^{2}/n\right]}}$$
(5)

Statistical evaluations were performed to compare the results of in situ and laboratory tests performed on the fine-grained units of the study area and to reveal any potential correlations between them. It is determined that the physical and mechanical properties of the examined specimens depict dissimilarities in the laboratory and in situ tests. These differences are also recognized in the relationships between these parameters. To start the analysis, the single-variable linear regression of the in situ pressuremeter readings and the parameters obtained through laboratory measurements was performed first.

Firstly, regression analyses were performed to obtain empirical relations between the EM and the σpc. The results of the regression analysis are shown in Table 5 and Fig. 5a. The equation with the highest coefficient (R2 = 0.83) of the regression between EM and σpc is represented by a power function (Eq. 9). In this equation, EM and σpc values are in kg/cm2. Evaluation of the changes in σpc and EM values with depth has shown that as depth increased, both of these values increased as well (Fig. 4b−c).

Fig. 4
figure 4

Depth-dependent changes of data; SPT-N60 (a), EM (b), σpc (c), and c (d)

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The other regression analyses were then performed to obtain empirical relations between the EM and the cohesion (c) derived from triaxial compressive strength (Fig. 5b). The equation with the highest coefficient (R2 = 0.73) of the regression between EM and c is represented by a power function (Eq. 10). Cohesion (c) values are influenced by factors like the grain size and water content of the soil, making it challenging to obtain correlations between in situ and laboratory data. Evaluation of the c value with depth has revealed that in general, as the depth increased, the c value increased as well (Fig. 4d).

Fig. 5
figure 5

Relationships between in situ and laboratory data; EM and σpc (a), EM and c (b), EM and SPT-N60 (c), and PL and SPT-N60 (d)

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Evaluation of the relationship between the PL and SPT-N60 values has shown only a low determination coefficient between the parameters (R2 = 0.58) (Fig. 5d). In general, the PL value was found to increase as the depth increased. Similarly, the SPT-N60 value increased as the depth increased in most cases (Fig. 4a).

When the correlation between EM and SPT-N60 values was investigated, a high determination coefficient was determined (R2 = 0.90) (Fig. 5c). Evaluating the relationship between depth and EM and SPT-N60 values has shown that, in general, these values increased as the depth increased (Fig. 4a−b). We considered that the data from the point closer to the hanging-wall side of the Van Fault (BL-1 and BL-6) affect the correlation between all the data, due to the presence of deformation structures in the soil caused by the fault. When the data from these boreholes are ignored, the regression coefficient increases from R2 = 0.75 to R2 = 0.90 (Eq. 11).

Adaptation between the measured and EM values computed exponentially through Eq. 11 was determined, except for data from BL-6. Similarly, the margins of error between measured and calculated values were also found to be low, once again with the exception of BL-6. The fact that deformations related to the fault nearby the BL-6 point are high is causing the margins of error in these data to rise beyond thresholds.

When all data groups are evaluated overall, it was found that the data nearby the hanging-wall side of the fault (BL-1 and BL-6) show increased variation due to deformation structures in the soil, which influence the regression results. In almost every variable inspected, ignoring these data resulted in higher harmony and determination coefficients. This is indicative of the significance and importance of data set selection, particularly in thrust fault deformation areas.

Multiple regression analysis

The relationship between one dependent variable and more than one independent variable can be examined in the regression model. The multiple linear regression model has the form:

$${Y}_{i}={\beta }_{0}+\sum_{j=1}^{n}{\beta }_{j}{X}_{ij}+{e}_{i}$$
(6)

Yi is the real-valued response for the ith observation, β0 is the regression intercept, βj is the jth predictor’s regression slope, Xij is the jth predictor for the ith observation, and ei is an error term.

The statistical analysis indicates that the non-linear multiple regression approach is more suitable than the linear regression analysis. In the multiple regression steps of the statistical studies, the relationships between the EM and PL with the SPT-N60, σpc, c, w, PI, and PL values were evaluated together. Generally, the equations with the high determination coefficients were obtained from multiple regression analysis.

The first step is to define the independent variables of the SPT-N60, σpc, c, PI, PL, and the w value as the function of EM dependent variable:

$$\left.\begin{array}{c}{E}_{M}=f\left({\mathrm{SPT}-\mathrm{N}}_{60},w\right) \\ {E}_{M}=f\left({\mathrm{SPT}-\mathrm{N}}_{60},{\upsigma }_{pc}\right)\\ {E}_{M}=f\left({\mathrm{SPT}-\mathrm{N}}_{60},{\upsigma }_{pc},w\right)\\ {E}_{M}=f\left({\mathrm{SPT}-\mathrm{N}}_{60},PI, w\right)\end{array}\right]$$
(7)

After this definition, the relationships between the different combinations of SPT-N60 and w% values were evaluated. A, B, C, and D represent the coefficients of the equations. Multiple regression experiments were carried out by creating linear and non-linear different equation groups. The non-linear multiple regression equations obtained are given as follows;

$$\left.\begin{array}{c}{E}_{M}=A+B {N}_{60}+C w \\ {E}_{M}=A+B {\sigma }_{pc}+C {N}_{60}\\ {E}_{M}=A+B {\sigma }_{pc}+C w\\ {E}_{M}=A+B w+C {N}_{60}\\ {E}_{M}=A{N}_{60}^{B}+C \mathrm{ln}(w)+D\end{array}\right]$$
(8)

When the differences between the measured EM values and EM values that were calculated through linear multiple regression analyses that contained these parameters were evaluated, it was found that the values are close to each other at the 95% confidence interval (CI) (Fig. 6a). The margin of error is small in all points except BL-6. We considered that the errors associated with BL-6 data are due to the deformation in soil influencing SPT values. Table 3 presents the results of the statistical analyses that were performed with these parameters.

Fig. 6
figure 6

Correlation between the dependent variable EM and the EM value calculated using the independent variables SPT-N60 and w (Eq. 26) and the distribution of error margins (a), the independent variables SPT-N60 and σpc (Eq. 25), and the distribution of error margins (b)

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Table 3 Statistical values of linear multiple regression analysis including EM dependent variable and SPT-N60 and w independent variables (Eq. 26)
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Investigation of Table 3 reveals that the R2 value between the EM variable and the independent variables SPT-N60 and w is 0.919. However, adjusted R2 value should be considered valid for multiple regression analyses. Accordingly, the R2 value should be taken as 0.845. This means that 84.5% of the change that occurs in the dependent variable (EM) can be explained by the independent variables (SPT-N60 and w). Furthermore, the results of the ANOVA (variance analysis) test have revealed the meaningfulness of the model as a whole through the results of F tests. The significance value here is essential. In case the F test finds a value meaningful, this means that our model is statistically meaningful as a whole. The significance value for this analysis was found as 0.000, meaning it is smaller than 0.05, and even 0.01, indicating that our regression model as a whole is statistically significant.

When the significance level of the regression model is inspected in Table 3, the intercept coefficient was determined as 18.893, and p value was determined as 0.007. This means that the constant term is also significant. The regression model coefficient for SPT-N60 was calculated as 0.341. T test results show that the significance level is 0.000, and since this value is below 0.05, the analysis is meaningful at a significance level of 5%. The regression model coefficient for the natural moisture content (w) was calculated as − 0.365, and the p value for it was 0.174. Here, the regression model coefficient is negative and w has an inverse relationship with the variables.

Another multiple regression analysis was performed between the dependent variable EM and independent variables SPT-N60 and σpc (Fig. 6b, Table 4). The evaluation of the difference between the measured EM value and the EM calculated in the linear multiple regression analysis that contains SPT-N60 and σpc independent variables has shown that these values are close to each other (Fig. 6b). The margin of error between measured and calculated values is low in all points except for BL-6. Like the case was in other analyses, we believe the errors with BL-6 are caused by the deformation constructs in the soil or the stress release beyond the fault area influencing the SPT values.

Table 4 The statistical values for the linear multiple regression analysis containing the dependent variable EM and the independent variables SPT-N60 and σpc (Eq. 25)
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Investigation of Table 4 has revealed the R2 value between the dependent EM value and independent SPT-N60 and σpc variables as 0.904. Yet, the adjusted R2 value should be considered in multiple regression analyses, and the R2 value should thereby be considered 0.804. This can be interpreted as 80.4% of the change in the dependent variable (EM) can be explained by the independent variables (SPT-N60 vs. σpc). ANOVA test also has revealed the F test results which display the statistical meaningfulness of the model as a whole. Here, the significance value is of importance, as in case the F test finds this value significant, this would mean our model would be statistically significant as a whole. The significance value for this analysis was found to be 0.000, which is smaller than 0.05, and even 0.01. This is indicative that our regression model is statistically meaningful.

Evaluation of the coefficients of the regression model in Table 4 and the significance value has shown that the coefficient for the intercept term was 9.032, and the p value was found as 0.005. This means that the intercept term is also meaningful. The SPT-N60 regression coefficient was calculated as 0.323. T test results show the significance level as 0.000, and since this value is lower than 0.05, the analysis is meaningful at a 5% significance level. σpc regression model coefficient was calculated as 1.614, and the p value was found as 0.312. The regression coefficient is positive, indicating that the σpc variable is directly proportional with other variables. In this analysis, when σpc as the weakest independent variable (with the lowest p value) was removed from the model and the regression was calculated again, the model was not suffered any additional weakening, and the independent variable was integrated back into the model.

All simple and multiple regression analysis results obtained as part of this study have been presented in Table 5. As can be seen from their inspection, certain correlations with high determination coefficients were determined between in situ and laboratory tests. The correlations with relatively lower determination coefficients are around the thresholds that are suitable for future tests with increased data volume.

Table 5 The results of simple and multiple variable regression analyses performed in the study (units for the variables are as follows: EM: kg/cm2, PL: kg/cm2, σpc: kg/cm2, c: kN/m2, w: %, LL: %, PI: %)
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The highest correlation level in this study was determined between EM and SPT-N60 values, which are also frequently worked in literature. However, the correlations between the laboratory tests are unusual in literature. The reason for this could be the fact that soil structure usually contains factors that influence the laboratory test results like the grain size, plasticity, and water content, which were also encountered in the study. Variables like consistency and water content, which are used in multiple regression analyses, often yield p values that are higher than 0.05. While this situation has lowered the level of correlation, it had a minor impact on the determination coefficients. Of the equations presented in Table 5, Eqs. 1, 3, 15, 17, 18, 20, and 23 have been determined as equations with relatively higher correlations with more statistically significant relationships.

All data obtained from in situ and laboratory tests were analyzed and put into graphs using spreadsheets prepared in Excel. The statistical correlations between the parameters obtained from the analyses were evaluated in the JMP-8v and SPSS-22v software (2020).

The EM (predict) data derived from the equation (Fig. 7) and the EM (measured) values correlated with the basic regression analysis results in a regression coefficient (R2) of 0.83–0.87, which is greater than the coefficient of determination of Eqs. 9–15, except for Eq. 11 (Table 5).

Fig. 7
figure 7

Graph showing the relation between EM(measured)–EM(predicted) with two variables in Eq. 11 (a), Eq. 25 (b), Eq. 26 (c), and Eq. 28 (d)

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While similar coefficients were obtained in certain simple regression analyses, the high consistence results obtained in multiple regression analyses where multiple variables were considered are of great importance. Equations that have been produced based on a higher number of variables reflect a concordance in data. Due to this concordance in multiple regression analyses, the results obtained are quite suitable. The nature of this relationship shows that the linear and non-linear multiple regression analyses are more suitable than the simple regression analyses.

When some literature equations were examined, it was determined that there were different distributions (Fig. 8a). The samples taken in this study, unlike other studies, generally have overconsolidated clay properties. After, 50 blow counts in the SPT values were obtained using automatic pile driver.

Fig. 8
figure 8

SPT-N60 and EM data obtained in the literature and in this study (a), comparison of the regression model between EM and SPT-N60 developed in this study with those proposed in the literature (b)

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Considering the trend between SPT-N60 and EM data obtained in this study, Yagiz et al. (2008) and Bozbey and Togrol’s (2010) studies were observed to be closer to their trends (Fig. 8b). It can be said that the equations obtained between SPT-N60 and EM in this study can give the best result for overconsolidated clays among other equations in the literature.

Conclusions and recommendations

In this study, the changes in physical and mechanical properties of clays in different depths in the study area including the Van thrust fault have been investigated with in situ and laboratory tests, and the test results were statistically evaluated.

In this study, some statistically significant empirical equations with high determination coefficient and 95% confidence interval between in situ and laboratory tests are proposed. Thus, new contributions and suggestions were made to the relations between in situ and laboratory findings of overconsolidated soils, which are limited in the literature. In addition, there is no correlation between MPT data and consolidation data in the literature, and a correlation obtained between these parameters was suggested in the study.

The statistical analysis indicates that the non-linear multiple regression approach is more suitable than the simple regression analysis. In the multiple regression steps of the statistical studies, the relationships between the EM and PL with the SPT-N60, σpc, c, w, PI, and PL values were evaluated together. Generally, the equations with the high determination coefficients were obtained from multiple regression analysis.

Linear multiple regression analyses containing the dependent variable EM and independent variables SPT-N60, natural water content (w), c, and σpc have been performed, and the suggested equations are provided below. In the suggested equations, significant results were obtained when there are parameter value intervals for all SPT values in the 95% confidence interval, w is greater than 10, σpc is greater than 50, and c values are between 0.1 and 0.4.

Suggested equations

R2

Equation no

EM = 24.016 (SPT-N60)0.6555

0.90

Equation 11

EM = 75.501 (σpc)1.9005

0.83

Equation 9

EM = 24.886 (c)0.7938

0.73

Equation 10

PL = 2.8661(SPT-N60)0.5734

0.58

Equation 16

EM = 18.893 + 0.3413 SPT-N60 − 0.365w

0.85

Equation 26

EM = 43.925 (SPT-N60)0.519 + 78.104 ln(w) − 86.031

0.87

Equation 28

EM = 9.032 + 1.614 σpc + 0.323 SPT-N60

0.85

Equation 25

EM = 17.017 + 0.344 SPT-N60 + 0.112PI − 0.458w

0.85

Equation 18

EM =  − 8.163 + 74.739 σpc + (7.457c/98.1)

0.79

Equation 23

PL = 1.2297 + 0.0247 SPT-N60 + 0.0136w

0.51

Equation 31

PL = 1.112 + 10.994 σpc + 0.332 c/98.1

0.52

Equation 33

Correlations with high determination coefficients are found in this study as the results of various simple and multiple regressions. The authors suggest that the correlations with relatively lower determination coefficients be tested again with the increased number of data. We suggest increased borehole and number of data for future studies so that more accurate evaluations with lower error margins could be performed. We also suggest further analysis for clay levels of different properties.