The paper presents the results of laboratory tests of plastic limit _{P}_{L}_{om} and calcium carbonate content CaCO_{3}. Comparison of the liquid limit _{L}_{LC}_{L}_{60} and 30° _{L}_{30} is shown. Based on statistical analysis of the test results, single- and two-factor empirical relationships for evaluating the plastic limit _{P}_{L}_{om} and/or calcium carbonate content CaCO_{3} are presented in this study.

#### Keywords

- plastic limit
- liquid limit
- Eemian gyttja
- Casagrande cup
- cone penetrometer
- statistical analysis

In engineering practice, Holocene organic soils are considered to represent difficult geotechnical conditions for structure foundation due to their high compressibility with creep effects, low undrained shear strength, significant changes in permeability with porosity changes, and nonlinear variability of material characteristics with spatial variability [22,24,44,45]. Organic soils formed during the Eemian Interglacial of the Pleistocene reveal slightly better index properties and higher stiffness and strength than Holocene organic soils [23,25]. In the past, Eemian organic soils were overloaded and subjected to long-term creeping; therefore, they have the behavior of preconsolidated soils [31]. Eemian gyttja is an example of such organic soils. However, the composition of the Eemian gyttja skeleton displays significant variability, especially regarding the organic matter content _{om} and the calcium carbonate content CaCO_{3}, which considerably affects the physical and mechanical properties. It is, therefore, necessary to take into account the nature of the geotechnical properties in procedures and interpretation of field and laboratory testing and calculation methods for geotechnical design. Currently, the physical and mechanical properties of Eemian gyttja and its behavior under complex stress conditions are being investigated, as well as work on elaborating design methods for structure foundation on the subsoil with the Eemian gyttja is conducted [15,23].

In addition to the basic properties of organic soils determined in engineering practice, such as bulk density _{s}_{n}_{P}_{L}_{om} and the calcium carbonate content CaCO_{3} [9,31]. Currently, the liquid limit _{L}

The experiments carried out by Wasti [42] on natural cohesive soils from various locations in Turkey have shown that the liquid limits determined by the Casagrande and the cone methods were in good agreement for liquid limit values up to about 100%. Research conducted by Di Matteo [6] on natural cohesive soils characterized by a liquid limit _{L}_{L}_{2} solutions, Mishra et al. [30] received comparable liquid limit values for values of _{L}

Existing reports supply empirical relationships between the liquid limit _{L}

Relationships between the fall cone liquid limit and the Casagrande liquid limit for cohesive soils in the literature.

Linear relationships | ||||

85%–200% | 60°–60 g | Danish Eocene clays | Grønbech et al. 2011 [16] | |

13%–117% | 60°–60 g | Fine-grained soils | Matusiewicz et al. 2016 [28] | |

30%–390% | 60°–60 g | Fine-grained soils, kaolin–bentonite mixtures | Mendoza and Orozco 2001 [29] | |

30%–350% | 30°–80 g | |||

<150% | 30°–80 g/100 g60°–60 g | Fine-grained soils | Shimobe 2010 [36] | |

27%–110% | 30°–80 g | Turkish natural soils | Wasti 1987 [42] | |

80%–150% | 30°–80 g | Soil–bentonite mixtures | Mishra et al. 2012 [30] | |

13%–117% | 30°–80 g | Fine-grained soils | Matusiewicz et al. 2016 [28] | |

Power relationships | ||||

>100% | 30°–80 g | Natural clays | Schmitz et al. 2004 [34] | |

Up to approx. 600% | 30°–80 g | Fine-grained soils | O’Kelly et al. 2018 [27] | |

<120% | ||||

Up to approx. 600% | ||||

<120% |

Note: _{L(FC)}_{L30}, liquid limit using 30°–80 g fall cone; _{L60,} liquid limit using 60°–60 g fall cone; _{LC}_{L, BS cup}, BS Casagrande cup liquid limit; _{L, ASTM cup}, ASTM Casagrande cup liquid limit; ^{2}, determination coefficient;

Relationships between the Atterberg limits and the clay and organic matter contents in the literature.

Fine-grained soils with organic content below 6% | De Jong et al. 1990 [5] | |

Holocene gyttja _{om} = 0.6%–73.1% CaCO_{3} = 2.0%–88.4% | Długaszek 1991 [8] | |

Note: _{LC}_{P}_{om}, organic matter content in %; ^{2}, determination coefficient;

In Poland, as in other European Union countries, geotechnical design according to EN 1997-1 [12] has been in force since 2010. According to EN 1997-2 [13], the cone penetrometer method is preferred for determination of the liquid limit _{L}_{L}_{L}

The aim of this work was to analyze the results of comparative studies of the plastic limit _{P}_{L}_{om} and calcium carbonate content CaCO_{3}. A comparison of the liquid limit _{L}_{LC}_{L}_{60} and 30° _{L}_{30} is presented. In addition, analysis of the test results allowed to develop single- and two-factor relationships of the plastic limit _{P}_{L}_{om} and/or the calcium carbonate content CaCO_{3}.

The studied organic soil was gyttja from the Eemian Interglacial of the Pleistocene, collected from the Żoliborz channel – one of the parts of Warsaw with very complex geotechnical conditions. The Żoliborz channel is located in the western part of Warsaw and currently extensively developed (metro station and tunnels, residential and office buildings with two- or three-floor basements). The channel is about 12 km long and nearly 800 m wide in its central part. In the Żoliborz channel, organic soils, that is, organic mud and gyttja, reach thicknesses up to 10 m. The first subsurface layer in the tested subsoil is formed by fills with thicknesses varying between 0.5 and 4.0 m. The fills are underlain by sand and mud deposits of the Vistulian glaciation to a depth of approximately 4–6 m below the ground level. Sand and mud layers cover a continuous layer of gyttja and organic mud from the Eemian Interglacial. The top of this layer was found to be at a depth of approximately 6 m with the bottom reaching down to 16 m below the ground level. Organic soils of the Eemian Interglacial are overconsolidated, with an overconsolidation ratio (OCR) varying in the range of 2.0 and 3.5. The grain size composition of the mineral part of gyttja points to silts without both the fine silt and clay fractions. The bottom of the channel is filled with moraine deposits from the Odranian Glaciation, represented mainly by sandy clays, followed by sand deposits of the Mazovian Interglacial, represented by dense fine, medium, and silty sands. Free ground water occurs in the sand layer from the Vistulian Glaciation at a depth of about 3 m. In the sand layer from the Mazovian Interglacial at a depth of 20–21 m, the water pressure is artesian, reaching up to 5 m below the ground level.

Samples of Eemian gyttja for laboratory tests were taken as block samples during deep excavations made for the construction of Płocka station of the II metro line and residential buildings along the Skierniewicka Str. in Warsaw. The collected samples were used to study deformation, creep, and strength characteristics and parameters of Eemian gyttja. Laboratory tests included oedometer tests, triaxial tests, and torsional shear hollow cylinder tests [15,25]. The following physical properties were determined in the tested samples: bulk density _{n}_{P}_{L}_{om}, calcium carbonate content CaCO_{3}, specific density _{s}

The liquid limit _{L}_{L}

The organic matter content _{om} was determined by combustion at a temperature of +440°C. The calcium carbonate content CaCO_{3} was determined by the gasometer method [44]. The results of index properties of the 16 tested samples of Eemian gyttja are shown in Table 3. The tested gyttja had an organic matter content _{om} at 7%–24% and calcium carbonate content CaCO_{3} at 30%–82%. The liquid limit _{LC}_{P}

Laboratory test results of the index properties of Eemian gyttja.

_{n} (%) | _{p} (%) | _{L}(%) | _{3} (%) | _{om} (%) | ||||
---|---|---|---|---|---|---|---|---|

_{LC} | _{L60} | _{L30} | ||||||

1 | Gyttja ( | 62.3 | 50.9 | 81.0 | 76.7 | 81.5 | 29.6 | 7.44 |

2 | 67.8 | 62.4 | 88.0 | 86.4 | 87.2 | 31.7 | 9.41 | |

3 | 61.3 | 60.7 | 80.9 | 75.1 | 78.1 | 34.9 | 7.69 | |

4 | 58.5 | 56.6 | 82.3 | 81.5 | 85.5 | 37.9 | 7.92 | |

5 | Gyttja ( | 74.4 | 68.0 | 104.5 | 101.5 | 105.5 | 31.1 | 12.0 |

6 | Gyttja ( | 102.1 | 119.2 | 150.4 | 148.5 | 163.6 | 54.7 | 17.8 |

7 | 98.7 | 122.2 | 136.1 | 135.5 | 137.5 | 60.9 | 18.6 | |

8 | 98.9 | 100.8 | 140.0 | 137.1 | 143.6 | 63.8 | 18.1 | |

9 | 110.1 | 116.8 | 156.2 | 156.8 | 159.0 | 66.7 | 18.4 | |

10 | 115.6 | 130.7 | 152.5 | 154.8 | 160.1 | 70.4 | 23.3 | |

11 | 87.1 | 130.9 | 159.2 | 166.1 | 171.0 | 77.7 | 20.6 | |

12 | 100.3 | 125.9 | 155.2 | 159.5 | 162.0 | 74.0 | 20.2 | |

13 | 97.7 | 97.7 | 121.3 | 125.4 | 130.6 | 65.4 | 20.7 | |

14 | 118.5 | 110.5 | 164.5 | 171.6 | 173.8 | 73.6 | 23.8 | |

15 | Marl ( | 90.6 | 114.3 | 139.1 | 131.6 | 140.1 | 81.0 | 18.1 |

16 | 79.9 | 110.1 | 131.0 | 130.8 | 133.4 | 82.1 | 16.2 |

Note:

low calcareous mineral gyttja;

high organic lacustrine marl;

high calcareous mineral-organic gyttja;

low calcareous mineral-organic gyttja.

Statistical analysis of the test results was carried out using Statistica software version 12 [37,38,39]. Comparison of the liquid limits of the studied Eemian gyttja determined by the Casagrande method _{LC}_{L}_{60} and 30° _{L}_{30} was carried out using the significance of average differences nonparametric Kruskal–Wallis test (as a nonparametric equivalent of variance analysis) [38]. The null hypothesis that the differences between the average liquid limit _{L}_{L}

Regression analysis was performed and single-factor models of linear or nonlinear regression equations were obtained, expressed by the formulas [11,38]:

A multiple linear regression analysis was carried out, with _{om} and CaCO_{3} taken as independent variables, and two-factor linear regression models were obtained, expressed by the formula [11, 39]:
_{0}, _{1}, and _{2} are the empirical coefficients.

One of the assumptions of regression analysis is the absence of collinearity of two explanatory variables (weak correlations with each other). The most common collinearity is estimated by two parameters: tolerance and variance inflation factor (VIF). The smaller the tolerance for an explanatory variable, the more redundant is its contribution to the regression equation. The variable is unnecessary when the tolerance is less than 0.1. In the case when VIF = 1, there is no collinearity of variables, and when VIF > 10, collinearity has a disturbing effect on the parameters of the regression model [39].

In order to assess the quality of prediction by means of regression equations, the determination coefficient (^{2}), relative error (RE) of the cases, and standard error of estimation (SEE), expressed by the following formulas, were used:
_{i}

Comparison of the liquid limits of the studied Eemian gyttja determined by the Casagrande method _{LC}_{L}_{60} and 30° _{L}_{30} is shown in Figure 3.

Figure 3 shows that in the tested range, the average values and standard deviations of the liquid limit determined by individual methods are similar to each other. The calculated values are: _{LC}_{L}_{60} = 127.43 ± 33.17, _{L}_{30} = 132.03 ± 33.91, where the average value of _{L}_{30} is slightly higher than the average values of _{LC}_{L}_{60} (by about 3%) and the standard deviation of the _{LC}

Table 4 shows single-factor regression relationships of the liquidity limit _{L}_{LC}_{L}_{60}, _{L}_{30}), with the Casagrande method being considered as the reference one. Reliable conversion formulas (25)–(27) shown in Table 4 were obtained. Their high accuracy of about max RE = 5%–7% indicates that these methods can be used interchangeably, and the results can be calculated using the proposed formulas.

Linear and power regression models of relationships between the liquid limit _{L}

^{2} (−) | ||||
---|---|---|---|---|

0.989 | 16 | 3.62 | ±5 | |

0.990 | 16 | 3.58 | ±7 | |

0.990 | 16 | 3.41 | ±5 |

Note: RE, relative error; SEE, standard error of estimation.

Single-factor regression relationships (25) and (26) are shown in Figure 4. The dispersion of the points of both studied relationships is clearly arranged along straight lines. The relationship (25) does not differ much from the line of equality, which indicates that the cone 60° method is almost equivalent to the Casagrande method. The relationship (26) coincides with the line of equality in the _{LC}_{LC}

A comparison of the relationships _{L}_{60} = f(w_{LC}) and _{L}_{30} = f(_{LC}

Figure 5 shows that for the relationships _{L}_{60} = f(_{LC}_{L}_{30} = f(_{LC}

_{P}

_{LC}

_{om}and CaCO

_{3}

Based on the calculated matrix of linear correlation coefficients according to Stanisz [38], it was found that the liquid limit _{L}_{P}_{om} and the calcium carbonate content CaCO_{3}. Higher _{om} or CaCO_{3} values result in higher liquid limit _{L}_{om} and CaCO_{3} taken as independent variables, and two-factor linear regression models were obtained, expressed by the formula (21).

The simple and multiple linear regression relationships of _{P}_{LC}^{2}, SEE, and maximum RE are given in Table 5.

Single- and two-factor linear regression models of the plastic limit (_{P}) and liquid limit (_{L}_{C}) relationship versus the organic matter content (_{om}) and/or calcium carbonate content (CaCO_{3}) relationship for Eemian gyttja.

^{2} (−) | |||
---|---|---|---|

0.833 | 12.15 | ±17 | |

0.786 | 13.75 | ±20 | |

0.876 | 11.13 | ±20 | |

0.731 | 16.39 | ±20 | |

Note: RE, relative error; SEE, standard error of estimation.

For two-factor models, statistical indicators were checked to detect the redundancy of the explanatory variables introduced in the models: the tolerance of CaCO_{3} is 0.267 and the VIF = 3.74, which allows to conclude that the collinearity of _{om} and CaCO_{3} variables is not disturbing and both independent variables can enter the model.

Based on Equations (28)–(33) in Table 5, it can be stated that in the case of the studied Eemian gyttja, there are positive correlations of the Atterberg limits _{P}_{L}_{om} and CaCO_{3} (positive equation coefficients for the variables _{om} and CaCO_{3}), which means that _{P}_{L}_{om} and CaCO_{3}. The _{P}_{L}_{om} content than CaCO_{3}.

Using single-factor linear regression models, _{P}_{om} or CaCO_{3} contents with a lower accuracy of around 17% and 20%, respectively (Table 5), than with the two-factor linear regression model with an accuracy of around 16% (Figure 6a).

Using single-factor linear regression models, _{LC}_{om} or CaCO_{3} with a lower accuracy of about 20% (Table 5) than using the two-factor linear regression model with an accuracy of around 15% (Figure 6b).

A comparison of the relationships _{P}_{om}) and _{LC}_{om}) for Eemian gyttja obtained by the authors (presented in Table 5) with Długaszek relationships taken from the literature for Holocene gyttja (presented in Table 2) is shown in Figure 7. Figure 7 shows a significant difference between the test results obtained for Eemian gyttja and the relationships obtained by Długaszek for Holocene gyttja.

The following conclusions can be drawn based on the statistical analysis of test results of Eemian gyttja with the organic matter content _{om} = 7.44%–23.8% and the calcium carbonate content CaCO_{3} = 29.6%–82.1%:

The results of the determination of the liquid limit _{L}_{LC}_{L}_{60}, and the cone penetrometer with an apex angle of 30° _{L}_{30} were compared. It is concluded that in the examined range of results, the three analyzed liquid limit testing methods can be used interchangeably for the material studied because the differences among the results are very small. Formulas allowing for conversion of the liquid limit _{LC}

For liquid limit _{LC}_{L}_{60} = f(_{LC}_{L}_{30} = f(_{LC}_{LC}

The plastic limit _{P}_{om} and the calcium carbonate content CaCO_{3}. The developed two-factor linear regression model allows for assessing the plastic limit _{P}_{om} and CaCO_{3} with a maximum RE of ±16% for the material studied.

The liquid limit _{L}_{LC}_{om} and CaCO_{3} with a maximum RE of ±15% for the material studied.

#### Relationships between the Atterberg limits and the clay and organic matter contents in the literature.

Fine-grained soils with organic content below 6% | De Jong et al. 1990 [ | |

Holocene gyttja _{om} = 0.6%–73.1% CaCO_{3} = 2.0%–88.4% | Długaszek 1991 [ | |

#### Single- and two-factor linear regression models of the plastic limit (wP) and liquid limit (wLC) relationship versus the organic matter content (Iom) and/or calcium carbonate content (CaCO3) relationship for Eemian gyttja.

^{2} (−) | |||
---|---|---|---|

0.833 | 12.15 | ±17 | |

0.786 | 13.75 | ±20 | |

0.876 | 11.13 | ±20 | |

0.731 | 16.39 | ±20 | |

#### Laboratory test results of the index properties of Eemian gyttja.

_{n} (%) | _{p} (%) | _{L}(%) | _{3} (%) | _{om} (%) | ||||
---|---|---|---|---|---|---|---|---|

_{LC} | _{L60} | _{L30} | ||||||

1 | Gyttja ( | 62.3 | 50.9 | 81.0 | 76.7 | 81.5 | 29.6 | 7.44 |

2 | 67.8 | 62.4 | 88.0 | 86.4 | 87.2 | 31.7 | 9.41 | |

3 | 61.3 | 60.7 | 80.9 | 75.1 | 78.1 | 34.9 | 7.69 | |

4 | 58.5 | 56.6 | 82.3 | 81.5 | 85.5 | 37.9 | 7.92 | |

5 | Gyttja ( | 74.4 | 68.0 | 104.5 | 101.5 | 105.5 | 31.1 | 12.0 |

6 | Gyttja ( | 102.1 | 119.2 | 150.4 | 148.5 | 163.6 | 54.7 | 17.8 |

7 | 98.7 | 122.2 | 136.1 | 135.5 | 137.5 | 60.9 | 18.6 | |

8 | 98.9 | 100.8 | 140.0 | 137.1 | 143.6 | 63.8 | 18.1 | |

9 | 110.1 | 116.8 | 156.2 | 156.8 | 159.0 | 66.7 | 18.4 | |

10 | 115.6 | 130.7 | 152.5 | 154.8 | 160.1 | 70.4 | 23.3 | |

11 | 87.1 | 130.9 | 159.2 | 166.1 | 171.0 | 77.7 | 20.6 | |

12 | 100.3 | 125.9 | 155.2 | 159.5 | 162.0 | 74.0 | 20.2 | |

13 | 97.7 | 97.7 | 121.3 | 125.4 | 130.6 | 65.4 | 20.7 | |

14 | 118.5 | 110.5 | 164.5 | 171.6 | 173.8 | 73.6 | 23.8 | |

15 | Marl ( | 90.6 | 114.3 | 139.1 | 131.6 | 140.1 | 81.0 | 18.1 |

16 | 79.9 | 110.1 | 131.0 | 130.8 | 133.4 | 82.1 | 16.2 |

#### Linear and power regression models of relationships between the liquid limit wL determined by Casagrande method and fall cone methods for Eemian gyttja.

^{2} (−) | ||||
---|---|---|---|---|

0.989 | 16 | 3.62 | ±5 | |

0.990 | 16 | 3.58 | ±7 | |

0.990 | 16 | 3.41 | ±5 |

#### Relationships between the fall cone liquid limit and the Casagrande liquid limit for cohesive soils in the literature.

Linear relationships | ||||

85%–200% | 60°–60 g | Danish Eocene clays | Grønbech et al. 2011 [ | |

13%–117% | 60°–60 g | Fine-grained soils | Matusiewicz et al. 2016 [ | |

30%–390% | 60°–60 g | Fine-grained soils, kaolin–bentonite mixtures | Mendoza and Orozco 2001 [ | |

30%–350% | 30°–80 g | |||

<150% | 30°–80 g/100 g60°–60 g | Fine-grained soils | Shimobe 2010 [ | |

27%–110% | 30°–80 g | Turkish natural soils | Wasti 1987 [ | |

80%–150% | 30°–80 g | Soil–bentonite mixtures | Mishra et al. 2012 [ | |

13%–117% | 30°–80 g | Fine-grained soils | Matusiewicz et al. 2016 [ | |

Power relationships | ||||

>100% | 30°–80 g | Natural clays | Schmitz et al. 2004 [ | |

Up to approx. 600% | 30°–80 g | Fine-grained soils | O’Kelly et al. 2018 [ | |

<120% | ||||

Up to approx. 600% | ||||

<120% |

_{L}_{L}

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