State-of-the-Art Review on Determining One-Dimensional Consolidation Parameters Based on Compression and Distribution of Pore Water Pressure: Coefficient of Consolidation (c_v), End of Primary (EOP) Consolidation

Abstract

Predicting the time rate of consolidation is one of the major aspects of structure design, founded on compressible fine-grained soil. The time to achieve the required advancement of the consolidation process is proportional to the coefficient of consolidation (c_v). In practical applications, the settlement rate is directly related to the excess pore water pressure dissipation rate. A plethora of interpretation methods have been proposed for determining consolidation parameters from laboratory one-dimensional consolidation test in the past decades. This state-of-the-art review presents a comprehensive literature study of available approaches for establishing both coefficient of consolidation and end of primary (EOP) consolidation using compression and pore water pressure laboratory data. The classification of the methods has been made to set in order interpretation approaches for future selection and comparisons. The first part of the paper describes approaches based on graphical curve-fitting. This part includes five approaches: square root of time fitting approach, Semi-logarithmic fitting approach, Differential methods, Hyperbolic approach, and approach based on excess pore water pressure dissipation. In addition, a method comparison study has been performed to evaluate the degree of agreement between selected methods statistically. For this purpose, simple regression and Bland & Altman differences analysis have been used. The second part refers to the computational-based approach, covering a wide range of methods centred on full-matching treated by least-squares, correlational equations linking c_v with index properties and soft computing approaches. A thorough insight into recently published literature on machine learning and physics-informed deep learning incorporated to derive the representative value of c_v has also been compiled.

1 Introduction

1.1 Background

Geotechnical engineering plays a crucial role in most civil engineering undertakings when the interaction between construction and soil or rock masses is considered. For shallow foundations, it is necessary to design them accordingly to the two fundamental criteria. The geotechnical stability related to the bearing capacity problems [141, 168, 222] and settlement problems [15, 86, 219] constitutes the first one. The structural strength belongs to the second. This extended state-of-art review revolves around one of the most complicated aspects of the settlement analysis: reliable prediction of the settlement rate of structures founded on the ground using appropriate consolidation theory. From the engineering point of view, the term consolidation refers to the time-dependent volumetric change of soil in response to increased loading, involving squeezing water from the pores, decreasing volume, and increasing effective stresses [23]. Figure 1 shows a definition sketch of the one-dimensional consolidation problem for a homogeneous saturated clay layer. The effects of loading on saturated soil are seen as changes in the state of stress in both the solid and liquid phases in proportion to their stiffness. The pressure increase in the pore space fluids and hydraulic gradients causes their flow through the soil mass, and the resulting excess pore pressure transfer to the soil skeleton.

Fig. 1

Definition sketch of the one-dimensional consolidation problem: soil element in consolidating stratum and changes in stress during the process

One of the earliest observations of the consolidation process appeared in the technical literature thanks to Telford [218]. However, the substantial development of research on the consolidation of fine-grained soils, especially clays, is due to the founder of the science of soil mechanics—Karl von Terzaghi. From the late’20 s of the twentieth century until today, it is assumed that the settlement rate is directly related to the excess pore water pressure dissipation rate in practical applications, and the time to achieve the required consolidation percentage is proportional to the coefficient of consolidation (c_v). The coefficient of vertical consolidation (L²/T) is the obligatory parameter in the analysis of deformations of porous media, especially in predicting the settlement rate both in designing new structures and evaluating existing constructions. This parameter controls the rate at which the consolidation process takes place and depends on the anisotropy of permeability and the definition of compressibility. Thus, c_v can be expressed by the following equation:

$${c}_{v}=\frac{{k}_{v}}{{m}_{v}{\gamma }_{w}}$$

(1)

where k_v is the vertical coefficient of permeability, m_v is the coefficient of volume compressibility and γ_w is the unit weight of water.

The c_v is commonly determined from an incremental loading oedometer [31, 220] or hydraulic consolidation cell test [181]. Probably, one of the first researchers who used an oedometer apparatus to simulate the one-dimensional deformation and drainage conditions of the soil subjected to external load was Frontard [224]. Shortly after the pioneering formulation of the Classic Theory of Consolidation [218], the oedometer test has become a laboratory standard. Crawford [54] detailed four possibilities for assessing consolidation behaviour guided by loading conditions as follows: normal incremental loading (standard 24 h oedometer test); slow incremental loading (long-term oedometer test); rapid incremental loading (EOP test); controlled rate loading (constant rate of strain (CRS) [82], constant rate of loading (CRL) [101]); controlled gradient (CG) [80, 124]).

Taylor [216], during his research at MIT in the 1940s, pointed out two possibilities of determination of c_v using the method based on strains (deformation) and the method connected with the pore water pressure measurements. These possibilities have advantages and disadvantages that will be thoroughly analysed and discussed in this paper. Depending on the size of the deformation (range of deformation for which the problem will be solved), many theoretical descriptions of the consolidation process have been proposed. In the mechanism of the continuum, there are small and large deformations. The scope of the former applies to the case when the displacement of particles is much smaller than any significant dimension of the soil body. On the other hand, the scope of large deformations refers to the situations where the deformation size is substantial, and there is a change in the geometry and the constitutive properties of the material. The comprehensive summary followed by extensive bibliographies concerning formulations for small and large strain consolidation can be found in [2, 56, 169, 183,184,185, 188]. The interpretative approaches for consolidation tests discussed in this paper will refer to the one-dimensional governing consolidation equation developed by Terzaghi and Fröchlich [223]:

$$\frac{\partial u}{\partial t}={c}_{v}\frac{{\partial }^{2}u}{\partial {z}^{2}}$$

(2)

where u is an excess pore water pressure, t is the time, c_v is the coefficient of consolidation and z is the vertical coordinate.

The derivation of this equation is systematically presented in most of the soil mechanics textbooks that appear on the publishing market [12, 152, 234]; thus, it will not be repeated in this paper.

The consolidation problem of cohesive fine-grained soils is commonly studied for vertical and horizontal radial flow conditions. Therefore, knowledge of the vertical and horizontal components of c_v is needed for the analysis. The radial water flow through the soil medium is extremely important when dealing with soil improvement and land reclamation projects [18, 19, 30, 44,45,46, 51, 92, 94, 95], especially in the issues associated with accelerating consolidation by the use of prefabricated vertical drains (PVD) [34, 71, 85, 97, 112, 116, 147, 150, 174, 236, 237].

The main difficulty in predicting the settlement rate is the discrepancy between laboratory and field values of the c_v [157]. Possible reasons for the large discrepancies, such as a sample disturbance, errors in laboratory testing procedures, errors in field measurements, and three-dimensional consolidation effects, were highlighted and discussed by Bromwell and Lambe [26]. There are two essential conditions for c_v to be theoretically the same in both the field and the laboratory. Regarding the first, if c_v were a fundamental soil property, its value would be independent of the measurement conditions. However, one can suppose that c_v is not a fundamental soil property. In this case, the same value can be expected theoretically only if the k_v and m_v were of the same magnitude in the laboratory and in the field. Particularly, if the boundary conditions in the field and the laboratory were the same, one could expect k_v and m_v to have the same values theoretically. On the other hand, different boundary conditions would provide different values for k_v and m_v. It is essential to remember that the m_v depends on the nature of the surface loading and the boundary conditions of the soil layer. Thus, bearing in mind that the measured conditions in most cases are different, the c_v should not be regarded as a fundamental soil property.

Over the past decades, time-consuming attempts have been made to develop appropriate methodologies for standardising compression–time (δ–t) data and deriving consolidation parameters. Generally, methods for determining the c_v can be classified as graphical curve-fitting methods (GCFMs), which can be further categorised and separated according to the mathematical structure of the δ–t relationship and the computational methods (CMs). Figure 2 presents an overview of available methods for interpreting consolidation tests and determining consolidation parameters.

Fig. 2

Overview of methods for determining consolidation parameters (AM – Analytical method, APPD – Asaoka pore pressure dissipation, CM – Casagrande method, DCM – Diagnostic curve method, DPM – Deviation point method, ETM – Extended Taylor method, FMSRM – Full-match settlement rate method, GCM – Generalised Casagrande method, GTM – Generalised Taylor method, IMSA – Iterative method of successive approximations, IPM – Inflection point method, IRHM – Improved rectangular hyperbola method, IVM – Improved velocity method, ITM – Improved Taylor method, LLM – Log Log method, LRM – Least residuals method, LSRSM – Logarithmic settlement versus rate of settlement method, MCM – Modified Casagrande method, MPM – Mid-plane point method, MSM – Modified slope method, OPM – One point method, PPDM – Pore pressure dissipation method, RCM – Revised Casagrande method, RHM – Rectangular hyperbola method, RSA – Rate of settlement approach, SIRSM – Settlement versus inverse rate of settlement method, SM – Slope method, SNM – Scott N-method, SRSM – Settlement versus rate of settlement method, STM – Simplified Taylor method, TM – Taylor method, t₁TM – Simplified t₁ Taylor method, VAM – Variance method, VM – Velocity method)

The most general classification seems to take into account the size of the consolidation period from which the c_v is extracted. The plethora of methods is based on a graphical curve fitting procedure in which the measured consolidation settlement δ–t data is compared to the theoretical progress of consolidation [55, 67]. In this procedure, c_v is calculated based on matching a single experimental point to the specific consolidation progress, e.g., 20% or 30% and so on. In contrast, CMs are based on mathematical models used to numerically study the soil’s behaviour by means of a computer simulation. This approach allows the full experimental record of the study to be analysed with a theoretical solution. Depending on the method used for c_v determination, the generalised form of the expression can be written as follows:

$${c}_{v}=\frac{{T}_{i}{H}^{2}}{{t}_{i}},$$

(3)

where T_i is the time factor corresponding to the arbitrary assumed time t_i and H is the length of drainage path.

The primary consolidation region on the δ–t curve can be isolated by identifying and excluding the initial and secondary compression regions of the overall settlement curve [121]. On the contrary, the computational-based approach uses a representative period of the consolidation progress or the entire consolidation period as a whole. This class of methods allows for assessing the theoretical model’s predictive ability. The value of c_v can be averagely determined by thoroughly combining the theoretical and experimental relationships. Despite a wide range of available approaches and methods, it is challenging to select a method that provides the most accurate way to determine the reliable value of the c_v. This is mainly because not all methods are suitable for all types of soils (i.e. the shape of the consolidation curve), and the sensitivity of the methods changes with the soil’s mineralogy and grain sizes. The selection of the applied method also depends on the type of consolidation test in terms of the duration of load increment [205]. The most common standards (i.e. British Standards Institution and American Society for Testing and Materials) adopted for the consolidation test require that each load increment be held constant for 24 h. However, the necessity to shorten the time of the entire test, which usually requires six to eight steps of loading, resulted in the introduction of rapid loading methods. Nevertheless, this type of test does not allow for obtaining secondary consolidation characteristics, which, from the point of view of the direction in which soil mechanics develops (constitutive modelling), may lead to an inappropriate, or at best incomplete, description of the soil behaviour.

Traditionally, volume changes induced by one-dimensional loading are entailed in the form of initial (immediate) compression, the primary and secondary consolidation [136, 139]. The first of them is related to the pseudo-elastic behaviour of the soil, which results from the load application due to getting rid of air from the pore space. During primary consolidation, the soil deforms due to the squeezing of water out of the pores resulting from excess pore water pressure dissipation. The gradual dissipation is accompanied by the simultaneous increase of effective stress. When the stress transfer is completed, the soil reaches the so-called end of primary (EOP) consolidation state, and further deformations develop at a rate controlled by soil viscosity during the secondary consolidation [58, 81, 104]. There is no compliance with the terminology of individual components of the soil deformation process subjected to compression among the authors. For practical reasons, the consolidation process is often distinguished between primary consolidation related to the dissipation of the excess pore water pressure and secondary consolidation that occurs when dissipation is completed. However, despite its transparency, this division does not have a full constitutive justification, as it relates to settlement caused by time only due to the secondary consolidation phase. Buisman [27] suggested direct compression to describe the deformation caused by changes in the effective stress and secular compression as the results of time only. Another approach was presented by Taylor [216], in which primary compression was referred to as the results of changes in effective stresses, and secondary compression concerning changes in compressibility, which may occur even if there is no change in pore water pressure. As Den Haan [57] correctly emphasised, such a defining secondary compression can be used for its constitutive importance and as the consistent significance associated with the temporary consequence. In turn, Leonards and Deschamps [117] have proposed to use primary and secondary consolidation terms for the hydrodynamic part of the process (dissipation of the excess pore water pressure) as a result of the changes in the effective stress and the duration of the loading, respectively, whereas for post-hydrodynamic part—secondary consolidation term. Another terminology was provided by Bjerrum [22] in which terms of instant compression and delayed compression were not concerned with the temporary consequences of subsequent phases of the occurring process but were concepts describing constitutive phenomena. In light of the matter, primary and secondary consolidation terms will be used in the following sections of this paper. At this point, it is convenient to introduce one additional term—creep, which is sometimes misunderstood as secondary consolidation. Tavenas et al. [215] emphasised distinguishing between these two phrases. In accordance with the findings, secondary consolidation is caused by pure creep; however, creep may also act during the dissipation of the excess pore water pressure. In this connection, creep is the process in which soil deformation will happen as a function of time, and its rate is associated with a solid’s viscosity. From a phenomenological manner, the deformation rate during dissipation is larger than the creep rate at constant effective stress. Moreover, to a minor extent, creep strains contribute to deformation relative to the total strain during the primary consolidation. This may lead to a perception of neglect of the creep strains in the primary consolidation modelling [115, 135, 213]. Formally, this issue was discussed in [115]; nonetheless, any deviations from the theory for field conditions do not have to result from the assumed creep hypothesis A or B. When predicting settlements of such a variable medium as sediment, it is essential to pay attention to the spatial variability of permeability and stiffness, local moisture (sand inserts), flow anisotropy and other non-linear soil characteristics.

1.2 Organisation of This Paper

This study aimed to discuss methods for determining consolidation parameters (i.e. c_v, strain, and time at EOP consolidation) using laboratory data on compression and pore water pressure. Based on a review of the world technical literature, the available methods have been classified and evaluated regarding their limitations and practical use. Unfortunately, the only available paper that reviewed the state-of-the-art for determining c_v was made by Shukla et al. [197]. The publication mentioned above paid particular attention to the GCFMs related to compression. Many computational-based methods were not taken into account. Hence, this work is the first attempt to systematise available approaches for interpreting consolidation tests, especially the computational-based approaches invented in the last decade. To keep the review within manageable limits, the following assumptions have been made:

• The review applies only to methods for determining the vertical c_v (many methods for determining the radial coefficient of consolidation (c_r), especially graphical methods, use the same principles as their c_v counterparts).

• Field dissipation methods for piezocone penetration test (CPTu) and dilatometer test (DMT) are not described in detail.

• The coefficient of consolidation is related to small strains only, and solutions to the large strain theories were not included in the review.

• Time and time-dependency are assumed to be related to the viscous effects in the soil skeleton, such as creep. Therefore, the consolidation process is not regarded as a true time effect.

A brief outline of this state-of-the-art review is as follows. In Sect. 2, five approaches, such as the Square root of time fitting approach, Semi-logarithmic fitting approach, Differential methods, Hyperbolic approach and Excess pore water pressure dissipation methods for evaluating consolidation test results, are concisely described. This section also compares c_v values determined by various selected GCFMs. Section 3, entitled “Computational-based methods”, discusses optimisation problems associated with interpreting the consolidation test treated by least-squares. Moreover, the error norms are introduced in the section. Section 4, entitled “Correlations”, consists of a systematic review of correlational equations linking c_v with index properties and state-stress parameters. In Sect. 5, soft computing approaches based on Artificial Intelligence (AI) or Machine Learning (ML) and Physics-informed deep learning have been deeply reviewed. Finally, the conclusion is made in Sect. 6. Several prospective directions for future research are also pointed out.

1.3 Outline of the Study Cope and Research Criteria

In order to extract relevant documents pertaining to methodological aspects of consolidation from those related to other consolidation issues, suitable keywords were defined. The publications were collected and compiled using the leading web search engines, such as Google Scholar, ScienceDirect, SpringerLink, and ACM Digital Library. The Boolean operators, namely AND and OR, were used as conjunctions to improve the focus in search and to combine or exclude keywords in a searching process, e.g. primary AND coefficient AND consolidation; coefficient AND consolidation; coefficient AND excess OR settlement; time AND settlement/pore pressure AND consolidation/dissipation curve. In the evaluation stage, the software program Publish or Perish [88] was also used to retrieve and analyse citations. The full texts were read after screening the eligible publications, and their reference lists were also reviewed at the inclusion stage. The publication date limits were not set when compiling the literature database on the investigated topic. As per the survey, selected papers and conference proceedings covered a broad range of publication timeframes from 1923 to the present.

2 Graphical Curve Fitting Methods

2.1 Square Root of Time Fitting Approach

It is commonly accepted that one of the first methods invented for consolidation analyses was based on the square root of time approach. Terzaghi [220], in his paper on “Principles of final soil classification”, incorporated into analytical procedure elements of the so-called “t₉₀ method” [162]. Several years later, Gilboy [77] discussed improvements in some soil testing methods and alluded to a refinement of the square root plot to linearise the initial region of the consolidation curve, which Taylor had made. The present form of the square root of the time fitting method appeared in later Taylor’s works [217, 218]. The most simple form of the method was introduced by Capper and Cassie [29] as a “t₁ method”. Over the subsequent decades, the approach has been repeatedly modified or adapted to solve exceptional cases of various consolidation problems.

2.1.1 Square Root of Time Fitting Method

In the square root of time fitting method, denoted as Taylor’s method (TM), a plot of settlement or compression against the square root of time is considered. The method is based on the assumption of dominating the filtration process up to 90% of consolidation and utilizes the knowledge that the initial portion of the theoretical consolidation curve is parabolic (i.e., U_v = 1.128, √T_v for U_v < 52.1%). Graphical procedure concerns drawing a straight line in the initial segment of the δ–√t, which is back extrapolated to t = 0 to obtain the corrected initial thickness representing 0% of primary consolidation (δ₀). The initial linear portion of the curve has a slope m (first straight line is of the form δ′ = mt^B + δ₀ where m is the gradient of that line). For a given U_v = 90%, the slope m can be expressed in terms of δ₉₀ and t₉₀, as follows:

$$m=1.153\frac{{\delta }_{90}}{\sqrt{{t}_{90}}}$$

(4)

where δ₉₀ is the settlement at 90% of primary consolidation, and t₉₀ is the corresponding time.

The factor of 1.153 is used along with the initial linear portion of the settlement-time curve to compute c_v at 90% consolidation. This factor (i.e., 1.153) can be interpreted as the ratio of the secant slope, m₅₀ at 50% consolidation to the secant slope, m₉₀ at 90% consolidation. The ratio of the secant slopes (ROSSi) is very useful when one evaluates the value of c_v for later stages of consolidation, even beyond 90% [9]. Thus, the ratio of secant slopes is expressed by:

$$\frac{{m}_{50}}{{m}_{90}}=\frac{\frac{{\delta }_{50}}{\sqrt{{t}_{50}}}}{\frac{{\delta }_{90}}{\sqrt{{t}_{90}}}}=\frac{{M}_{50}}{{M}_{90}}=\frac{50}{90}\frac{\sqrt{{T}_{50}}}{\sqrt{{T}_{90}}}$$

(5)

where M₅₀ is the theoretical slope of the consolidation curve at 50% consolidation, M₉₀ is the theoretical slope of the consolidation curve at 90% consolidation, T₅₀ is the dimensionless time factor at 50% consolidation and T₉₀ is the dimensionless time factor at 90% consolidation.

Figure 3a shows variation of theoretical ROSS_i with various values of U_v beyond 52.6 percentage of consolidation. For determining EOP consolidation, a second straight line is drawn through the 0% compression ordinate so that the abscissa of this line is 1.153 times the abscissa of the first line through the data. The time corresponding to the intersection of this second line with the curve represents 90% primary consolidation. For interpretation purposes, a third point taken at U_v = 90% is used to estimate the EOP settlement; thereby, this method is theoretically valid when only three data points are used. In the TM, after the point of U₉₀ is obtained, the point where the primary consolidation is assumed to finish (U_v = 100%) is determined on the compression axis by the relative compression at U_v = 100% (δ₁₀₀). Any settlement below the time corresponding to δ₁₀₀ is considered as secondary compression. Al-Zoubi [9] noticed that this method inherently includes limitations due to fitting the experimental settlement-time curve. The actual time to EOP consolidation exhibits a definite value to the Terzaghi theory, in which the theoretical time to EOP consolidation is infinity. Moreover, the method’s weakness may come from the unknown effect of secondary compression in the later test stages.

Fig. 3

Theoretical plots for square root of time fitting approach: a variation of ROSS_i with U_v (A_U is the percentage increase in the abscissa for different values of U_v in TM, A_U(ITM) is the percentage increase in the abscissa for different values of U_v in ITM); b rate of consolidation curve with its characteristic features for TM, t₁TM, ITM, STM, SM, DPM and GTM

TM also depends upon identifying the initial straight line portion of the δ–t curve; therefore, this method cannot be applied for soils that exhibit continuous curvature in the initial curve portion. This method is also prone to interpreter judgement in the way that depends on whether the straight line is drawn through the first few points or as an “average line” through more points. The following formula gives the value of c_v in TM:

$${c}_{v}=\frac{\pi }{4}{\left(\frac{1.153{\delta }_{90}H\times 0.90}{\sqrt{{t}_{90}{\delta }_{90}}}\right)}^{2}=\frac{0.848{H}^{2}}{{t}_{90}}$$

(6)

Raheena and Robinson [170] have proved that TM can be successfully applied in the accelerated consolidation testing. The time required to complete the test could be as low as 2–5 h for most of the soils, compared to 10–14 days in the case of the conventional consolidation test.

2.1.2 t₁ Method

In the simplified t₁ method (t₁TM) the portion of the consolidation curve up to U_v = 50% was assumed as the parabola that have the following equation:

$$U=\frac{2}{\sqrt{\pi }}\sqrt{{T}_{v}}=1.13\sqrt{{T}_{v}}$$

(7)

In the δ–√t curve, the point representing final compression was extrapolated to the compression axis, and the straight line in the initial segment of the curve was drawn. This segment is the distance from the point where t = 0 to the intersection of the two straight lines representing the square root of the time t₁, which is required for complete consolidation. After decomposing the expression of the dimensionless time factor, for U_v = 100% and t = t₁, c_v is calculated as follows:

$${c}_{v}=\frac{\pi }{4}\frac{{d}^{2}{U}_{v}^{2}}{t}=\frac{\pi {d}^{2}}{4{t}_{1}}$$

(8)

where d is the length of drainage path.

2.1.3 Improved Square Root of Time Fitting Method

Tewatia and Venkatachalam [232], based on the properties of the slope of the U_v–√T_v curve (see Fig. 3b), proposed an extension to the square root of time fitting method to determine compression corresponding to any U_v above 70% consolidation for evaluation of the c_v. The method is called the improved square root of time fitting method or improved Taylor’s method (ITM). The authors have shown that the slope of the U_v–√T_v curve at any degree of consolidation is directly proportional to the slope of the δ–√t curve; hence, this method allows the generalisation of the δ–√t plot in the latter stages of the consolidation. As reported by the authors in the proposed method, data collection can be stopped as soon as the δ–√t curve shows enough curvature for drawing the tangent (generally after going a little beyond the tangent point). In the case of U_v = 70%, the calculated c_v is higher and closer to field values when compared with TM c_v, and the time taken to determine c_v is almost half that of the original TM. While consolidation range between 75 and 95% U_v is selected, it shall be possible to determine the c_v value between TM c_v and CA c_v. Moreover, the effects of the secondary consolidation on the c_v value at different consolidation percentages can be studied. Al-Zoubi [9] used similar assumptions in the extended Taylor method (ETM) to improve the EOP consolidation and c_v estimation. In the ETM, Taylor’s procedure is repeated twice at any two U_v values greater than 52.6% instead of the one used in the TM at 90%, and the value of c_v explicitly relates to the slope of the initial linear portion of the δ–√t curve. As indicated by Al-Zoubi, the ETM can be applied with a minimum of four consolidation data points to estimate EOP consolidation and c_v. Two of them determined from the early stages of consolidation (U_v ≤ 52.6%) are used to back-calculate initial compression and the initial slope of the δ–√t curve. Two more data points from the later stages of consolidation (U_v ≥ 52.6%) are used to compute the EOP settlement. Values of c_v from this method yield similar values to those of Casagrande’s method (CM) and are lower than those of the TM.

2.1.4 Simplified Taylor Method

Feng and Lee [71] proposed another so-called simplified version of the TM (STM). Authors assumed that the theoretical U_v–√T_v curve is linear up to 60% consolidation, and the experimental δ–√t may also present a linear segment that ends at 60% consolidation. The deviation point is recognised from the lower end of the linear segment and is used with the T_v = 0.286 to determine c_v. Unfortunately, Robinson and Allam [180] showed that the theoretical slope of the linear segment (m = 1.128) changes at about U_v = 45%. Therefore, discussers indicated that when U_v = 60% is taken as the characteristic point, there is an error of as much as 40% in the T_v value.

2.1.5 Slope Method

The slope method (SM) proposed by Al-Zoubi [5] is based on a fitting procedure in which the slope of the initial linear segment of the δ–√t curve is fitted to the corresponding slope of the U_v–√T_v relationship, which is constant and is equal to 1.128. It is shown that c_v can explicitly be expressed in terms of the slope of the initial linear segment of the δ–√t curve and the EOP settlement independently of any specific U_v value in the range where the SM is valid. This method requires the determination of the initial and final compressions that correspond to 0% and 100% consolidation, respectively, for estimating the EOP consolidation. Originally, Sivaram and Swamme’s [190] computational procedure was adopted for determining 0% consolidation. The compression (δ_e) at which the δ–√t curve deviates from the initial linear portion was used for estimating EOP consolidation. Theoretically, this compression corresponds to exactly U_v = 52.6%. The deviation point with the corresponding time t_e = t_52.6 was also used by Ming et al. [142] to compute c_v using a square root plot and the standard expression for the c_v. This one-point fitting procedure was designated as the square root deviation point method. This work simplified this name to the deviation point method (DPM). The procedure for determination of the EOP consolidation incorporated in the SM was modified several years later by the same author [8] and is based on a unique relationship between the ratio of the secant slope (ROSS) of δ–√t curve at 50% consolidation to the secant slope at any time arbitrarily selected beyond the deviation point and U_v. The modified slope method (MSM) requires a minimum of four compression-time data points for calculating c_v and has the same basis as ETM. It should be remembered that selecting points that relate to significant compression in the later stages of consolidation may lead to unrealistically large consolidation EOP values. The methods particularly vulnerable to this are the MSM and ETM.

2.1.6 Generalised Square Root of Time Fitting Method

The generalised square root of time fitting method (also named time exponent method) denoted herein as generalised Taylor’s method (GTM) was introduced by Lovisa [120] to account for any non-uniform initial excess pore water pressure distribution (u_i) by introducing so-called adjustment factor (f_T). The adjustment factor depends on the type of u_i distribution and can be determined by approximating separate regions of the U_v–T_v curves using the exponential function as follows [120]:

$${f}_{T}={\left[{T}_{90}exp\left(\frac{0.1054+lnA}{B}\right)\right]}^{B}$$

(9)

where A, B are the approximation function constants, which are unique to each u_i distribution.

Values of the adjusting factors are provided in Table 1. The initial corrected dial reading is determined on the same basis as in the TM. However, the second straight line is of the form d′ = (g/f_T) t^B + d₀ and is drawn until it intersects the actual d–t^B curve to determine the time t₉₀ and to calculate c_v. For the sinusoidal u_i distribution and the single-sided drainage, T₉₀ is taken as = 0.866, while for the double-sided drainage, T₉₀ is equal to 0.233. In the case of non-uniform cases for u_i distribution, the inflection point method (IPM) is also suitable for calculating c_v [137].

Table 1 Adjusting curve-fitting factors f_T for various initial excess pore water pressure distributions

2.1.7 δ–t/δ Method

Sridharan and Prakash [207] proposed to represent the U_v–T_v relationship in the form of the U_v–T_v/U_v curve instead of U_v–√T_v curve. The theoretical U_v versus T_v/U_v plot shows a clear, initial straight line portion as in the U_v versus √T_v plot. If a straight line whose slope is 1/1.33 times that of the identified initial straight line is drawn through the origin, it can be observed that the line intersects the U_v–T_v/U_v curve at the 90% primary consolidation point. The resulting methodology is similar to the TM but uses δ–t/δ curve with higher precision to obtain EOP consolidation. The results from the δ–t/δ curve compare well with the results by the TM. Because of the flatter slope of 1/1.33 (δ–t/δ curve) as against 1/1.15 (δ–√t curve), the estimation of the 90% consolidation point is more accurate.

2.1.8 Log–Log Method

Another attempt to improve the determination of the end of primary consolidation was made by Sridharan and Prakash [207]. They developed a log–log method (LLM) that allows both to determine the time at EOP and c_v. It was shown that the slope (M) of the initial linear portion of the theoretical log U_v–log T_v is constant up to U_v = 56%. The relationship between M and U_v bears some resemblance to that of TM. The graphical procedure of LLM involves extending the initial linear portion of the log U_v–log T_v curve to cut the horizontal U_v = 100% line. The resulting intersection occurs at T_v = 0.79, corresponding to U_v = 88.3%, which forms the basis for determining c_v. Note that LLM does not require the determination of initial compression to obtain the c_v value.

2.2 Semi-logarithmic Fitting Approach

A second prevalent approach for the consolidation analysis is to plot settlement or compression against the logarithm of time. Arthur Casagrande probably invents the semi-logarithmic approach during his work at MIT. From 1930 through 1932, his research concentrated on shear strength and consolidation tests [240]. As a result, Casagrande established a procedure for identifying the preconsolidation pressure of clays and evaluating time-settlement curves through semi-logarithmic plots [239]. In 1940 at Harvard University, Casagrande and Fadum published the “Notes on soil testing for engineering purposes” under the Harvard Soil Mechanics Series, and the logarithm of time fitting method’s principles was first presented to the broad public [32]. The semi-logarithmic fitting approach, similar to the square root of the time fitting approach, went through many modifications and refinements.

2.2.1 Logarithm of Time Fitting Method

The fitting method, which uses a logarithmic time plot, was developed by Casagrande and is denoted after its inventor as a Casagrande method (CM). Casagrande observed that the plot of U_v versus T_v (see Fig. 4) has three distinct portions: an initial parabolic-shaped portion, a linear middle portion and a final portion asymptotic to the horizontal.

Fig. 4

Semi-logarithmic plot for CM, RCM, MSM, IPM and GCM

Two constructions have been devised to eliminate initial and secondary consolidation from the experimental curve. In this method, it is necessary to establish the points representing 0% and 100% primary consolidation to evaluate the c_v. The time-settlement relationship in the early stages of consolidation is parabolic. Thus, corrected initial thickness (0% of consolidation) is obtained by arbitrarily selecting times t₁ and t₂ on the initial curved portion such as t₂ = 4t₁. The CM assumes that the following lines are plotted on the graph: the straight line passing through the final points, which exhibit linear trend and constant inclination and tangent to the steepest part of the curve. The intersection formed by the final straight line projected backwards, and the tangent to the curve at the point of inflection represents 100% of the primary consolidation, e.g. EOP settlement. The coefficient of consolidation is related to the reference point on the curve, corresponding to the time t₅₀ required to archive 50% of the primary consolidation and is defined by the following formula:

$${c}_{v}=\frac{0.197{H}^{2}}{{t}_{50}}$$

(10)

Many researchers highlighted this method’s limitations [67, 121, 154]. For example, it is unreasonable to use CM for soils that do not exhibit theoretical S-shaped time-settlement curves or when the recorded data from the test produce a ‘flat-shaped’ curve. Besides that, if the time-settlement curve exhibits no inflection point, the secondary consolidation part is absent, or secondary consolidation starts at the initial consolidation phase, CM is not applicable.

2.2.2 Revised Logarithm of Time Fitting Method

Robinson and Allam [179] suggested a revised CM (RCM) in which the early stage of the time-compression data in the semi-logarithmic plot is used to determine the c_v. In terms of theoretical assumptions, this method utilises the tangent through the point of inflection (U_v = 70.52% and T_v = 0.41), which is extended to intersect the U_v = 0 line. The point of intersection corresponds to U_v = 22.14% and T_v = 0.0385. This feature of the U_v–log T_v relationship, together with the characteristic time t_22.14 that is obtained by the point of intersection between the tangent passing through the inflection point and the line passing through the corrected initial thickness (0% of consolidation) are used to calculate c_v. The procedure for determining the 0% consolidation in this method is the same as for CM, and the c_v is calculated with the following formula:

$${c}_{v}=\frac{0.0385{H}^{2}}{{t}_{22.14}}$$

(11)

One of the advantages of the method is that the secondary effects are less severe compared to other methods due to utilising smaller T_v values for computing c_v. This means that the obtained c_v value is higher than those obtained from TM and CM and more similar to those obtained from direct permeability measurements. Moreover, if characteristics of the secondary compression are not required, the test can be conducted with a shorter duration.

2.2.3 Su’s Maximum Slope Method

The maximum slope method (MSM) also known as steepest-slope method proposed by Su [212] can be used in situations where the curve’s end portion explained in the log of time method cannot be clearly distinguished. The graphical procedure involves drawing a tangent to the steepest part of the consolidation curve with the slope originally denoted as (h) and selecting the point for any required average degree of consolidation. Based on this point, the time is determined from a consolidation curve, and the c_v is calculated. MSM is more applicable for consolidation curves that do not exhibit the typical S-shape.

2.2.4 Inflection Point Method

The theoretical basis of the inflection point method (IPM) was first given by Matlock and Dawson [129], and further, the method was refined by Cour [53], Robinson [175] and Mesri [138]. The IPM can also be categorised as a differential method because it uses gradient M = dU_v/d(log T_v) [162]. According to the invention, the graphical procedure involves the identification of an inflection point on the semi-log plot (see Fig. 5). There are two ways in which the inflection point can be determined by visual observation or using the tangent method, where the point of inflection is selected as the point at which the absolute value of the slope of the tangent to the time-settlement curve is the maximum [120]. Note that if the finite-difference approximation is used to locate the inflection point, the results are sensitive to the time-step size. Matlock and Dawson [129] observed that the slope of the U_v–log T_v curve at the inflection point is 68.4% per log cycle of T_v. In the construction used by the authors mentioned above, the tangent line is drawn to the time-compression curve, and the average settlement change for one log cycle of time is measured. Subsequently, this value is divided by 0.68 to obtain the amount of primary compression (δ₁₀₀–δ₀), and this value is added to δ₀ to get δ₁₀₀. For the computation of the c_v reference point corresponding to U_v = 68% and T_v = 0.38 was used. If the consolidation curve has no inflection point, it is impossible to apply IPM. Such situations may occur for soils with significant susceptibility to creep deformations, small pressure increment ratios, and pressure increments spanning the preconsolidation pressure [118, 136]. Cour [53] presented a simplified approach in which settlement at the point of inflection is used to determine the time t₇₀, corresponding to 70% consolidation. As indicated by Lovisa [120], without another reference point on the consolidation curve, it is impossible to calculate the U_v for the remaining time-compression data and conduct a subsequent comparison between experimental and theoretical results.

Fig. 5

Relationship between gradient M and time factor T_v

Further, Robinson [175] demonstrated that the inflection point exactly corresponds to U_v = 70.15% and T_v = 0.405, with the maximum slope of M_i = 0.6868 at this point. Indicated theoretical features of the consolidation curve were used to obtain c_v. The main advantages of IPM are that it does not require the definition of the beginning and end of the primary consolidation stage to compute c_v, and the inflection point at the average degree of consolidation of 70% is within the midrange of consolidation progress (commonly U_v = 62–78%) [163] and is least affected by initial compression and secondary consolidation [137, 138]. Independently, Robinson [175] pointed out that IPM can also be used effectively to obtain both the coefficient of secondary compression (c_α) and the total primary consolidation settlement without any graphical construction.

Cheng and Yin [38] observed that estimation EOP point are generally influenced by the non-linear secondary compression, in which the c_α has a non-linear relationship with the logarithm of time. This method utilises selected principles of the TM and IPM to isolate the primary and secondary consolidation. The TM was applied to determine 0% of consolidation, while from the inflection point, a horizontal line was drawn, and the length between horizontal lines of U_v = 0% and U_v = 70.15% was obtained. Finally, the second vertical line was moved to intercept the time-settlement curve. At the interception, the time at the end of primary consolidation (t_EOP) on the time-axis corresponding to U_v = 100% was obtained. This way, the proposed method eliminates the arbitrary curve-fitting induced by the non-linear secondary consolidation. Another successful combination of characteristic features of both TM and CM was discussed by Al-Zoubi [10]. For the primary consolidation range, it was shown that c_v could be directly related to the ratio of the steepest slopes of the δ–log t and δ–√t curves without the need to estimate the initial compression and secondary consolidation and without the explicit use of any specific U_v value. The slope of the initial linear portion of the δ–√t curve was expressed in terms of c_v and EOP compression (δ_EOP) as follows:

$${m}_{\delta -\sqrt{t}}={\delta }_{EOP}\left(\frac{4}{\pi }\frac{{c}_{v}}{{H}^{2}}\right)$$

(12)

The EOP compression was estimated based on the steepest tangential slope of the δ–log t curve at the inflection point as follows:

$${\delta }_{EOP}=1.456{m}_{\delta -logt}^{IP}$$

(13)

For the accelerated consolidation testing (rapid consolidation tests), Raheena and Robinson [171] used IPM to estimate EOP consolidation and c_v. It was assumed that the subsequent loading could be applied for the accelerated consolidation testing once the inflection point on the consolidation curve is observed. This method cannot determine c_α because the secondary consolidation phase is not recognised.

2.2.5 Generalised Logarithm of Time Fitting Method

The generalised form of the logarithm of the time fitting method denoted in this review as generalised Casagrande’s method (GCM) was described by Lovisa and Sivakugan [122, 123]. GCM uses the adjustment factor f_C (see Table 2) to account for any non-uniform initial excess pore water pressure distribution (u_i). The method utilises the points of 0% and 100% primary consolidation to assess the time t₅₀. To estimate the EOP point, the authors extended the linear tail of the curve back toward the y-axis and drew a tangent to the point of inflection in the central portion of the curve. These two line’s intersection points are deemed EOP (U_v = 100%). The determination of 0% of consolidation was done by selecting time in the initial part of the curve and by calculating ty such that t_y = f_Ct_x. Note that f_C is a factor dependent upon u_i and drainage conditions. The value of δ₀ was determined by simple subtraction as δ₀ = δ_x − Δδ (Δδ = δ_y-δ_x). For computing c_v, the same step was utilised as in CM. Note that, for non-uniform cases where the power approximation only captures a small portion of the U_v–T_v curve, it may be challenging to use the corresponding modified curve-fitting procedure objectively.

Table 2 Adjusting curve-fitting factors f_C for various initial excess pore water pressure distributions

2.3 Differential Methods

Among the methods for determining consolidation parameters, the differential methods significantly contributed to develo** tools for interpreting consolidation tests. The issue of using consolidation rate or its inverse rate, in particular, has been at the core of the time resistance concept [99, 100]. The concept of resistance (rate inverse) is commonly used in almost all technical fields. All materials have resistance against enforced changing equilibrium conditions and can, therefore, be determined by measuring the incremental response to a given incremental action. For instance, the essential elements of this concept can be found in Newton’s Statute of Dynamics of the seventeenth century. Parkin [161, 162] used the concept of consolidation rate to develop a double logarithmic plot that utilises the relationship between consolidation rate (dU_v/dT_v) and T_v for the consolidation analysis. Fundamentals of this concept also formed the basis for the Rate of settlement approach (RSA) [225].

2.3.1 Velocity Method

The velocity method (VM) [161] is a fundamental and direct method of comparing theoretical and experimental consolidation curves. This method is based on the superimposition of consolidation velocities on a double-log graph, wherein scale changes are affected by a body displacement of the curves, and the analysis becomes independent of the initial and final conditions [125]. Velocity plots can be constructed concerning either strain rate or pore-pressure dissipation rate by differencing consecutive readings of strain (compression) or pore water pressure. Considering compression data, the theoretical velocity plot of compression rate generally has two parts with different features. Figure 6 compares theoretical double-logarithmic plots of dU_v/dT_v versus T_v and U_v/T_v versus T_v. The U_v/T_v–T_v relationship is invoked here because it reflects some resemblance to the velocity curve when plotted on a double-logarithmic plot. The main difference between the diagnostic curve method (DCM) and VM was the presentation of the data.

Fig. 6

Double-logarithmic plot for consolidation analysis by the methods based on consolidation rate

The bilinear character of this curve, together with the intersection of the extrapolated straight lines, might be used for determining time t corresponding to T_v = 0.793 (U_v = 88.5%), enabling the calculation of c_v [159]. As can be seen from Fig. 6, both curves initially exhibit similar rates, but around T_v = 0.2, the velocity plot (solid line) is steeper, indicating a drastic change in the theoretical consolidation rate. In the velocity plot, there is an initial 1:2 slope, corresponding with the early parabolic part of the settlement-time curve up to U_v = 50%, whereafter, the slope increases continuously to infinity. It was suggested that a velocity plot could be used as an overlay to any experimental curve, giving a direct correlation between real time t and T_v. In the VM, the observed compression rate is determined between successive intervals of time and plotted against time in a log–log scale. In the initial portion of the experimental plot, a line with a slope of 1:2 is fitted. In VM, the transition from the initial linear portion has to be identified to select time t corresponding to T_v = 1, and the theoretical curve is then superposed on the experimental curve. The method’s main drawback is the problem with the correct superimposition of the theoretical curve on the experimental curve resulting from the scattering of data obtained from the test [159]. Singh [189] also used a double logarithmic plot previously developed by Parkin to analyse laboratory consolidation data.

Moreover, VM requires a straight line to be fitted to the initial points, while DCM does not, and DCM allows a straightforward determination of the primary consolidation range. However, the essential principle of the method was the same as VM. The so-called diagnostic theoretical curves were proposed: “Diagnostic Curve without a Peak” and “Unimodal Diagnostic Curve”. The main reason for develo** the “Unimodal Diagnostic Curve” was to identify the non-ideal consolidation conditions (i.e. non-standard shape of the consolidation curve, scattering in test data). Both variants of DCM do not require 0% and 100% of primary consolidation in the estimation of c_v.

2.3.2 Improved Velocity Method

Pandian et al. [160] proposed an improved VM variant, which involves identifying the characteristic segments on the experimental velocity curve. It was observed that it is frequently challenging to locate the transition point on the velocity plot due to the scatter of points. In the improved velocity method (IVM), a definite point of intersection (T_v = 0.524) is obtained to evaluate c_v. Using the generalized dδ/dt–log t curve, the authors found that the curve has three distinctly linear portions separated by one transition part. Therefore, the end of the primary consolidation has been identified. IVM involves the identification of straight lines but not of particular slopes. Hence, the difficulties of matching curves are eliminated, and the method is unaffected by loading duration, soil type, or loading intensity.

2.3.3 Rate of Settlement Approach

The rate of Settlement Approach (RSA) includes several tools for analysing various consolidation problems. In the RSA, a dependent variable, Y, is plotted versus its differential dY/dX, inverse dX/dY or both of them [230]. Note that a dependent variable, Y, is acclaimed as settlement, and X is an independent variable, e.g. settlement rate. In the case of compression, different formats of the theoretical consolidation curves were used: U_v–dU_v/dT_v, U_v–dT_v/dU_v, U_v–log dU_v/dT_v, U_v–log dT_v/dU_v, U_v–dU_v/d√T_v, together with corresponding experimental curves: δ–dδ/dt, δ–dt/dδ, δ–log dδ/dt, δ–log dt/dδ, δ–dδ/d√t. Tewatia et al. [227] stated that many consolidation problems could be solved using RSA. Several other worth mentioning advantages of RSA are given by Tewatia [226]. Characteristics of linear and semi-log relationships can be used to determine the true value of c_v, determining how far c_v is calculated from the true c_v and for the isolation of secondary compression from the experimental consolidation curve. Moreover, semi-logarithmic plot δ–log dδ/dt can be applied to specify the characteristic feature of primary consolidation assessed by a qualitative indicator named universal constant of primary consolidation (µ). This indicator is the measure of the trueness of the c_v. It has been shown that for any soil following true Terzaghian behaviour, in the case of pure primary consolidation the maximum value of µ = 0.0802. However, if rate-dependent viscoplastic strains develop during primary consolidation, which means that creep runs concurrently with the primary consolidation, µ is less than 0.0802 [229]. It has been shown that this property can be used to detect the presence of secondary consolidation in the range of primary compression before U_v = 50%. In the case of secondary consolidation, a concept of instantaneous c_v was introduced [230]. Therefore, six consolidation phases with their quantitative beginnings and mathematical characteristics were established. For instance, Tewatia et al. [229] separated three phases of primary consolidation. The first primary phase was characterized by the slightest impact of secondary consolidation effects, and the calculated values of the c_v are the highest. After that, the transition from the first primary to the second primary phase occurs. A constant c_v value for a considerable percentage of the total settlement characterizes the second primary phase in many soils.

Basically, three methods included in the RSA are intended to analyse vertical or horizontal consolidation with fast loading. Tewatia et al. [228] observed that the subsequent load increase can be applied at 50–60% of consolidation progress for vertical consolidation. In addition, rapid loading methods can be used when the settlement or load increment time is unknown. In the first method, pattern similar to TM is used. The so-called settlement vs rate of settlement method (SRSM) is based on a fitting procedure where the later linear part of the experimental curve δ–dδ/dt is compared with the corresponding part of the theoretical U_v–dU_v/dT_v curve (see Fig. 7). The graphical procedure involves increasing the abscissa of the straight linear portion of the curve by 1.224 times. Thus the second line is drawn, which cuts the theoretical curve at 30% consolidation. The δ–dδ/dt plot is a straight line having a slope of m₂ and intercept on the δ-axis of δ₁₀₀. Therefore, the abscissa of the first line is increased by 1.224 times, and the resulting line cuts the consolidation curve at δ₃₀. In contrast to the TM, neither δ₁₀₀ nor δ₀ are required for computing c_v.

Fig. 7

Linear plots for settlement vs rate of settlement method (SRSM) and settlement vs inverse rate of settlement method (SIRSM)

The second method uses an inverse plot of U_v–dT_v/dU_v and is denoted as settlement vs inverse rate of settlement method (SIRSM). The theoretical U_v–dT_v/dU_v curve (see Fig. 7) has an initial linear portion originating from U_v = 0%. Like the TM method, a second line is drawn so that the abscissa of the first line is increased by 1.217 times, and this line cuts the theoretical curve at U_v = 70%. This corresponds to the δ₇₀ point on the experimental curve. Based on this finding, one can estimate the amount of primary compression.

The semi-logarithmic variant of SRSM was first introduced by Tewatia [225], who discussed the concept of instantaneous consolidation parameters. A basic assumption of the logarithmic settlement vs rate of settlement method (LSRSM) is to find a very short interval for which a constant value of the instantaneous c_v can be assumed that satisfies the constancy condition resulting from the theory. The instantaneous values of c_v and true c_v are quantified by µ-parameter using the U_v–dU_v/dT_v relationship. Figure 8 presents a semi-logarithmic plot for this method. It was demonstrated that theoretical and experimental rates of consolidation could be related as follows:

Fig. 8

RSA approach: a theoretical semi-logarithmic plot of U_v versus dU_v/dT_v for determining the true value of c_v; b theoretical linear plot of dU_v/dT_v versus U_v for studying the variation of instantaneous c_v due to the secondary consolidation and for isolation of secondary consolidation at the tangent point

$$\frac{d\delta }{dt}=-\frac{{c}_{v}\left({\delta }_{0}-{\delta }_{100}\right)}{{H}^{2}}\frac{d{U}_{v}}{d{T}_{v}}$$

(14)

In this case, using characteristic features of the theoretical curve (symmetrical S-shaped curve with an inflection point at U_v = 50%), initial thickness (d₀) is obtained by drawing a tangent at any point in the upper portion (0 < U_v < 40%) of the curve, and calculating its slope. Then the slope value (i.e. s₅₀) of the second tangent at the inflection point divided by 1.009 and subtracted from d₀ gives the end of primary consolidation. To determine the „true” c_v, finding the settlement rate at U_v = 16.19% is necessary. It could be done by utilising the point of the intersection of the tangent, at the point of inflection, with the horizontal d₀ line (initial thickness line). Table 3 summarises the basic characteristics of RSA methods.

Table 3 Characteristics of RSA methods

RSA, specifically LSRSM, was extensively used to evaluate consolidation test results. Unfortunately, some works appear throughout the literature without adequately referencing the original paper [225] that introduced RSA. In most cases, some adopted equations had changed notations relative to predecessors (see [6, 7, 119, 131, 238, 250, 251]).

2.4 Hyperbolic Approach

Many results of structural and mechanical tests demonstrate the hyperbolic character. The equation for the rectangular (equilateral) hyperbola was given by Ayrton and Perry [13] and later developed by Southwell [200] to explain the behaviour of eccentrically loaded thin elastic struts. Tan [214] and Chin [39] observed the hyperbolic relationship between settlement and time. Further, Chin and Vail [40] used rectangular hyperbola to approximate the settlement results of vertical piles. It has been shown that the hyperbolic approach was also suitable for diagnosing the condition of driven piles [41]. This approach had also been used earlier to model the non-linear stress–strain relationship in triaxial tests to account for the loss of shear stiffness [108, 109].

2.4.1 Rectangular Hyperbola Method

The rectangular hyperbola method (RHM) was developed by Sridharan and Rao [211] and improved by Sridharan and Prakash [206]. Here, the improved rectangular hyperbola method is denoted as IRHM. In the RHM and IRHM, the U_v–T_v relationship is approximated by a rectangular hyperbola over 60% < U_v < 90% [202]. The methods use a normalised theoretical plot of the form of T_v/U_v versus T_v in the arithmetic scales (see Fig. 9).

Fig. 9

Principles of hyperbolic approach for the consolidation analysis

Both methods, in principle, are iterative. The coefficient of consolidation and ultimate compression (EOP consolidation settlement) can be obtained from the time-compression data, plotted in the form of a curve of t/δ against t. Note that the values of t and δ preferably should be taken at equal intervals. As demonstrated by Sridharan and Rao [211] the experimental plot has an initial curved portion, followed by a well-defined straight line (60% < U_v < 90%). The equation of this linear portion is given by t/δ = mt + c. A procedure for determining c_v involves obtaining a straight line joining the origin to the various points on the linear portion of the experimental curve, which can be represented by t/δ = Amt (where A is a function of the degree of consolidation) [206]. For the computation of c_v, the slope and intercept of the straight line are necessary as follows:

$${c}_{v}=B\left(m{H}^{2}/c\right)$$

(15)

The value of B for various degrees of consolidation in the range of 60% < U_v < 90% is unique and is equal to 0.2972343 (see Fig. 10). Originally, the RHM and IRHM were designated for the soils for which the TM and CM could not be used. Moreover, the authors pointed out that the hyperbolic approach significantly improves the estimation of the 90% primary consolidation point compared to TM. This is because the slope of the U_v–T_v/U_v curve decreases more rapidly than that of the U_v–√T_v curve.

Fig. 10

Changes of A and B parameters with T_v/U_v

2.5 Excess Pore Water Pressure Dissipation Methods

According to the phenomenological observations of the consolidation, there are two strategies to ascertain the rate of the process: the strain method and the pore water pressure method. Taylor [216] has attracted the attention that the first strategy does not provide sufficient data for an accurate interpretation of consolidation behaviour and a reliable comparison between test results and theory. To overcome the unsatisfactory results, the research direction has turned to the permeability and pore water pressure measurements during consolidation testing. The permeability of soil controls the rate at which water is expelled from the pore space, while compressibility controls the evolution of excess pore water pressures. Then, a combination of two phenomena can be used to establish the rate of strain at any time and the duration of consolidation. The pore pressure in consolidation studies is typically measured at the sample’s lower end at the cell’s impermeable base. This measurement relates to the threshold pore pressure distribution at the base of the sample. Taking measures at different sensor positions allows for simultaneous records of the pore pressure at different depths. Sonpal and Katti [199] measured pore pressures at four different depths, i.e. one-quarter, one-half, three-fourths and the lower edge of the sample. Based on the pore pressure records, the dissipation curves were obtained, expressed as the ratio of the pore pressure to the stress increase, C_IL = u_b/∆σ and the T_v. The dimensionless C_IL parameter is a modification of the Skempton parameter C, referred to as the pore pressure parameter. Typical C_IL–T_v curves for one load stage (78.4 kPa) are shown in Fig. 11a and Fig. 11b. As depicted in the figures, the peak value of the pore water pressure parameter increases with the measurement depth, reaching its maximum value at the lower edge of the sample. For two sizes of the clay fraction (0.002 and 0.005 mm), there was a tendency to decrease the value of T_v corresponding to the required time of pressure mobilization with the measurement depth. The increase in the required mobilization time of the maximum value of the pore pressure in terms of the physical properties of the soil can be attributed to the increasing stiffness of the soil sample (increase in volume density and decrease in porosity).

Fig. 11

Distribution of pore pressure parameter C_IL for: a remoulded artificial soil with 0.002 mm fraction obtained during test in a large size Rowe cell; b remoulded artificial soil with 0.005 mm fraction obtained during test in large size Rowe cell (Data after Sonpal and Katti [199])

In the second half of the twentieth century, research focused on the dominant influence of the pore water pressure measuring system flexibility on the early stages of consolidation [76, 151, 239]. The results indicated the delay in pore water pressure mobilisation, e.g. time lags in pore water measurements, characterised by an increase in the pore water pressure until its stabilised maximum value was reached. The phenomenon of pore pressure increase during the early stages of consolidation shows some similarity to the Mandel-Cryer effect [128]. The Mandel-Cryer effect describes the non-monotonic evolution of the pore pressure in response to loading or changed stress conditions and is a characteristic feature of the hydraulic-mechanical coupling occurring in porous systems, which is described by the theory developed by Biot [21]. In the theory of thermoelasticity, a sudden change in pressure should generally cause a monotonic reaction: the initial pressure will gradually dissipate towards a new equilibrium state. However, in a porous medium, an increase in pressure associated with the diffusion process can be observed for some time before the overall decrease becomes dominant. Despite the time-lag effect, several studies reported contradictions in relation to the classical assumptions of instantaneous and complete transmission of an external load to the liquid phase of the sample [20, 50, 90, 114, 196, 241, 247, 249].

2.5.1 Mid-Plane Point Method

Dissipation plots are obtained from each test stage with pore water pressure measurements. Direct analysis of the dissipation curve allows for a precise definition of endpoints (0 and 100% dissipation) and to derive the time corresponding to 50% or 90% primary consolidation (t₅₀ or t₉₀) [88]. The t₅₀ point is preferable because the middle portion of the curve best fits the theoretical curve, and c_v value is less affected by secondary compression [181]. Depending on whether the 50% or the 90% consolidation is assumed the c_v is calculated from Eq. (2) using appropriate multiplying factor (i.e. T_v = 0.379 and T_v = 1.031, respectively).

2.5.2 Pore Pressure Dissipation Method

Robinson [176] investigated compression development during excess pore water pressure dissipation for determining c_v emerged earlier by Crawford [54]. The method, hereafter called the pore pressure dissipation method (PPDM), is based principally on the uniqueness of excess pore water pressure and compression using the linearity concept of consolidation characteristics. The theoretical relationship between the average degree of consolidation (U_v) and the degree of dissipation (U_ub) has been studied to recognise the development of compression during excess pore water pressure dissipation. PPDM uses U_ub–δ–√t plot to determine c_v. From the U_ub–δ relation, the beginning and end of the primary consolidation are found by extrapolating the rectilinear end segment of the curve to the intersection with the lines of U_ub = 0% and U_ub = 100%. The basis for determining the t₅₀ necessary for computing c_v was the relationship between compression and time plotted with a parabolic scale. It should be noted that Charlie [35] points out that the late time pore water pressure taken to determine c_v could overestimate the results. Moreover, it has been highlighted that the considerable time lag in the pore pressure response should be reduced by back-pressure saturation and a stiff pore water pressure measuring system. A subsequent study by Robinson [177] revealed that the secondary compression starts during the dissipation of excess pore water pressure, and its beginning strongly depends on the load increment ratio, LIR. Thus, the observed settlement is due to a combination of pore water pressure dissipation during primary consolidation and secondary compression, which was afterwards confirmed by Mesri [140].

2.5.3 Asaoka Pore Pressure Dissipation Method

Vinod and Sridharan [235] extended the conventional Asaoka [11] method to evaluate c_v and EOP consolidation from the pore water pressure data. The method is denoted as the Asaoka pore pressure dissipation method (APPDM). The original Asaoka method was intended for interpreting and extrapolating field settlement observations based on the Bayesian inference of a non-stationary stochastic process and simple graphical procedure [167]. The Asaoka method requires a time versus compression plot. Then a set of settlements δ₁, δ₂,..., δ_n, δ_n+1 at elapsed times t₁, t₂,..., t_n, t_n+1 is recorded such that t_n+1—t_n = Δt is a constant. The difference between settlement readings is relatively great for the initial readings, but the difference between readings becomes much less approaching the end of primary consolidation. The transformed data is then plotted in the form of δ_n+1 versus δ_n to produce a straight line with the slope β. By extending this straight line to intersect the 45° line through the origin of the plot, the EOP point is obtained. Vinod and Sridharan used the same graphical procedure for pore water pressure data, whereby the slope of the straight line, in this case, was signed by λ. This parameter was used to compute c_v using the following equation:

$${c}_{v}=\frac{-4{H}^{2}}{{\pi }^{2}}\frac{ln\uplambda }{\Delta t}$$

(16)

It was recommended that the pore water pressure data for U_ub > 50% be selected for calculating c_v.

Overall, according to results obtained by MPM, PPDM and APPDM, the c_v values are higher than those determined by methods utilising compression data and are practically free from secondary compression effects.

2.5.4 Determining EOP by Pore Pressures

Zeng et al. [247] studied change law in the primary consolidation time (time at EOP) determined by the pore pressure dissipation (t_p(u)) and compression (t_p(TM) and t_p(CM)) with increasing stress level. The EOP times for compression data were obtained by TM and CM. The values of t_p(u) for the 40 mm height specimens were four times smaller than those of the 20 mm height specimens and indicated that the effect of the drainage path on the primary consolidation time does not follow the Terzaghi theory. The authors observed that the primary consolidation time determined based on the compression is smaller than that specified by the pore pressure dissipation. Moreover, analysed pore pressure was not completely dissipated at the primary consolidation time determined by the TM and CM. In view of the above, several practical terms have been introduced:

• Degree of additional settlement induced by the remaining pore pressures at t_p(TM) determined by the TM (D_stp(TM) = (Δs_tpu − Δs_tp(TM))/Δs_tpu × 100%)

• Degree of additional settlement induced by the remaining pore pressures at t_p(CM) determined by the CM (D_stp(CM) = (Δs_tpu − Δs_tp(CM))/Δs_tpu × 100%)

• Degree of remaining pore pressure at t_p(TM) determined by the TM (D_utp(TM) = u_btp(TM)/Δσ′ × 100%)

• Degree of remaining pore pressure at t_p(CM) determined by the CM (D_utp(CM) = u_btp(CM)/Δσ′ × 100%)

Note the Δs_tpu, Δs_tpt and Δs_tpc mean the settlement under the step load increment Δσ′ at the consolidation time of t_pu, t_p(TM) and t_p(CM), respectively. The u_btp(TM) and u_btp(CM) represent the base pore pressure measured at t_p(TM) and t_p(CM) determined by the TM and CM, respectively. Figure 12 presents example values of the magnitude of remaining pore pressure at the primary consolidation time determined by the TM and CM. The authors concluded that the settlement determined by the TM and CM might be significantly underestimated due to the dissipation of remaining pore pressure. Similar results were obtained by Abbasput et al. [1] using the end-of-arc method (EAM). Note that the EAM EOP is directly obtained by extrapolating the time at which the excess pore pressure is dissipated to the compression records.

Fig. 12

Magnitude of remaining pore pressure at the primary consolidation time determined by the Taylor method (TM) and Casagrande method (CM) (Data after Zeng et al. [247])

The previous analysis was extended by Zeng et al. [248] and Olek [155] to account for experimental relationships correlating the degrees of consolidation determined by the compression and pore pressures. It was found that the relationship between the average degree of consolidation and the degree of dissipation is not unique. Instead, it consists of a cluster of curves depending on different step stress increments. Two key causes are found to be responsible for the substantial discrepancy between experimental and theoretical results: (1) the nonlinear development of deformation during the dissipation of excess pore pressure; (2) the degree of consolidation determined by excess pore pressure measurements lower than 100% when the degree of consolidation determined based on the compression-time curve becomes 100%. The results were also assessed by the amount of consolidation degree (100 − U_EOP) happening within the period from the end of primary consolidation based on the compression-time curve to the end of primary consolidation determined by the excess pore water pressure observations. It has been shown that this amount can also be expressed by the difference between the time at the end of primary consolidation based on the compression-time curve and the time when dissipation is fully completed.

2.6 Similarities in the Characteristic Features of GCFMs

Sridharan and Prakash [208] studied characteristic features of the theoretical and experimental curves expressed in different modes of presentation. The main feature considered was the slope (M) of the theoretical relationship between U_v and T_v. It was observed that similarity might be present in the initial stages of consolidation when comparing the slope of the U_v–√T_v curve with the slopes of log dU_v/dT_v–log T_v, U_v–T_v/U_v, and log U_v–log T_v curves. In the case of the latter stages of the consolidation, similarities may be identified in U_v–log T_v and T_v–U_v/T_v plots. Figure 12 demonstrates the ranges of U_v over which the characteristic features of different representation modes of the consolidation process exist. Considering the initial stages of consolidation, one may observe that the value of slope changes from the constant value with the following decreasing order: log U_v–log T_v up to U_v = 56% > U_v–T_v/U_v curve up to U_v = 48% > U_v–√T_v curve up to U_v = 43%. The second feature of theoretical curves worth paying attention to is the nature of the slope change in the latter portion of the curve. This is important from the perspective of determining the end of primary consolidation based on the square root of time fitting approach. Based on the inspection of the curves (see Fig. 13), it can be concluded that the precision of determining the EOP point increases with the flattening of the last section of the curve.

Fig. 13

Slopes for different modes of theoretical consolidation curves

2.7 Comparison c_v Values Determined by GCFMs

In this subsection, as a summary of the GCFMs overview, a method comparison study is attempted. Due to the lack of a “gold standard”, i.e. a reference method recognized according to current practice as the best and most reliable, the focus was on assessing the degree of agreement of the two most frequently used methods, i.e. TM, CM and other selected methods. In the case of a consolidation study, when two methods are compared, neither provides an unequivocally correct calculation of c_v and therefore, assessing the degree of agreement [75] between these methods might be helpful. A study of the mean difference and construct limits of the agreement have been performed to assess the degree of agreement between the considered methods. The limits of agreement [confidence interval (CI)] represent the range of values in which agreement between methods will lie for approximately 95% of the sample. This kind of statistical approach was first presented by Eksborg [68]; later, it was popularized by Bland and Altman [4]. In general the B&A analysis may be used to compare two “new” methods or one method against a reference standard [63]. The Bland–Altman (B&A) analysis is based on the computation of the statistical limits by using the mean and the standard deviation (SD) of the differences between two measurements or calculated variables [144]. The famous B&A scatter plot XY [the Y-axis shows the difference between the two paired values (A–B), and the X-axis represents the average of these values ((A + B)/2)] is used to check the assumption of normality of differences. The difference in values obtained with the two compared methods represents the bias as a measure of accuracy [84]. Comparison between c_v values determined by the TM and those obtained with CM was carried out using more than 500 data from different sources. For the construction of the database, search engines such as Crossref, Google Scholar, OpenAlex, Semantic Scholar, Scopus and Research Gate were used through Publish or Perish software [88]. The search comprised indexed journals, conference proceedings, thesis dissertations and symposium papers. Figure 14 depicts the summary of the statistical analysis carried out on the investigated database.

Fig. 14

Method comparisons of TM and CM: a regression analysis; B&A plot where differences are presented as units of c_v (Note: The representation of confidence interval limits for mean and agreement limits are as follows: upper CI of mean = 7.88; lower CI of mean = 4.22; upper CI of mean + 1.96SD = 50.22; lower CI of mean + 1.96SD = 43.95; upper CI of mean –1.96SD = –31.84; lower CI of mean –1.96SD = –38.11) (Data after: [6, 52, 67, 72, 93, 111, 154, 159, 179, 191, 203])

The linear regression that describes the relationship between two data sets does not give any information about their agreement; instead, it only quantifies the goodness of fit. Globally, as seen in Fig. 14a, the data is scattered around the 1:1 line in a wide range. The obtained regression equation (for all data points n = 503) in the form of c_v(TM) = 1.8998c_v(CM) − 1.5084 with R² = 0.89 and SE = 12.14 points a quite high degree to which two variables are related. Some data points lie above and below the 1:1 line, and the results indicated no definite trends in the values of c_v relative to each other determined by the two methods. In engineering practice, it is customary to assume that the c_v values obtained by the TM method are higher than those determined by the CM method. However, comparing the data for various worldwide soil types and test conditions, this statement seems to be a significant simplification.

The statistical study of the behaviours of the differences between one variable and the other was used to evaluate the agreement between the two methods. The B&A plot is shown in Fig. 14b. The mean of the differences is 6.055 units of c_v, which means that, on average, the TM determines 6.055 units more than the CM. The lack of global agreement between methods was assessed by the bias, estimated by the mean difference and the standard deviation of the differences (SD = 20.93). Based on the analysis, the upper and lower bounds (solid black lines in Fig. 14b) were respectively: mean + 1.96SD = 47.09 and mean –1.96SD = –34.98. It can be concluded that c_v values determined by CM maybe 35 units below or 47 above TM.

Further analysis was carried out for two groups of methods, taking into account the mathematical structure of the δ–t relationship and the scale of the plot. In this respect, the c_v values calculated based on TM were compared to those determined by RHM, STM, SM, MSM and OPM, while c_v values obtained by CM were confronted with those established by RHM, IPM, SRSM, RCM and OPM. Figures 15 and 16 show the resulting regression scatter plots. As can be seen, it is impossible to definitively say whether a given method always returns lower or higher values than TM or CM. Considering both graphs, it is possible to identify c_v ranges for which the c_v values determined by TM or CM are comparable to other methods. For example, for a c_v range of 0.1 to 0.3 (1 × 10⁻⁸ m²/s), TM and RHM give close results. A similar tendency was observed for the c_v range of 30–120 (1 × 10⁻⁸ m²/s) for TM and STM. In turn, for CM, similar c_v values are returned by IPM for the range of c_v 2–20 (1 × 10⁻⁸ m²/s). However, this information is of little use from a practical point of view. Among all the methods considered, IVM, VM, IPM and SM have the most similar results (see Fig. 17).

Fig. 15

Regression scatter plot comparing values of c_v determined by TM and those from RHM, STM, SM, MSM, SRSM and OPM (Data after [6, 138, 179, 191, 203])

Fig. 16

Regression scatter plot comparing values of c_v determined by CM and those from RHM, IPM, SRSM, RCM and OPM (Data after [6, 138, 179, 191, 203])

Fig. 17

Regression scatter plot comparing values of c_v determined by IVM and those from VM, IPM, RHM, CM and SM (Data after [159])

As mentioned earlier, GMFCs use different reference points on the consolidation curve for the fitting and calculating the c_v are performed. The use of each of them is also sensitive to the shape of the consolidation curve (soil type), disturbances resulting from sampling, test conditions (saturation, sample height, loading pattern) or the judgment of the interpreter. Due to the multitude of factors influencing the determination of the c_v, it isn’t easy to evaluate the graphic methods clearly. It seems that methods giving a higher c_v value should be preferred because the higher the laboratory c_v value, the closer it is to the in-situ value. As a rule, in-situ c_v values are, on average, two orders of magnitude higher than those determined on a laboratory sample [156].

3 Computational-Based Approach

Develo** reliable and robust computational technologies and mathematical procedures opens the possibility of solving optimization problems associated with interpreting the consolidation test. Regression analysis and curve-fitting are ordinarily used to obtain the approximate analytic expression of discrete data. Nowadays, the curve fitting technique is used in varying degrees in almost every geotechnical topic. Given its major importance in the context of the interpretation of laboratory tests and evaluation field works, curve fitting based on least squares delimits the degree of subjectivity. CMs have the advantage over graphical methods in that they allow automatic interpretation of the test. Such an approach is effective in conjunction with an automated data-logging system, resulting in a completely integrated system for investigating the soil behaviour subjected to the consolidation process. The possibility of continuously recording simultaneous records of compression and pore water pressure is also not without significance while evaluating soil’s behaviour [61]. Regression formulation implemented in spreadsheet programs makes interpretation less time-consuming, and the need for human intervention is kept to a minimum [16]. Moreover, the computational-based approach considers all measured points in the analysis, which provides c_v to be related to a wider range of settlement or pore water pressure than one reference point on the consolidation curve. Indeed this enables to more efficient determining the range and the end of primary consolidation. Consequently, the resulting correlations between consolidation parameters and other soil parameters could be more consistent for further constitutive modelling. This section has compiled existing previously developed CMs for determining consolidation parameters.

3.1 Scott N-Method

Scott [186] used family of U_v(T_v)/U_v(NT_v)–T_v curves for determining consolidation parameters. The author assumed N to be any number greater than one and observed that as N tends to infinity, U_v(NT_v) becomes unity, and the plot coincides with the theoretical consolidation curve. The Scott N-method (SNM) uses a definition of U_v and so-called compression ratio F(T_v) = U_v(T_v)/U_v(NT_v) for computing the theoretical values of F(T_v) for different assumed values of N and then on this basis construct nomogram of F(T_v)–N–T_v. The determination of c_v consists in calculating F(T_v) using the initial dial reading (δ₀) at time t = 0 and the dial readings δ_t and δN_t at two subsequent times t and Nt and then reading the T_v value from the nomogram. Note that the trial and error procedure for computing corrected initial dial reading was used. Shukla et al. [196] mentioned that SNM could not be used with low values of t because the U_v(T_v)/U_v(NT_v)–T_v curve is relatively flat, making defining T_v difficult. They also pointed out that sufficient care is needed in selecting the two times t and N_t required in the procedure.

3.2 Analytical Method

Sridharan et al. [210] showed that the theoretical consolidation rate could be expressed in the form of log₁₀H²/t–U_v curve for the known value of c_v. The analytical method, abbreviated herein as (AM), was intended to compare the experimental and theoretical consolidation rates and verify the validity of proposed c_v values, estimated using different methods. Moreover, AM can be used for assessing the impact of initial and secondary compression on the time rate of consolidation. The procedure uses a single diagram with experimental values, represented by δ–t measurement coordinates in relation to model log₁₀(H²/t_theor)–U_v curves. By superimposing experimental log₁₀(H²/t_exp)–U_v curves and model log₁₀(H²/t_theor)–U_v curves, established for the various c_v and correspondent time values, a direct check of compliance with the theory is possible. The theoretical time (t_theor) required in the procedure is calculated from the following equation:

$${t}_{theor}=\frac{{T}_{v}{H}^{2}}{{c}_{v}}$$

(17)

It was shown that experimental data uncorrected for initial and secondary compression effects match well with theory in the range of 40% < U_v < 60%. Subsequently, Sridharan [209] used a simple one-point method (OPM) to determine c_v. The compression corresponding to the end of the loading period was considered as so-called final compression (δ_final). Based on δ_final, the compression corresponding to 50% of consolidation (i.e. δ₅₀, = 0.50δ_final) along with the corresponding time, t₅₀, were used to calculate c_v. The other possibility is that the several triplets (i.e. δ, t, T_v) can be taken for calculating c_v values in the range 40% < U_v < 60% and then averaged them. Cortellazzo [52] noticed that AM could be a preferable method for comparing laboratory c_v values with those determined by in situ measured settlement. Hence, it is possible to estimate the c_v of the whole layer in the field. Figure 18 shows a regression scatter plot comparing c_v values calculated by OPM with those determined by TM and CM. As depicted, the c_v from OPM is within the range of that produced by TM and CM, whereby the vast majority of points lie around the 1:1 line. Similar results were observed when comparing other CMs, such as VAM, LRM or FMSRM, with the results from the two most commonly used methods.

Fig. 18

Regression scatter plot comparing values of c_v determined by OPM and those from TM and CM (Data after: [52, 166, 191])

3.3 The Least Squares-Based Approach

The least squares-based approach is a class of calculational methods to solve the problem of finding a line or curve that best fits a set of data points. The concept of the least squares method is commonly used in various branches of engineering to find the best fit for a set of data points by minimising the variance (the sum of squares of the errors) between the actual values of the dependent variable and the predicted values. By itself, the least squares method is part of a more complex concept of inverse modelling. For an uncomplicated Terzaghi’s model, inverse analysis is based on optimisation iterative algorithms such as the gradient method [118]. As stated by Calvello and Finno [33], the gradient method is a parameter identification technique based on calibration by iteratively changing input values until the simulated output values match the observed data. The fitness between measurements and the associated numerical result is assessed by the well-defined error function (a statistical measure of discrepancy that is minimised).

To overcome the most negative aspects of GCFMs, i.e. the judgement in selecting points on the consolidation curve for establishing 0% and 100% of primary consolidation, the exact analytical solution of the Terzaghi consolidation equation can be used for the determination of c_v from the results of a consolidation test using the least squares error adjustment procedure [37]. This approach generally refers to utilising a triplet of data points (i.e. d₀, d₁₀₀ or in the equivalent notation δ₀, δ₁₀₀ and c_v) by directly matching them with the theory; hence, methods in this group could also be called 3-point matching methods (or three-variable matching methods) [187].

One of the most traditional published paper of relevance concerning the application of the least squares-based approach to determine consolidation parameters was developed by McNulty et al. [132]. Although such an idea and principles of the method used has almost 75 years old [146]. The authors have written a computer program in FORTRAN to precise determine primary consolidation based on compression measurements. Originally the method is called as an iterative method of successive approximations (IMSA). The Terzaghi’s solution to the differential equation was plotted with the natural logarithm of (1–U_v) versus T_v, then a straight line was obtained for the portion of the curve lying between 60 and 80% of consolidation (0.2 < 1–U_v < 0.4) [133]. It was shown that the d₀ and d₁₀₀ could be assumed and then corrected by an iterative process of successive approximations until a sufficient match is obtained. In like manner, they concentrated on the advancement in automation of TM and CM and statistical procedures for evaluating the representativeness of the sample data groups.

More recently, Robinson and Allam [178] expressed the experimental time-compression data using initial compression (δ_i) and the definition of an average of consolidation U_v. In their case, a minimum of three time-compression data points are needed to evaluate three unknowns (δ_i, δ₁₀₀ and c_v) by solving three equations using the least-squares method. Chan [36] and El-Gehani [69] also adopted the least-squares approach to the primary consolidation settlement for c_v determining. The issue of using an appropriate approximation to the exact analytical solution of the consolidation equation was also raised. From the discussion, it is concluded that Fox’s approximation [74] is significantly more accurate than the Hansen approximation [87], and furthermore, a more accurate representation of Fox’s approximation was proposed. Besides, Day and Morris [66] used the classical Fourier series solution and directly solved the least-squares minimisation problem using a weighted objective error function.

3.3.1 Variance Method

Another three-variable matching method called the variance method (VAM) was developed by Lovisa [120] and Lovisa and Sivakugan [121]. They were concerned with evaluating the end of primary consolidation point and took advantage of the matrix manipulation capabilities of MATLAB combined with the trial-and-error approach. The four-step procedure includes: (1) establishing the limits for wide range of possible values for d₁₀₀ selection (first N-point array); (2) converting the settlement data to percentage consolidation for each selected value of d₁₀₀ (matrix of total number of data points xN values for experimental U_v); (3) determining the value of T_v from Terzaghi’s U_v–T_v curve that corresponded to each value of experimental U_v (second array of U_v–T_v values); (4) selecting appropriate value of T_v by matching calculated theoretical U_v to the particular value of experimental one. The VAM considered the calculation of instantaneous c_v for each data point using each previously determined T_v. The “true” end of primary consolidation is chosen as one that produced the least variance in c_v, and the “true” c_v is taken by averaging the entire array of c_v that corresponded to that particular value of d₁₀₀ [120].

3.3.2 Least Residuals Method

Likewise, Sebai and Belkacemi [187], have described a CM based on a combination of the probabilistic method and least residuals method. Originally the procedure is denoted as least residuals method (LRM). The key point of the LRM is defining probable ranges of values with uniform probability distributions for d₀ and d₁₀₀ and further extracting a probable range of c_v values. Then probable range is forced to minimisation with predefined objective function. The computer program used for the analysis was written in FORTRAN. The tool utilised a loop function to select in random manner values of δ₀, δ₁₀₀ and c_v and to compute the objective function (error norm) value for each triplet.

3.3.3 Full-Matching Methods

Chung et al. [49] introduced a full-match method denoted herein as full-match settlement rate method (FMSRM) for determining c_v in which all the recorded compression data were utilised for matching with the theoretical solution. The theoretical time-settlement relationship was partially illustrated by using the RSA (i.e., the secant slope U_v/T_v). The authors found that the relationship between U_v/T_v and U_v^1/2 has two rapidly changed curvatures and a linear portion expressed by the intercept, slope, and so-called upper-bounded U_v value. When this relationship is multiplied by the factor M₁ (i.e., M₁ = 0.02, 0.1, 0.5, 1), the upper-bounded U_v does not change its value, and this feature can be used for establishing EOP. Thus, EOP settlement in the FMSRM is determined based on the variations in the U_v/T_v–U_v^1/2 and δ/t–δ^1/2 graphs. In turn, the c_v value was extracted from back-analysed δ–t and δ/t–δ^1/2 curves using a tool “Logic” in the Function Library of Microsoft Excel Solver.

Independently, Chow et al. [41] and Olek [155] presented another full-matching method, denoted as the gradient method (GM) and applied it to the compression and pore water pressure data for which the value of c_v has been directly obtained based on inverse modelling. In the first case, the authors used the theoretical expression for excess pore water pressure, and in the second, the model caller was the degree of dissipation. However, regardless of the model’s caller, the only unknown was the c_v among all the variables in the equation; therefore, the optimisation problem was a mono-objective. This way, the c_v with the lowest error function value was selected and appraised as the optimal value for experimental results. In the GM, the average difference between the recorded and the simulated results was expressed using the least square method. Basically, FMSRM and GM are very similar to the full-matching piezocone methods for determining the radial coefficient of consolidation. The difference comes from the formulation of an error function and the model caller used. The reader is referred to [28, 47, 48, 96, 102, 110] for more details.

3.4 Checking Quasi-constant c_v Criterion

One of Terzaghi’s theory assumptions is that the coefficient of consolidation is constant during the primary consolidation stage. Tewatia [231] came up with an idea concerning the criterion of checking the compliance of the process with the theoretical assumptions by testing the quasi-constant value of c_v as a function of the degree of consolidation of U_v. It has been supposed that any experimental δ–t point on the consolidation curve could be converted into the degree of consolidation U_v, and the corresponding time factor T_v divided by t/H² indicated c_v. Depending on the distribution of the initial excess pore water pressure in the axial cross-section of the consolidating sample, the U_v–T_v relation takes a different course [217]. Dobak and Pająk [62] pointed out that in the advanced stages of consolidation, where significant excess has been dissipated, the actual excess distribution is parabolic or triangular rather than rectangular. The wrong assumption takes the effect of c_v underestimation. This results in an extension of the predicted soil settlement time.

In summary, the initial excess pore water pressure distribution selection influences the value of T_v and, hence, the value of c_v. Wherever the plotted c_v–U_v curve is horizontal, the constancy criterion is fulfilled, and wherever the curve exhibits any slope, the c_v value is no more constant, and the effects of initial and secondary compression on the value of U_v become more apparent. Based on the consistent research conducted by Dobak [60], Dobak and Pająk [62], Majer and Białobrzeski [127], Olek and Woźniak [156], Olek [153], Pająk and Dobak [158], Woźniak et. al. [242] it was confirmed that the effect of initial compression increases c_v values while the impact of secondary compression decreases c_v values.

3.5 Evaluation Full-Matching Methods Treated by Least Squares

In the present review, full-matching CMs for the consolidation analysis have been described. Based on the comprehensive literature survey provided above, the main similarities for included methods in this group, along with their advantages and drawbacks, might be drawn. The mathematical optimisation embodied in consolidation studies refers to determining the parameter values that best fit the experimental data. Each of the methods, in principle, is based on the back-calculation procedure, in which, depending on an inventor, the triplet of δ₀, δ_EOP, c_v or single target c_v are to be optimised. Therefore, these methods should give practically similar values of the consolidation parameters. However, in reality, the parameter values determined by these methods differ slightly from each other. The reason behind this situation lies in the type of objective function used to be optimised. Table 4 shows the comparison between methods for directly checking the compliance between experimental results and unique theoretical solution based on the model caller, an objective function representing the goodness of the fit and the matching range. An undeniable advantage of methods based on optimization is the reduction of judgment in the interpretation of results to a minimum. Note that the optimisation accuracy depends on a number of measurement points, the scale effects on the fitness between the experimental and the simulated results, and the mathematical model that describes the modelled phenomenon. Therefore, these methods are most effective when interpreting consolidation curves composed of many measurement points (δ–t). The considerable amount of measurement points also makes it easier to convert primary data for differential methods such as RHM or RSA. Based on previous discussions and comparisons of individual GCFMs and CMs, a combined interpretive approach for laboratory consolidation testing can be proposed. Such an approach should take into account the possibility of examining a convergence between theoretical and experimental consolidation curves and a simple check of the assumptions of the consolidation model used, which in the case of the present consideration is the constancy of c_v, that should be valid throughout the entire pore water pressure dissipation process. The description of the recommended procedure can be reduced to writing down the following seven steps:

• 1. Plot the consolidation data in H²/t–U_v space. In this way, it is possible to plot a family of theoretical curves superimposed on an experimental curve. This allows us to compare different interpretation methods because each theoretical curve has assigned a specific value of c_v.

• 2. For each pair of (δ–t) points on consolidation curve calculate average degree of consolidation (U_v) and corresponding values of time factor (T_v) together with instantaneous c_v. At this stage it is possible to apply OPM for quick determining c_v.

• 3. Check quasi-constant criterion using instantaneous c_v–U_v relationship. Wherever the plotted log c_v–U_v curve is horizontal, the constancy criterion is fulfilled, and a range of primary consolidation is established (say quasi-filtration phase of consolidation).

• 4. Take the mean value of instantaneous c_vs from the primary consolidation range as a target value for optimal c_v.

• 5. Use any full-matching method based on optimisation procedure (i.e. VAM, LRM, FMSRM or GM) to find optimal value of c_v. The optimal value of c_v is evaluate by analysing the topology of error function (J). Since the J is minimised, the smaller values are better.

• 6. Compare the value of c_v from step 4 with those determined in step 5.

• 7. Follow again steps 1–6 for the pore water pressure data.

Table 4 Summary of the different methods for direct check the compliance between experimental results and theoretical solution

As an example of presenting the results from the Rowe cell test [89, 182], the record of the consolidation course for remoulded marine clay from the Polish Carpathian Foredeep was used. The tested soil had the following physical properties: specific gravity (G_s) = 2.72 (–), water content (w = LL) = 65 (%), plasticity index (PI) = 40 (%) and clay fraction (CF) = 58 (%) with main clay mineral as kaolinite. The results refer to one individual load increase in the rapid incremental loading test (EOP test) with the following conditions of the experiment: consolidation pressure (σ_c) = 400 kPa, load increment ratio (LIR) = 1, one-way drainage conditions with permeable top boundary and impermeable bottom boundary (PTIB). Figure 19 depicts the plots necessary for interpretation and a schematic of the optimisation technique based on GM. As can be seen, both plots directly provide an indication of the rate of consolidation progress as assessed from the compression records and the pore pressure measured at the base of the sample. This makes it possible to determine the predominance of a given factor that governs the consolidation process (compression/structure or dissipation factor). For example, for quantitative influence evaluation of straining and permeability in the consolidation process, Dobak and Gaszyński [61] introduced the parameter (η) defined as:

Fig. 19

Consolidation analysis by the combined approach (Note For compression and pore water pressure data instantaneous c_v were calculated based on uniform and sinusoidal initial pore water pressure distribution u_i, respectively. Based on the c_v value determined by GM, a specific range of acceptable c_v values has been established, beyond which the results may be rejected as having too much error. The range limits are the lower (b_min) and higher (b_max) values of c_v, with a fixed confidence level of 95%

$$\eta =\frac{{{c}_{v\left(compression\right)}-c}_{v\left(dissipation\right)}}{{c}_{v\left(dissipation\right)}}$$

(18)

Note the value η = 0 means complete model compliance of consolidation progress characteristics analysed based on the compression and pore pressure dissipation. Positive values of η point to the delay of the pore pressure dissipation process in relation to the compression, whereas negative values of η indicate a more effortless and faster pore pressure dissipation than compression progress. The first case commonly occurs at the initial load stages during the intensive rebuilding of a solid skeleton. In turn, the second case might be connected with a decreasing pore pressure contribution in load transfer.

4 Correlational-Based Approach

It may be seen from the literature that many engineering properties are correlated with index properties. Particularly noteworthy are correlations linking Atterberg limits with other geotechnical properties [43, 59, 64, 65, 107, 134, 145, 201, 202, 243, 244]. Shimobe and Spagnoli [194] emphasized that fines content (FC), as well as clay mineralogy, contribute to the liquid limit (LL), plastic limit (PL), specific surface area (SSA), cation exchange capacity (CEC), amount of water retained and indirectly the shear strength of the soil. They considered over 1000 data from various sources and correlated the plasticity ratio of clays (R_p), defined as the ratio of plastic limit to liquid limit to the clay fraction of clays, and the liquidity index (LI). Subsequently, they provided a general overview of the intercorrelation among the engineering properties of clays. Similarly, the same scholars presented an overview of the correlation of the compression index (C_c) with index properties [195]. Sharma and Bora [192] presented a study on the correlation between Atterberg limits and some consolidation properties of fine-grained soils. Feng and Vardanega [73] investigated the correlation of the vertical hydraulic conductivity (k_v) with the water content ratio (WCR [113]). Over the past decades, attempts have also been made to predict c_v based on parameters derived from index tests [221]. Table 5 summarizes the findings of these studies conducted on various worldwide soils. It should be noted that the above-mentioned correlational equations for the c_v can only be applied in specific soil conditions and only for the limited soil types or regions of the occurrence.

Table 5 Summary of the different correlational equations for coefficient of consolidation

5 Soft Computing Approach

With the acceleration of industrialisation and urbanisation, Artificial Intelligence (AI) or Machine Learning (ML) methods, known as soft computing approaches, have been developed and applied to solve various technological and scientific problems introduced to the industry by Revolution 4.0. ML (a subset of AI) is a technique that allows computer systems and smart devices to autonomously learn from the environment around them as they perform tasks, collect data through measurements, and analyse and interpret it [79]. The soft computing approach assumes that the system can change its own behaviour through algorithms and logical rules to optimise the functioning and operation of a given task [250]. This approach makes it possible to analyse large amounts of information (Big Data) to determine patterns and capture the potential correlations among information without any prior assumptions [78, 233]. Moreover, soft computing can be understood as an evolutionary form based on AI, which integrates the sub-domains of Artificial Neural Networks (ANN), Genetic Algorithms (GA), Evolutionary Computation (EC) and Fuzzy Logic (FL). ANNs [129] are commonly referred to as nonlinear statistical data models that replicate the role of central nervous systems of biological neural networks (NNs) [98, 130]. ANN can be described as a set of interconnected nodes used to solve modelling problems with complex relationships between determinants and responses. ANNs consist of vertically stacked components called layers. ANN is an ML method that uses past data to predict future trends, while GA is an algorithm that can find better subsets of input variables for importing into ANN, enabling more accurate prediction by its efficient feature selection [24, 252]. Another family of algorithms for global optimization and continuous optimisation inspired by the biological process is EC. On the other hand, FL is a reasoning method based on the Fuzzy set theory developed by Zadeh [246]. It is similar to human reasoning and decision-making, which allows making definite decisions based on imprecise or ambiguous data. FL makes decisions based on the raw and ambiguous data given to it, whereas ANN tries to incorporate the human thinking process to solve problems without mathematically modelling them. In essence, FL deals with fixed and approximate reasoning, taking the values of variables from 0 to 1 that contradict the traditional binary sets, which take the value of either 1 or 0 and since it can take any value in the range 0 to 1 [83].

In the previous section, many empirical equations correlating the consolidation properties to some basic physical properties have been compiled. However, when the relationships between them are highly nonlinear, the empirical approach becomes impractical or burdensome to interpret, and the techniques incorporating AI are more suitable [103]. Among them, inverse analysis using optimisation methods constitutes an efficient parameter identification method [245].

5.1 Machine Learning Methods

There is an increasing interest in applying ML in soil consolidation studies. Here a systematic review of selected concepts in ML and practical optimisation techniques with representative application examples concerning the prediction of c_v value has been presented.

Zhu [253] proposed a multi-layer error back-propagation (MLEBP) feed-forward neural network model in which the initial void ratio, the preconsolidation pressure, and the consolidation pressure were chosen as the inputs to predict c_v. Bao and Liu [14] used the Updated Hybrid Simple Genetic Algorithm (UHSGA) to evaluate c_v and pointed out that GA can effectively eliminate human judgement and improves the speed and accuracy of the calculation.

Sharma et al. [193] adopted Multi-Gene Genetic Programming (MGGP), a distinct form of an artificial neural model for predicting c_v, and highlighted that integration of nodes in the form of tree structures which is also called GP tree, may develop a compact and explicit prediction equation in terms of the model variable with optimised coefficients. The main advantage of the developed MGGP model was the possibility of updating to obtain the equation for different soil by presenting new training examples as new data become available.

Nguyen et al. [149] developed an AI approach based on the random forest (RF [25] method for the prediction of c_v. RF algorithm consists of many decision trees [91] (i.e. a decision support tool that forms a tree-like structure), and the so-called “forest” is trained through bagging or bootstrap aggregating [90]. The model was built around plasticity and consistency parameters like clay content (%), moisture content (%), liquid limit (%), plastic limit (%), plasticity index (%), and liquidity index (%). One hundred sixty-three soil samples originating from Vietnam have been considered for generating datasets and modelling. Eight different models were analysed based on the various combinations of input variables and used to predict the c_v. Each model showed good predictive capabilities; however, researchers stressed that care has to be taken and that model input parameters should be based on the soil condition of similar lithology of the study area.

Pham et al. [165] determined the value of c_v from index soil properties employing a hybrid machine learning model based on a multi-layer perceptron neural network and biogeography-based Optimisation (MLP-BBO). The MLP that constitutes the addendum of the feed-forward neural network, composed of the layered structure, has been utilised to predict coefficients of consolidation. On the other hand, the BBO was used to optimise the weights of the MLP. For the building of the MLP-BBO model, a database consisting of 164 soil samples collected from the two highway project sites in Vietnam was prepared. The input parameters used for the modelling included clay fraction (%), moisture content (%), liquid limit (%), plastic limit (%), plasticity index (%), and liquidity index (–). The analysis presented by Pham et al. is valuable in terms of cognitive because the prediction accuracy and applicability of the MLP-BBO model were compared with other AI methods. For the comparison following methods were considered: back-propagation multi-layer perceptron neural networks (Bp-MLP Neural Nets), radial basis functions neural networks (RBF-Neural Nets), Gaussian process (GP), M5 tree and support vector regression (SVR). The results showed that MLP-BBO had the highest predictive capability among the considered methods, whereby other methods also gave good results.

Subsequently, Pham et al. [164] used 188 experimental results to construct the datasets for the Artificial Neural Network (ANN) model. The database consists of 13 input variables (i.e. depth of samples (m), clay content (%), water content (%), bulk density (g/cm³), dry density (g/cm³), specific gravity, void ratio (-), porosity (%), degree of saturation (%), liquid limit (%), plasticity index (%), liquidity index (–)), and the prediction target was c_v. The results derived from 600 simulations showed that the ANN structure containing 14 neurons in the hidden layer was superior to the ANN with 26 neurons in the hidden layer. The prediction capability of the ANN algorithm was assessed using root-mean-square error (RMSE) and mean absolute error (MAE). The results proved that the ANN model could be a good predictor of c_v, with satisfying accuracy performance.

Nguyen et al. [148] applied various soft computing approaches, namely biogeography-based optimisation based artificial neural networks (ANN-BBO), artificial neural networks (ANN), adaptive network-based fuzzy inference system (ANFIS), and support vector machines (SVM) for c_v prediction. For the predicting of c_v, several relevant input parameters were exploited [i.e. depth of the sample, clay content (%), specific gravity (–), degree of saturation (%), plastic limit (%), plasticity index (%) and liquidity index (–)]. The authors concluded that the investigated AI models performed well, but the performance of the ANN-BBO model was the best in predicting the c_v of soil compared with other studied models.

Mittal et al. [143] proposed an AI-based approach to predict the c_v using various ML models [i.e. Multiple Linear Regression (MLR), ANN, Support Vector Regression (SVR), Adaptive Network-based Fuzzy Inference System (ANFIS)]. Three feature selection techniques have been included in the proposed methodology to investigate dimensionality reduction. As a rule, dimensionality reduction relies on feature selection and rejection of features that are overly correlated with each other. For example, the authors used the least absolute shrinkage and selection operator (LASSO) to rank feature importance and to improve model accuracy. The support vector machine-recursive feature elimination (SVM-RFE) and mutual information (MI) were used to evaluate feature's importance and check mutual dependence between variables. Experiments are performed on the dataset collected on the 534 soil samples from Vietnam. It has been observed that the predictive ability of the ANFIS model combined with the LASSO feature selection method was higher than other models.

Ly et al. [126] also focused on the dimensionality reduction issue. They proposed a hybrid model of random forest coupling with a relief algorithm (RF-RL) to predict the c_v. The relief approach used by the authors evaluated the feature’s importance and minimised input space. Based on sensitivity analysis, liquid limit (%), water content (%), liquidity index (–) and plasticity index (%) were the most significant parameters affecting c_v. Consequently, simulations were based on a database that consisted of 188 soil samples from Vietnam. The results revealed that all constructed models achieved high performance, and the RF-RL model with 13 variables has the highest prediction accuracy.

Nowadays, with the advent of the big data era, the perspectives of research tend to replace obsolete theoretical solutions with soft computing methods to solve complex geotechnical issues. From this point of view, determining soil parameters like c_v using algorithms and AI will gain an increasingly clear advantage over conventional methods, especially those based on time-consuming graphical procedures. Soft computing methodologies are tolerant to imprecision and vagueness, allowing solving problems with an element of uncertainty and can derive approximate solutions to problems. Furthermore, soft computing effectively solves nonlinear problems described by advanced constitutive models, which better characterise soil behaviour than simple Terzaghi’s model. This involves solving complex sets of differential equations using advanced physics-informed deep-learning techniques.

5.2 Physics-Informed Deep Learning

The new learning philosophy, such as physics-informed deep learning, exhibits a growing interest in varied applied domains. This type of learning allows training neural networks to solve learning tasks under the presence of a supervisor with respecting any law of physics described by partial differential equations (PDEs) [104, 105]. In other words, physics-informed neural networks (PINNs) is ML technique used to solve problems involving PDEs using deep neural networks. The effect of applying the PINNs approach is the approximation of the PDE solution by training the neural network to minimise the loss function. The loss function may contain, for instance, information about the initial and boundary conditions and PDE residual term, which behave as a physics-informed regularisation. Bearing in mind that PINNs are global function approximators, their use in solving complex tasks and describing various physical processes will be more and more common. Such techniques have been used and will probably continue to be developed in issues related to soil consolidation.

Bekele [17] examined a deep-learning model based on PINNs for one-dimensional consolidation. In his work, the author came from the general assumption that there are two types of problems associated with PDEs, i.e. forward problems and inverse problems. In the first case, the effects are calculated based on causes. At the same time, the inverse problem concerns the cause calculated based on the effect, i.e. the model parameters must be determined on the basis of the observed values. Thus, the inverse problem refers to the prediction of the c_v. In the context of issues studied, the deep learning model has been built as a fully-connected sequential neural network. Automatic differentiation implemented in the ML library TensorFlow was used to determine the partial derivatives in the governing equation. The simulations showed remarkable prediction accuracy of the model for both forward and inverse problems, irrespective of boundary conditions, i.e. permeable top, impermeable bottom (PTIB) and permeable top and bottom (PTPB).

6 Conclusions

The issue concerned with the consolidation behaviour of fine-grained soils has aroused great interest in the last hundred years. This has resulted in the development of multiple interpretation approaches for laboratory consolidation tests. The main contribution of this article is to present a detailed and updated state-of-the-art review concerning determining consolidation parameters like the c_v, primary consolidation range, and EOP consolidation, as well as to provide an overview of the potential new techniques based on soft computing approaches. Through the analysis of a substantial volume of work conducted in this domain, interpretation approaches used in connection with the prediction of the time rate of settlement can roughly be categorised as one of the two following types: (1) GCFMs and (2) CMs. The categorisation of GCFMs was based on two criteria: mode of consolidation curve according to the mathematical nature of the relationship used for analysis and scale of the plot (i.e., linear, logarithmic). In turn, CMs consist of FMM, correlational equations linking c_v with other geotechnical parameters, and methods based on a soft computing approach. Despite the development of various GCFMs, there is no clear consensus on which one should be the standard. For historical reasons, when proposing new interpretation solutions, comparing the values of c_v or compression and time at EOP with those defined by TM and CM has become a practice. In some cases, the proposed methods give similar values of consolidation parameters, as presented in this paper. However, the discrepancies between the TM and CM values can be significant, making them difficult to use as references. Moreover, two factors affect the determination of c_v: the magnitudes of initial and secondary compressions and the instant at which the secondary consolidation starts. The methods that depend on the initial stages of consolidation for curve fitting give higher values of c_v due to initial compression effects, and the methods that rely on the later stages of consolidation, which are dominated by the secondary compression effect, give lower values of c_v. Comparison of the c_v values obtained in the laboratory with those determined in field conditions indicates getting lower values for the former. Therefore, reasonable c_v values should refer to methods that give higher c_v values. As a rule, higher c_v values are obtained using FMMs compared with those determined by GCFMs.

Methods based on pore pressure measurements come in handy here. The advantage of the pore water pressure analysis is the direct reference to the partial differential consolidation equation, which describes the excess pore water pressure dissipation and does not need to consider secondary compression effects. Moreover, EOP consolidation can be clearly marked by inspecting the dissipation curve. An analysis of instantaneous consolidation parameters calculated on the basis on compression or excess pore water pressure may be helpful too.

It should also be remembered that some methods should be used following their original purpose. Examples might include rapid tests and RSA. In other cases, the shape of the consolidation curve makes it impossible to use some methods. A certain consensus might be using CMs based on the full-matching approach and least-squares. These methods are fast, easy to implement, do not require interpreter judgment, and constitute a unified computational approach based on minimising the error function characterising the discrepancy between the experimental data and the model solution.

Another, and as it seems, more adequate direction of determining geotechnical soil parameters is soft computing, which is now being used as an alternative to the deterministic approach. In the last decade, there has been an increase in available soil data, which opens the door to the efficient adoption of ML techniques to train new models. These methods successfully replace empirical correlations derived with the help of statistical methods and are commonly used to evaluate various engineering properties of soils. AI-supported modelling does not need prior information about the nature of the relationships among the data, which is in contrast to traditional empirical and statistical methods. However, these models should be used cautiously because their input parameters refer to specific soil conditions of similar geoenvironmental impact, geological origin and lithology of the study area. Therefore, as part of the future development and faster dissemination of DL techniques, it is proposed to create a networked knowledge exchange platform to collect data for various world fine-grained soils. On the one hand, this would enable further validation of the existing models and, on the other hand, invent new ones considering different international locations with distinct soil types. Such an idea would bring scientists, project contractors and working engineers closer, consolidating the geotechnical community and diffusing new developments in CMs.