1 Introduction

Reliability-based design is increasingly being applied to geotechnical engineering because the probabilistic approaches allow the robust consideration of various sources of uncertainty such as the inherent variability of soil properties (Lacasse 2016; Phoon 2017). In probabilistic methods like Monte Carlo simulation, the soil properties are modeled as probability distributions instead of single deterministic values. However, some soil properties are mutually dependent, meaning that there is a statistical relationship (which may or may not be causal) between them. The degree of cross-correlation between two soil properties can be quantified by means of correlation coefficient ρ, which can vary from − 1 to + 1. If ρ = 0, there is no cross-correlation between the soil properties (i.e., they are independent of each other). Any significant correlation between the soil properties in the model should be considered for the most accurate estimation of the probability of failure or of exceeding the serviceability limit state (Baecher and Christian 2003; Cheng and He 2020; Li et al. 2012; Nguyen et al. 2023).

In geotechnical applications, copula functions are often used to model these cross-correlations (Huffman et al. 2015; Li et al. 2012, 2013; Tang et al. 2013, 2015; Wu and Xin 2019; Zhang et al. 2018, 2019). One advantage of copulas is that the multivariate distribution (with any number of random variables) can be constructed using simple pairwise correlation coefficients. The Gaussian copula, which incorporates the product–moment (Pearson) correlation coefficient, is often adopted (Ching and Phoon 2012; Li et al. 2012; Tang et al. 2015). For instance, copulas have been proven to be useful when simulating load–displacement curves for piles because the two hyperbolic curve-fitting parameters are correlated (Li et al. 2013; Tang and Phoon 2018; Wu and Xin 2019). Li et al. (2013) concluded that ignoring this strong negative correlation between the fitting parameters would cause the overestimation of the probability of exceeding the allowed pile settlements.

Modeling involving copulas, however, has been scarcely applied in probabilistic settlement calculations for embankments on soft clays. In most of the past case studies, the uncertain variables were assumed to be independent of each other (or perfectly correlated) for the sake of simplicity (Ching and Hsieh 2009; Liu et al. 2018; Spross and Larsson 2021). On the other hand, copulas also require knowledge of the bivariate correlation coefficients; even though many empirical correlations have been established for predicting the compression index from the index properties (e.g., Shimobe and Spagnoli 2021; Yamada et al. 2023), little attention has been paid to the cross-correlation between the compressibility properties (e.g., compression index vs. preconsolidation pressure). Moreover, the need to model cross-correlation between soil parameters depends on the selected mechanical model. In soft clays, an appropriate mechanical model and carefully chosen parameters are crucial to for an accurate prediction of settlement response (e.g., Cui et al. 2023). For instance, in the case of soft Nordic clays, the tangential stiffness approach (Janbu method) is preferred over the compression index approach because it allows nonlinear compressibility in a semi-logarithmic space. Such nonlinearity is often encountered in soft marine clays (Di Buò et al. 2019; Larsson 1986; Tanaka 2002). Recent database analyses have shown that the curve-fitting parameters in the Janbu method are in fact cross-correlated (Löfman and Korkiala-Tanttu 2021a).

This study aimed to evaluate the influence of the applied correlation structure (i.e. cross-correlations or independence) on the uncertainty in stress–strain response modelled with Monte Carlo simulation. Two extensively studied soft clay sites were utilized to assess the site-specific correlation structures and marginal distributions for the compressibility parameters of the compression index (Cc) and Janbu settlement calculation methods. In addition, to simulate a situation where the site-specific data are limited, the correlation coefficients were also defined using a clay database, and marginal distributions were constructed using literature values for inherent variability. Next, Monte Carlo simulation with a Gaussian copula was performed, and the simulated sets of compressibility parameters were used to construct stress–strain curves that were then compared to the observed variability in oedometer curves. Finally, the simulated settlements in the clay sublayers were compared by means oof boxplots in order to quantify the influence of the assumed correlation structure between compressibility parameters.

2 Compressibility of Soft Marine Clays

The Cc method is often used to calculate the vertical strain caused by primary consolidation in clay soils. The total vertical strain εv is the sum of the overconsolidated (OC) and normally consolidated (NC) strains:

$${\varepsilon }_{v}={\varepsilon }_{OC}+{\varepsilon }_{NC}= \frac{{C}_{s}}{1+{e}_{0}}{{\text{log}}}_{10}\left(\frac{{\sigma }_{p}^{\mathrm{ {\prime}}}}{{\sigma }_{v0}^{\mathrm{ {\prime}}}}\right)+\frac{{C}_{c}}{1+{e}_{0}}{{\text{log}}}_{10}\left(\frac{{\sigma }_{v0}^{\mathrm{ {\prime}}}+ \Delta {\sigma }_{v}}{{\sigma }_{p}^{\mathrm{ {\prime}}}}\right)$$
(1)

where e0 is the initial void ratio, Cc is the compression index, Cs is the swelling index (or recompression index), σv0′ is the effective in-situ stress, σp′ is the preconsolidation stress, and Δσv is the external added vertical load. The indices Cc and Cs are defined as slopes of linear lines fitted to the oedometer curve plotted in a log(σv′)-e graph. Figure 1 shows a typical oedometer result of soft Finnish clay. As the stress–strain response after σp′ is nonlinear, the common practice in Finland is to determine Cc using two load increments after σp′ (which usually corresponds to the relevant stresses in a practical geotechnical design).

Fig. 1
figure 1

Typical oedometer curve of soft clay: a the Janbu method compared to the Cc method. b m1 modification in the Janbu method

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Unlike the Cc method, the tangential stiffness concept proposed by Janbu (1963) allows the modeling of the nonlinear compressibility encountered in most Nordic clays. In the Janbu method, the NC strain caused by stress increase from σp′ to (σv0′ + Δσv) can be calculated using the following equation (Helenelund 1974; Janbu 1963):

$${\varepsilon }_{NC}=\left\{\begin{array}{c}\frac{1}{{m}_{1}{\beta }_{1}}\left[{\left(\frac{{{\sigma }_{v0}}^{\mathrm{^{\prime}}}+ \Delta {\sigma }_{v})}{{\sigma }_{ref}}\right)}^{{\beta }_{1}}-{\left(\frac{{{\sigma }_{p}}^{\mathrm{^{\prime}}}}{{\sigma }_{ref}}\right)}^{{\beta }_{1}}\right] (if {\beta }_{1}\ne 0)\\ \frac{1}{{m}_{1}}{\text{ln}}\left(\frac{{{\sigma }_{v0}}^{\mathrm{^{\prime}}}+ \Delta {\sigma }_{v}}{{{\sigma }_{p}}^{\mathrm{^{\prime}}}}\right) (if {\beta }_{1}=0)\end{array}\right.$$
(2)

where σref is the reference stress (100 kPa) while m1 and β1 are the modulus number and stress exponent in the NC state and are defined from the oedometer curve using curve-fitting methods such as regression analysis. If β1 is negative, the oedometer curve is concave. Figure 1 illustrates an oedometer result where β1 = − 1.1 (m1 = 4.4; σp′ = 53 kPa; Cc = 2.99). If β1 = 0, the Janbu method corresponds to the Cc method (i.e., the stress–strain curve is linear in a semi-logarithmic space).

Equation (2) can also be used to calculate the strains in the OC state (stress increase from σv0′ to σp′), in which case the notations m2 and β2 are used for the modulus number and stress exponent, respectively (Vepsäläinen and Takala 2004). Parameters m2 and β2 are usually defined from the recompression curve of the unloading–reloading stage of the oedometer test, but the swelling curve can also be used. The example of such curve in Fig. 1 has been fitted to recompression (m2 = 50.4; β2 = 0.68). If β2 is set to 1, Eq. (2) applied to OC strain simplifies it to linear stress–strain dependency: εOC = (σp′− σv0′)/(100 m2).

Highly structured clays are characterized by negative β1, and the related steepness after σp′ in the stress–strain curve may induce settlement calculation errors. That is, if the Janbu parameters defined from an oedometer test are applied to a calculation layer with a lower σp′ (i.e., due to the strain rate effects or the conservative assumption of overconsolidation), the settlement is overestimated in the case of negative β1 due to the extrapolation of the oedometer curve. Länsivaara (1995, 2003) suggested modifying m1 to avoid this error, as follows:

$${m}_{1;calc}={m}_{1;test}{\left(\frac{{\sigma }_{p;test}^{\mathrm{^{\prime}}}}{{\sigma }_{p;calc}^{\mathrm{^{\prime}}}}\right)}^{-{\beta }_{1}}$$
(3)

where m1;test and t σp;test′ are the modulus number and preconsolidation pressure defined from the oedometer test, respectively; σp;calc′ is the preconsolidation pressure used in the settlement calculation; and β1 is the stress exponent defined from the oedometer test. A more negative β1 induces a greater change in m1. Figure 1b shows how the m1 modification (assuming σp;calc′ = 40 kPa) affects the Janbu curve: after the modification, the general shape of the oedometer curve is preserved while an unmodified m1 will lead to much greater strains. This modulus number modification is also recommended in the Finnish geotechnical design guidelines (Finnish Transport Agency 2012). It should be noted that Eq. (3) implies that m1 is unaffected if β1 = 0, meaning that Cc method is not characterized by this stress dependency.

3 Gaussian Copula

The Gaussian copula can be constructed from a multivariate normal distribution, and it is based on probability integral transform, which allows the transformation of any continuous probability distribution into a standard uniform distribution, and vice versa. That is, if random variable X is defined with cumulative distribution function (CDF) FX, the following applies:

$$Y={F}_{X}(X)$$
(4)

where random variable Y has a standard uniform distribution. To transform this uniform distribution into any other continuous distribution, the inverse CDF is applied to Y. Hence, the individual marginal probability distributions do not need to be normally distributed.

For the construction of a Gaussian copula, the multivariate normal distribution is first defined. The probability density function (PDF) of a multivariate normal distribution is defined by mean vector μ and covariance matrix C:

$$f\left(\mathbf{X}\right)=\frac{1}{\sqrt{{\left(2\pi \right)}^{k}\left|\mathbf{C}\right|}}{\text{exp}}\left(-\frac{1}{2}{\left(\mathbf{X}-{\varvec{\upmu}}\right)}^{T}{\mathbf{C}}^{-1}\left(\mathbf{X}-{\varvec{\upmu}}\right)\right)$$
(5)

where X = (X1, X2, …., Xk)T is a k-dimensional normal random vector (i.e., k is the number of individual random variables). The “T” symbol represents a matrix–vector transpose. In the case of three variables (k = 3), μ and C are defined as:

$${\varvec{\upmu}}=\left\{\begin{array}{c}{\mu }_{1}\\ {\mu }_{2}\\ {\mu }_{3}\end{array}\right\} \mathbf{C}= \left[\begin{array}{ccc}{\xi }_{1}^{2}& {\rho }_{12}{\xi }_{1}{\xi }_{2}& {\rho }_{13}{\xi }_{1}{\xi }_{3}\\ {\rho }_{12}{\xi }_{1}{\xi }_{2}& {\xi }_{2}^{2}& {\rho }_{23}{\xi }_{2}{\xi }_{3}\\ {\rho }_{13}{\xi }_{1}{\xi }_{3}& {\rho }_{23}{\xi }_{2}{\xi }_{3}& {\xi }_{3}^{2}\end{array}\right]$$
(6)

where μi is the mean of Xi, ξi is the standard deviation of Xi, and ρij is the Pearson’s correlation coefficient between Xi and Xj. In the case of standard normal variables with a zero mean and a standard deviation equal to 1, the covariance C matrix is simplified into:

$$\mathbf{C}=\mathbf{R}= \left[\begin{array}{ccc}1& {\rho }_{12}& {\rho }_{13}\\ {\rho }_{12}& 1& {\rho }_{23}\\ {\rho }_{13}& {\rho }_{23}& 1\end{array}\right]$$
(7)

where R refers to the correlation matrix. Hence, with the Gaussian copula, the multivariate probability distribution may be constructed using two easily definable elements: (1) pairwise Pearson’s correlation coefficients and (2) marginal probability distributions defined for the individual random variables. However, it should be noted that the Pearson’s correlation coefficient measures linear dependency. If the dependency between the variables is strongly nonlinear or asymmetric, a different copula model should be chosen as the selected model may have a considerable effect on the simulation results (Tang et al. 2015).

Gaussian copula was chosen because it allows a straightforward comparison between different correlation coefficient inputs, and various applications have demonstrated its adequacy in geotechnical problems. Nonetheless, it should be noted that more sophisticated methods to construct multivariate distributions have been suggested recently. For instance, the method proposed by Ching and Phoon (2020) incorporates not only cross-correlations but also spatial autocorrelation in soil properties.

4 Probabilistic Modeling of Compressibility

4.1 Site Descriptions

Two marine soft clays were selected as case studies: Haarajoki and Suurpelto clay. The basic geotechnical properties of the studied clay layers are presented in Table 1.

Table 1 Basic properties for the studied clays
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Haarajoki clay was investigated as a part of the Haarajoki test embankment constructed in 1997 by Finnish Road Administration near the city of Järvenpää in southern Finland. The Haarajoki clay deposit is approximately 20 m deep, and the clay layer below the dry crust is soft and moderately overconsolidated. The groundwater level is near the ground surface. Further details can be found from Yildiz et al. (2009), among others. The studied Haarajoki sampling profiles were located within a 20 m distance from each other. The homogeneous clay layer that was selected for this study was from depth z = 2.1–6.4 m. The depth profiles of the compressibility properties are shown in Fig. 2. The layer was later divided into two 2.1 m-thick calculation sublayers centered at depths z = 3.25 and 5.35 m (“Calc. depth” in Fig. 2).

Fig. 2
figure 2

Depth profiles of compressibility properties for Haarajoki clay

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A total of 39 oedometer tests were performed on the clay specimens from the studied layer: 13 incrementally loaded standard oedometer tests (“IL”), 15 constant-rate-of-strain oedometer tests (“CRS”), five IL tests, and six CRS tests performed on specimens rotated 90 degrees (“Horizontal IL” and “Horizontal CRS”). Figure 3 shows examples of the oedometer results, and although horizontal tests are marked with a somewhat smaller σp′, the overall stress–strain behavior seems to represent a homogeneous clay layer.

Fig. 3
figure 3

Examples of oedometer curves in the Haarajoki clay layer

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The CRS test results in Fig. 3 have not been rate-corrected. The apparent preconsolidation pressures σp′ defined from CRS were modified to correspond to the strain rate of the standard IL tests, as recommended by the Finnish Transport Agency (2012):

$${\sigma }_{p;calc}^{\mathrm{^{\prime}}}= \frac{{\sigma }_{p;test}^{\mathrm{^{\prime}}} }{k} ; k= {\left(\frac{{\dot{\varepsilon }}_{test}}{{\dot{\varepsilon }}_{calc}}\right)}^{B}$$
(8)

where \({\dot{\varepsilon }}_{test}\) is the strain rate in the CRS test, \({\dot{\varepsilon }}_{calc}\) is the target strain rate of the modification (10−7 1/s), and the recommended value for B is 0.0728. In addition, modulus number m1 was modified using Eq. (3) to correspond to the lower σp;calc′. The depth profiles (Fig. 2) show both unmodified and rate-corrected values for σp′ and m1. The OC parameters (m2, β2, Cs) were defined from the unloading or reloading curve of the oedometer tests. For some of the tests, β2 = 1 was assumed (i.e., linear stress–strain response). Cc was defined from the IL tests using the interpretation shown in Fig. 1a. From the CRS tests, Cc was calculated from the rate-corrected Janbu parameters using the conversion method suggested by Löfman and Korkiala-Tanttu (2019).

The Suurpelto clay basin is located in the city of Espoo in southern Finland, and it consists of clay, organic clay and clayey gyttja. The geological and geotechnical properties of Suurpelto basin have been reported by Ojala et al. (2007) and Pätsi (2009), among others. The three sampling points in this study were located within 400 m from each other, and due to the small variations in layering, the datasets could be combined (Löfman and Korkiala-Tanttu 2019). The Suurpelto depth profiles are shown in Fig. 4. The selected clay layer was from a 2–12 m depth, and most compressibility properties change linearly with depth. The layer was later divided into five 2 m-thick sublayers centered at calculation depths z = {3, 5, 7, 9, 11} m.

Fig. 4
figure 4

Depth profiles of compressibility properties for Suurpelto clay

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The Suurpelto dataset included 26 IL oedometer tests, four of which were accompanied by falling head permeability measurement. The oedometer curves were constructed using mainly the end-of-primary (EOP) stress–strain points estimated from time–settlement analysis, and the OC parameters were mostly interpreted from the recompression stage. Figure 5 presents the IL odometer curves, and it can be seen that the specimens with a lower initial void ratio (corresponding to the specimens from the deeper layer) form a separate group. However, as the depth-dependent spatial variability was considered in the analyses, the Suurpelto clay layer could be treated as one sample when deriving the soil statistics.

Fig. 5
figure 5

Oedometer curves in the Suurpelto clay layer

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4.2 Cases and Correlation Structures

Four analysis cases were considered to investigate how the applied correlation structure and properties of the marginal probability distributions affect the uncertainty in stress–strain response. Figure 6 summarizes the main differences between cases Correlated, Independent, Database, and Simplified. In the following sections, the correlation structures are first discussed, followed by a section where the definition of marginal distributions is described.

Fig. 6
figure 6

The considered cases in the Monte Carlo simulations. Note: in Suurpelto clay, the depth-dependent mean trends induced some correlation between parameters (despite ρ = 0)

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Cases Correlated and Independent were otherwise similar but were accompanied by either site-specific correlation coefficients or an assumption of independent parameters. In case Correlated, the Pearson’s correlation coefficients ρ were defined from site-specific Haarajoki and Suurpelto laboratory data using the formulae for the sample correlation coefficient. As illustrated in Figs. 12 and 17, the dependency between site-specific oedometer test data is linear, and hence copula model with Pearson’s ρ is appropriate (see also supplementary figures S1–S4). Table 2 presents the site-specific correlation coefficients for the Janbu parameters. For Haarajoki clay, the unmodified preconsolidation pressure σp′ and modulus number m1 from the CRS tests were used when deriving the correlation coefficients. The strength of cross-correlation may be evaluated by using the classification suggested by Evans (1996): there was a very strong correlation between σp′ and m1 at both sites (\({\rho }_{{m}_{1}{\sigma }_{p}{\prime}}\) ≈ − 0.88). This negative correlation was in accordance with the need to modify m1 using Eq. (3): a decrease in the value of σp′ used in calculation (compared to the value defined with oedometer test) should lead to an increase in m1. Strong to very strong correlation was also observed between m1 and β1 (\({\rho }_{{m}_{1}{\beta }_{1}}\) ≈ 0.73…0.86), which was expected because they were curve-fitting parameters describing the same stress–strain curve. Meanwhile, strong correlation was observed between σp′ and β1 (\({\rho }_{{\beta }_{1}{\sigma }_{p}{\prime}}\) ≈ − 0.64…− 0.73): an increase in σp′ led to a more negative β1 (i.e., a more concave shape of the stress–strain curve). The correlations related to the OC parameters were on average weaker for both Haarajoki clay and Suurpelto clay. Finally, Table 3 presents the correlation coefficients for the Cc parameters. For the Haarajoki clay CRS tests, the rate-corrected σp′ was used because Cc was calculated from the rate-corrected σp′ and m1. Strong to very strong correlation was observed between e0 and σp′ (\({\rho }_{{e}_{0}{\sigma }_{p}{\prime}}\) ≈ − 0.67…− 0.86), while there was a moderate correlation between e0 and Cc (\({\rho }_{{e}_{0}{C}_{c}}\) ≈ 0.5) and weak to very weak correlation between σp′ and Cc (\({\rho }_{{\sigma }_{p}{\prime}{C}_{c}}\) ≈ − 0.3…0.1).

Table 2 Cross-correlations between Janbu parameters
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Table 3 Cross-correlations between Cc parameters
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However, in typical geotechnical design situations, the site-specific data are too limited for the accurate estimation of the correlation coefficients or statistics like standard deviation. Hence, case Database was included to study how accurately the uncertainty in stress–strain response may be replicated by means of cross-correlations derived using a database and literature values for inherent variability. Database consisting of Finnish clays, FI-CLAY/14/856 (Löfman and Korkiala-Tanttu 2021a), was extended with additional sets of Janbu paramaters and then used to acquire estimates for the correlation coefficients for NC parameters. Cross-correlations related to OC parameters (m2, β2, Cs) were not considered since they were generally weaker and did not notably affect the simulation results. The FI-CLAY/14/856 database was first filtered to include only those oedometer tests that had negative β1, because the correlation between m1 and σp′ is related to cases with negative β1 specifically. The database analysis showed that the cross-correlations between Janbu parameters were stronger for clays with higher water content wn and lower values of m1. Hence, clays similar to the studied clay layers at Haarajoki and Suurpelto were identified by considering oedometer tests with m1 < 10; this limit is approximately double the average value in the studied clay layers. Likewise, specimens with wn > 75% were considered because 75% was the lowest water content observed in the studied layers. Figure 7 shows the correlation coefficients regarding both Janbu and Cc parameters for nine clay sites or areas extracted from the database together with Haarajoki and Suurpelto. The number of oedometer tests per database site was the greatest at Söderkulla-N. area (93) and the smallest at Murro test embankment site (10). Sites with less than 10 observations were not considered (as in e.g. Zhou et al. 2021). Figure 7 shows that the most consistent cross-correlations are related to variable pairs m1–σp′ and e0–Cc; other pairs are marked with greater differences between sites. In case Database, the average values for these database sites were used to define the correlation structure (see Tables 2 and 3). For instance, average \({\rho }_{{e}_{0}{C}_{c}}\) was 0.72 (Table 3, case Datab.) which is in accordance with the study by Zhou et al. (2021) where site-specific cross-correlations were summarized: they found a median value of 0.72 for the variable pair wn–Cc.

Fig. 7
figure 7

Site-specific cross-correlations between compressibility properties of various soft clays

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In case Simplified, the marginal distributions were mostly identical with case Database, but the parameters were assumed to be independent. In addition, there was one less random variable. In the Cc method, e0 was treated as a deterministic constant. In the Janbu method, m1 was not defined via marginal distribution but was instead defined through m1 modification (Eq. (3)) to tie m1 to σp′. As there was no single laboratory test value, the average rate-corrected σp′ was used as σp;test′, and the average rate-corrected m1 was used as m1;test. For β1 and σp;calc′, the simulated values were used.

4.3 Marginal Probability Distributions

The marginal probability distributions of compressibility parameters were defined to represent uncertainty in observed variability, which is the sum of inherent variability and measurement error (Orchant et al. 1988; Phoon and Kulhawy 1999; Müller et al. 2014). Statistical uncertainty was ignored for the sake of simplicity because the number of site-specific observations was relatively high, in the scale of n = 26–39. Transformation uncertainty, which is related to an indirect evaluation of a soil property (e.g., using an empirical correlation to estimate Cc from water content), was likewise ignored, since the compressibility properties were based on oedometer test data. The uncertainty related to the conversion of Janbu parameters into Cc (which was applied to Haarajoki CRS data) is relatively small and was thus considered to be included in the observed variability.

In Cases Correlated and Independent, the marginal distributions were fitted to site-specific observations. For the NC parameters in Haarajoki clay, the most suitable probability distribution was found using the Kolmogorov–Smirnov test, and was then fitted using the maximum likelihood method within SciPy Python library (scipy.org). Since the objective was to find PDFs that most accurately replicate the observed site-specific variability, also non-standard distribution types were used (see Table 4 and Fig. 12): Nakagami distribution (for σp′) is related to gamma distribution, Johnson SB (“system bounded”) distribution is for bounded data and hence was able to replicate the unique variability in β1, hyperbolic secant distribution (e0) resembles the standard normal distribution but has heavier tails while folded normal distribution (Cc) allows to cut off certain small values and to add that probability mass to the other side.

Table 4 Statistics and distributions for the compressibility properties
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For the OC parameters in Haarajoki clay, either a normal or a lognormal distribution was fitted to the data using the method of moments (i.e., fitted using the arithmetic mean and sample standard deviation). The distribution types and soil statistics are listed in Table 4. The 2nd and 98th percentiles for the PDFs used in cases Correlated and Independent are illustrated in Fig. 2.

In Suurpelto clay, the mean trend for the compressibility parameters was defined using ordinary least squares linear regression (shown in Fig. 4), and the standard deviation was estimated as recommended by Lacasse et al. (2007):

$${SD}_{de-trended}=\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}{\left({y}_{i}-({a}_{o}+{a}_{1}{z}_{i})\right)}^{2}}$$
(9)

where yi is the actual property value at depth zi, a0 and ai are the coefficients of the linear trend function, and n is the number of observations. The resulting constant SDde-trended among the trends is illustrated in Fig. 4, where the 2nd and 98th percentiles of the assumed normal distribution are shown (corresponding to mean ± 2.054 × SDde-trended). The soil statistics, including the trend coefficients, can be found in Table 4. At each calculation depth, the mean was calculated from the trend function, and the normal distribution was constructed from the resulting mean and SDde-trended. It is noteworthy that in regression analysis, it is also possible to model the uncertainty in the position of the trend line, which is the smallest near the gravitational center of the data (Ang and Tang 2007; Tang 1980). However, this study was limited to models of stationary variability, i.e., constant SD with depth (see e.g. Li et al. 2015) for the sake of simplicity and more straightforward comparison between the Haarajoki and Suurpelto sites.

Meanwhile, in Suurpelto clay the Janbu OC parameters m2 and β2 did not have a clear trend with depth. Hence, the PDFs were fitted simply using the method of moments. Lognormal distribution was used for m2 due to the notable positive skew.

In cases Database and Simplified, the marginal distributions were constructed using site-specific means and literature values for observed variability, defined as coefficient of variation COV. The standard deviation was defined as the mean multiplied by the literature COV. For Suurpelto clay, the trend value at the middle of the whole clay layers was used as the mean. As for the literature value, COV = 0.20 was selected, which corresponds to the average observed COV for the compressibility properties like (e.g., Cc and σp′) of Finnish clays (Löfman and Korkiala-Tanttu 2019, 2021b). The COV for compressibility generally varied from 0.10 to 0.30, although it should be noted that the study did not consider Janbu parameters. Nonetheless, the observed range corresponds to the approximate guideline of 0.10–0.37 for Cc and σp′ given by Uzielli et al. (2006). For void ratio e0, average observed COV = 0.06 was used (Löfman and Korkiala-Tanttu 2021b). Either a lognormal or normal distribution was used for the compressibility parameters, as suggested by Uzielli et al. (2006) and Löfman and Korkiala-Tanttu (2021b), and the selection between the two was based on rough visual assessment. For example, lognormal distribution was selected for σp′ in Haarajoki clay because the histogram (see Fig. 12a) was marked with a positive skew.

4.4 Monte Carlo Simulation

The applied Monte Carlo simulation code written in Python had the following steps. First, random samples where simulated from a multivariate standard normal distribution defined using Eq. (5). This multivariate PDF was defined using the correlation matrix R with the desired correlation coefficients between the compressibility parameters (Tables 2 and 3). Next, the samples from the multivariate normal distribution were transformed to produce uniform marginal distributions using the probability integral transform defined by Eq. (4). Finally, the uniform marginal distributions were transformed into the desired marginal distributions (defined in Table 4) using the inverse CDF of that specific distribution while preserving the intended correlation structure.

The number of simulation runs was 105 for each calculation depth (hence, 2 × 105 in all for Haarajoki clay and 5 × 105 for Suurpelto clay). One modification was applied to the simulated sets of compressibility parameters: if σp′ was smaller than effective in-situ stress σv0′, the clay was assumed to be normally consolidated and σp′ was replaced with σv0′.

The clay layers were divided into approximately 2 m-thick calculation sublayers, centered at calculation depths, to evaluate the settlements in the studied layers. It was conservatively assumed that no variance reduction occurs due to averaging effect because the averaging length L (i.e. thickness of calculation sublayer) was estimated to be roughly equal to the scale of fluctuation SOF (Vanmarcke 1983): L was approximately 2 m while the vertical SOF values for soft, sensitive or marine clays are in the scale of 1.6–1.7 m (Cami et al. 2020). Strain in each sublayer was computed by assuming constant added stresses Δσv. The values for Δσv were selected based on the degree of overconsolidation described via pre-overburden pressure POP (equal to σp′ minus σvo′). The first added load Δσv was 1.5 × POP, the second was 3 × POP, and the third was 6 × POP. For Suurpelto clay, the POP at the middle of the whole clay layer was used.

5 Results and Discussion

5.1 Haarajoki Clay

Using the simulated sets of Janbu or Cc parameters, the corresponding oedometer curves were constructed starting from in-situ stress σv0′ at calculation depth (where the strain was zero) and extending until 600 kPa using Eqs. (1)–(2). Figures 8 and 9 visualize the first 2000 simulation runs of the total 105 in each calculation depth together with the corresponding oedometer curves constructed using site-specific laboratory data. Some of the Haarajoki oedometer tests lacked OC parameters, and the mean values calculated from the rest were used. Figure 9, which shows the Cc method simulations, presents two sets of laboratory-based curves: those based on the fitted Janbu parameters and those based on the fitted Cc parameters. It should be noted that Cs leads to a slightly greater OC strain compared to Janbu (m2, β2) because Cs was interpreted as a linear line fitted to all unloading or reloading points instead of stress points between σv0′ and σp′ (see Fig. 1a).

Fig. 8
figure 8

The Janbu method for Haarajoki clay: oedometer curves in cases a Correlated, b Independent, c Database, and d Simplified

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Fig. 9
figure 9

The Cc method for Haarajoki clay: oedometer curves in cases a Correlated, b Independent, c Database, and d Simplified

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The simulated settlements are presented as boxplots for the Janbu and Cc methods in Figs. 10 and 11, respectively. The box boundaries represent the interquartile range (i.e., from the 25th to 75th percentiles), and the solid horizontal line is the median. The dashed line is the arithmetic mean. The whiskers represent the 2nd and 98th percentiles. The underlying simulated sets of compressibility parameters are illustrated as histograms and pairwise scatter plots in the Supplementary Information: simulated data for Haarajoki clay are collected to Figs. S1 and S2 for Janbu method and Cc method, respectively.

Fig. 10
figure 10

The Janbu method for Haarajoki clay: settlement boxplots for each case

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Fig. 11
figure 11

The Cc method for Haarajoki clay: settlement boxplots for each case

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Figure 8 shows that the Janbu method applied to Haarajoki clay was very sensitive to changes in the correlation structure. Case Independent was marked with much greater scatter about the laboratory curves compared to the other cases, where the dependency between the parameters was somehow taken into account. In case Independent, the simulated curves indicating larger strains were due to the extrapolation of the oedometer curve linked to the smaller σp′, as discussed earlier. Accordingly, the boxplots (Fig. 10) show a wide interquartile range with a positive skew caused by high settlement values. The 98th percentile of the settlement was up to 1 m greater than that of case Correlated. On the other hand, the mean settlement was approximately equal in all the cases.

Cases Database and Correlated produced quite similar results for the Janbu method. Figure 12 compares the simulated values of the NC parameters (m1, β1, σp′) in cases Correlated and Database, and it can be seen that the dependencies between the parameters were quite similar. However, the assumption of normal distribution for β1 in case Database fit the site-specific data quite poorly.

Fig. 12
figure 12

Cases Correlated and Database in Haarajoki clay: a σp′ versus m1; b m1 versus β1; and c β1 versus σp′

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For the Janbu method, case Simplified was quite similar to case Correlated at low added stress Δσv (Figs. 8 and 10), but the scatter decreased at higher stresses. In fact, Fig. 8 shows that at σv′ > 200 kPa, the simulated curves had less scatter than the laboratory curves. Accordingly, the Simplified boxplot in Fig. 10 corresponding to Δσv = 200 kPa was marked with unrealistically low uncertainty, because in this case m1 was calculated via Eq. (3) with constant (average rate-corrected) values for σp;test′ and m1;test.

Indeed, it was observed from the scatter plots (Fig. S1) that even though case Simplified adequately reproduced the correlation structure, it also excessively restricted the variability in m1.

Finally, Figs. 9 and 11 show that the Cc method applied to Haarajoki clay behaved rather similarly in all the cases despite the assumed correlation structure. The correlation between the Cc parameters was quite weak in the site-specific data, so considerable differences would not be expected either. Nonetheless, case Correlated was marked with the least scatter in the oedometer curves (Fig. 9), and the lowest uncertainty in settlement (Fig. 11). However, it should be noted that compared to the Janbu method, the Cc method overestimated the settlements at the highest added stress due to the nonlinearity of the oedometer curves. Nevertheless, in the lower added stresses, the Cc and Janbu methods had similar mean settlements.

Lastly, it should be noted that treating initial void ratio e0 as a deterministic constant in case Simplified did not affect the simulations much. In fact, the related uncertainty was generally the greatest in case Simplified, even though one uncertain variable was omitted. This intuitively contradictory result may be related to an issue discussed by Wesley (1988): even though the correlation between e0 and Cc is typically positive, meaning that a higher e0 implies larger compressibility, the resulting compression from Eq. (1) actually decreases if a smaller e0 is used (while Cc remains the same). Hence, in case Independent, the variability in e0 might inhibit the scatter in the resulting strain. Therefore, case Simplified might exhibit larger scatter partly because this inhibition effect had been removed.

5.2 Suurpelto Clay

Figure 13 shows the oedometer curve simulations using the Janbu method for Suurpelto clay, and Fig. 14 illustrates the Cc method simulations. The corresponding settlements are shown as boxplots in Figs. 15 and 16. The underlying simulated sets of compressibility parameters are illustrated as scatter plots in Figs. S3 and S4 in Supplementary Information for Janbu method and Cc method, respectively.

Fig. 13
figure 13

The Janbu method for Suurpelto clay: oedometer curves in cases a Correlated, b Independent, c Database, and d Simplified

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Fig. 14
figure 14

The Cc method for Suurpelto clay: oedometer curves in cases a Correlated, b Independent, c Database, and d Simplified

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Fig. 15
figure 15

The Janbu method for Suurpelto clay: settlement boxplots for each case

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Fig. 16
figure 16

The Cc method for Suurpelto clay: settlement boxplots for each case

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In Suurpelto clay, there were smaller differences between cases Correlated and Independent compared to Haarajoki clay. There are two main reasons for this. First, the β1 values in Suurpelto clay were generally closer to zero (less negative) than those in Haarajoki clay, meaning the possible errors related to the extrapolation of the oedometer curve are less severe (Länsivaara 2003). Accordingly, case Simplified behaved more consistently with increasing stress because the modification applied to m1 was less significant when β1 was less negative. Second, as most of the soil parameters in Suurpelto clay had a trend with depth, some correlation between the parameters was produced in case Independent also as a result of updating the mean values at the given calculation depths (see Fig. S3). Nonetheless, at the highest added stress, the interquartile range of settlement was almost two times wider in case Independent than in Correlated (Fig. 15).

Figure 13 shows that case Database was marked with greater scatter compared to Correlated. Figure 17 compares the simulated Janbu NC parameters in these two cases, and it can be seen that the assumed COV = 0.20 for m1 resulted in greater variability compared to the site-specific data. Indeed, Table 4 shows that the site-specific COV was smaller for both m1 and σp′. On the other hand, the site-specific variability in β1 was somewhat greater compared to the assumption of COV = 0.20. There was also a notable difference in correlation structure between β1 and σp′, which may be another explanation for the more significant scatter in case Database compared to Correlated; a high σp′ should lead to a more negative β1 (which would result in an oedometer curve with a more concave shape). It should be noted that in Fig. 17, the simulations for five calculation depths are combined, and the shown site-specific data were not de-trended. In addition, the blunt lines in the σp′ simulations resulted from assuming that clay is normally consolidated when the simulated σp′ is smaller than σv0′. All in all, due to the higher assumed uncertainty in most of the NC parameters in Suurpelto clay, the settlement boxplots for cases Database and Simplified Fig. 15 were marked with larger uncertainty compared to those for case Independent.

Fig. 17
figure 17

Cases Correlated and Database in Suurpelto clay: a σp′ versus m1; b m1 versus β1; and c β1 versus σp′

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Figures 14 and 16 show that the Cc method also exhibited very little difference between the cases in Suurpelto clay. The settlement boxplots Fig. 16 were very similar to each other, and the differences were partly caused by the different assumptions of soil variability rather than by a varied correlation structure. Figure 14 also illustrates how the Janbu- and Cc-based laboratory curves do not diverge until higher stresses are reached because β1 was generally close to zero (i.e., assumption of linear stress–strain behavior in the semi-logarithmic space was applicable in wide stress range). Accordingly, the settlement boxplots for the Janbu method Fig. 15 and the Cc method Fig. 16 had similar mean values despite the difference in settlement method.

6 Summary and Conclusions

In this study, the Gaussian copula was applied to the probabilistic modeling of clay compressibility using the Cc and Janbu settlement calculation methods. For soft marine clays, the Janbu method is often preferred because it allows a nonlinear stress–strain response in a semi-logarithmic space. A Gaussian copula was constructed using the site-specific correlation coefficients from the two clay layers and the correlation structure derived using a regional database. Marginal probability distributions were defined by both fitting to site-specific data and by means of literature values for inherent variability. Monte Carlo simulations were run to acquire sets of compressibility parameters, which were then used to construct a simulated group of oedometer curves. The settlements in the studied clay layers were then computed for three cases of added external load.

The results illustrated that the Janbu method leads to a considerable overestimation of uncertainty in settlement if the correlation between the parameters (i.e., modulus number m1, preconsolidation pressure σp′, stress exponent β1) is ignored. The best agreement between the simulated and laboratory oedometer curves was acquired via the site-specific correlation structure and soil statistics. Nonetheless, the correlation coefficients defined from the database and literature values for inherent variability also provided quite reasonable results. In addition, by applying a modification of m1, some correlation between m1 and σp′ could be introduced, and the simulation result was adequate despite the assumption of independent uncertain variables. Nevertheless, it is evident that a large amount of laboratory data enabling site-specific probabilistic characterization allows the most accurate modeling.

On the other hand, the Cc method was found to be rather insensitive to the assumed correlation structure. The site-specific correlation coefficients and soil statistics led to the smallest scatter in the simulated oedometer curves, but the difference was relatively small compared to an assumption of independent variables. Hence, the study results indicate that when using the Cc method, ignoring the cross-correlation between the soil parameters does not lead to significant errors.

To conclude, Cc method without copula (i.e., ignoring the cross-correlation) may be used in preliminary assessments and also in cases with clays characterized by linear stress–strain behavior in semi-logarithmic space (i.e., stress exponent β1 ≈ 0 meaning that Cc method can replicate the oedometer curve). Similarly, if Cc is interpreted from oedometer test data using the problem-relevant stress range, the risk of overestimating mean settlement is reduced. However, for increased model accuracy in soft clays, Janbu model with copula should be used. In such model, correlation coefficients defined via regional database provide a reasonable estimation, but accuracy can be further increased by collecting more site-specific data to be able to define site-specific correlation structure and marginal distributions.