1 Introduction

Soil compaction is routinely employed, often in conjunction with supplementary soil stabilization techniques, to satisfy earthworks construction requirements (e.g., Attom 1997; Benson et al. 1999; Sridharan and Gurtug 2004; Zhang et al. 2019; Soltani et al. 2022). The compaction characteristics are commonly measured in accordance with standardized soil laboratory tests. The prevalent methods include the standard and modified Proctor (SP and MP) compaction tests, which measure changes in the soil dry unit weight \({\gamma }_{\text{d}}\) with increasing moisture content w for defined compaction energy levels (CELs) of E = ∼600 and ∼2700 kJ/m3, respectively (e.g., BS 1377–4, 1990; AS 1289.5.1.1 2017; AS 1289.5.2.1 2017; ASTM D698 2021; ASTM D1557 2021). From the measured \(\gamma_{\text{d}}\):w plot, the optimum moisture content (OMC or wopt) and corresponding maximum dry unit weight (MDUW or \({\upgamma }_{\text{dmax}}\)) parameters, defined by the peak point of the said plot, can be visually deduced. Because of the time and labor demands associated with conventional/standardized laboratory compaction testing of fine-grained soils, especially for CELs > SP compactive effort, several attempts have been made to either (i) devise alternative compaction testing techniques or (ii) indirectly estimate the optimum compaction parameters through empirical-type correlations. The former mainly involves the application of miniature compaction testing equipment, thereby significantly reducing the amount of soil material needed for, and also alleviating the time and labor demands associated with, quantifying the OMC and MDUW parameters (e.g., Little 1948; Wilson 1950; Singh and Punmia 1965; Sridharan and Sivapullaiah 2005). In spite of generally promising results, to date, none of these alternate compaction testing methods has been standardized, nor have they gained widespread acceptance among practicing geotechnical engineers, likely owing to their relatively lower accuracy in replicating soil compactability for CELs > SP compactive effort (Santos et al. 2019; Loshelder et al. 2023).

Empirical-type correlations generally aim to eliminate the need for performing compaction testing, instead relying on data-driven mathematical expressions that link the OMC and MDUW parameters to common soil index properties, such as the Atterberg limits and grain-size distribution parameters. For a complete review of these empirical-type correlations, the reader is referred to the papers by Verma and Kumar (2020) and Shimobe et al. (2021). Because of the intricate nature of the compaction process, being governed by several inter-related soil attributes (e.g., gradation, plasticity, clay mineralogy and the soil’s chemical composition (Howard et al. 1981; Horpibulsuk et al. 2008)), efforts to establish a universally applicable empirical model capable of consistently producing reliable OMC and MDUW predictions (across rational CELs) based on common soil index properties still remain somewhat inconclusive. In other words, most existing empirical models have been observed to be overly dependent on the specific ranges of soil index properties used for their development, with their predictive performance often diminishing when applied outside of their original calibration domains (Shimobe et al. 2021; Di Matteo and Spagnoli 2023). Moreover, for cases where the MDUW and/or OMC parameters predicted by empirical correlations are overestimated, the theoretically deduced degree of saturation for the \({\gamma}_{\text{dmax}}\):wopt prediction often surpasses the physically limiting zero-air-voids (ZAV) saturation line (i.e., implying Sopt ≥ 100%), thus producing a materially meaningless optimum compaction state (Soltani et al. 2023).

The limitations described above have produced renewed interest in energy conversion (EC) models, defined as a special/separate category of empirical-type correlations that work on the premise of employing measured OMC and MDUW values obtained for a rational CEL (mainly SP) to predict the same for other lower and/or higher compactive efforts (e.g., Hamdani 1987; Blotz et al. 1998; Drew 2005; Horpibulsuk et al. 2008; Vinod et al. 2015; Gurtug et al. 2018; Khalid and Rehman 2018; Shaivan and Sridharan 2020; Shivaprakash and Sridharan 2021; Di Matteo et al. 2022; Di Matteo and Spagnoli 2023; Soltani et al. 2023). In other words, EC-type models eliminate the need to perform additional compaction tests that would otherwise be required to cover a desired/wide CEL range. In general, EC-type models can be classified into two groups, namely (i) aided [energy] conversion (AC) and (ii) direct [energy] conversion (DC) approaches. In addition to the measured OMC and MDUW parameters for a rational CEL, AC-based models also employ soil index properties (mainly the Atterberg limits), as supplementary inputs, to estimate the optimum compaction parameters for other lower and/or higher CELs. On the other hand, DC models postulate that the relationship between the optimum compaction parameters of fine-grained soils across rational CELs is somewhat unique, and hence, reliable conversions of the OMC and MDUW parameters (from one CEL to another) can be directly achieved without the need of acquiring any soil index properties. In fact, the authors’ recent findings, originally presented by them in the discussion paper of Di Matteo et al. (2022) and later substantiated in Soltani et al. (2023), have demonstrated that incorporating soil index properties (as inputs) into EC-based models has the general effect of restricting their predictive performance, causing similar limitations as those described above for conventional empirical-type correlations. The latest DC-based EC modeling framework, as proposed by the authors of the present study, can be expressed along these lines (Soltani et al. 2023):

$${w}_{{\text{opt}}}^{{\text{R}}}={w}_{{\text{opt}}}^{{\text{SP}}} {\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{SP}}}}\right)}^{-0.178} \ni {\text{R}}=\left\{{\text{RSP}},{\text{RMP}},{\text{MP}}\right\}$$
(1)
$${\gamma }_{{\text{dmax}}}^{{\text{R}}}={\gamma }_{{\text{dmax}}}^{{\text{SP}}} {\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{SP}}}}\right)}^{+0.068} \ni {\text{R}}=\left\{{\text{RSP}},{\text{RMP}},{\text{MP}}\right\}$$
(2)
$${\gamma }_{{\text{dmax}}}^{{\text{R}}}=\frac{{G}_{{\text{s}}}{\gamma }_{{\text{w}}}}{1+\left(\frac{{G}_{{\text{s}}}{\gamma }_{{\text{w}}}}{{\gamma }_{{\text{dmax}}}^{{\text{SP}}}}-1\right){\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{SP}}}}\right)}^{-0.178}}\ni {\text{R}}=\left\{{\text{RSP}},{\text{RMP}},{\text{MP}}\right\}$$
(3)

where \({{w}}_{\text{opt}}^{\text{SP}}\) and \({{w}}_{\text{opt}}^{\text{R}}\) are the OMCs for CELs of E = ESP and ER, respectively; \({\gamma }_{\text{dmax}}^{\text{SP}}\) and \({\gamma }_{\text{dmax}}^{\text{R}}\) are the MDUWs for E = ESP and ER, respectively; Gs is specific gravity of soil solids; γw is unit weight of water (= 9.81 kN/m3); RSP is reduced standard Proctor effort (here defined as ERSP < ESP); and RMP is reduced modified Proctor effort (here defined as ESP < ERMP < EMP).

Referring to Soltani et al. (2023), the fitting parameters −0.178 and +0.068 in Eqs. 1 and 2, which define the mean rates of decrease in OMC and increase in MDUW with increasing CEL, were deduced based on statistical analyses performed on a large and diverse database of 242 compaction test results obtained for 76 different fine-grained soils. Each of these soils had test results for at least three CELs, collectively covering E = 225–2708 kJ/m3. In examining the predictive performance of Eqs. 1 and 2, Soltani et al. (2023) noticed that in some instances where the converted MDUW and/or OMC parameters were overestimated, the theoretically deduced degree of saturation value associated with the \({\gamma }_{\text{dmax}}^{\text{R}}\):\({{w}}_{\text{opt}}^{\text{R}}\) conversion exceeded the ZAV line boundary (i.e., giving \({{S}}_{\text{opt}}^{\text{R}}\) ≥ 100%). It should be mentioned that this occurrence is not commonplace compared to employing conventional empirical correlations and even AC-based EC models. Nevertheless, towards addressing the inconsistency of \({{S}}_{\text{opt}}^{\text{R}}\) ≥ 100%, as well as for improving the accuracy of the MDUW conversions, Soltani et al. (2023) proposed the application of Eq. 3 (in lieu of Eq. 2). Note that Eq. 3 was established using basic soil volume–mass relationships (in conjunction with Eq. 1) and on the premise that the value of Sopt for a given fine-grained soil does not change significantly across different CELs (Blotz et al. 1998; Gurtug and Sridharan 2004; Horpibulsuk et al. 2008, 2009; Shimobe et al. 2021; Soltani et al. 2023).

Notwithstanding the promising results observed for the EC-based modeling framework given by Eqs. 1–3, its inherent data-driven nature warrants further validation studies based on additional independent datasets. Such validation studies would foster greater confidence within the geoengineering community in adopting this practical and user-friendly modeling framework in routine practice. To this end, the present study aims to critically examine the asserted accuracy of Eqs. 1–3 in converting the OMC and MDUW parameters of dissimilar fine-grained soils across rational CELs. This is achieved by performing comprehensive statistical analyses on a newly compiled database of 206 compaction test results, entailing 70 very different fine-grained soils and accounting for CELs ranging between 202.0 and 2723.5 kJ/m3.

2 Compiled Database of Compaction Test Results

A large independent database of 206 compaction test results, compiled from 38 different investigations (see Table 1), was employed to critically examine the validity of the EC-based modeling framework given by Eqs. 1–3. The database included compaction test results for 70 fine-grained soil materials (herein designated as S1–S70). Inclusive of SP measurements, test results for at least two CELs were available for each database soil, with 31 of them having data for at least three CELs. As is evident from Table 1, the database soils cover broad ranges of soil index properties, with CF (< 2 µm) = 5–97%, FF (< 75 µm) = 40–100%, LL = 19–365 %MC, PL = 7.9–50.0 %MC, PI = 5–333 %MC and A = PI/CF = 0.28–6.50 (the notations CF, FF, LL, PL, PI, A and MC represent the clay content, fines fraction, liquid limit, plastic limit, plasticity index, soil activity and gravimetric moisture content, respectively). Referring to Fig. 1a, which illustrates the database soils (excluding soils S13, S14, S32 and S41–S44 from Agus (2005), Tüfekçi et al. (2010) and Bhat et al. (2015) for which the Atterberg limits were not reported) plotted in the British-standard plasticity chart (BS 5930 2015), the database soils comprised of 54 clays and 9 silts, with a detailed classification analysis presented in Fig. 1b.

Table 1 Summary of the assembled database of NT = 206 compaction test results
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Fig. 1
figure 1

Plasticity characteristics of the database soils (excluding soils S13, S14, S32 and S41–S44 for which the Atterberg limits were not reported in the original sources): a database soils plotted on the British-standard plasticity chart (BS 5930 2015); and b detailed analysis of the soil classification results

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Figure 2a illustrates the variations of MDUW (i.e., γdmax = 11.7–22.3 kN/m3) against OMC (i.e., wopt = 6.3–40.0 %MC) for the compiled database. The optimum compaction parameters are strongly correlated with each other, exhibiting, for instance, the following exponential relationship (with R2 = 0.930 for NT = 206 compaction test results): γdmax = 23.655 exp(−0.019 wopt), which defines the blue chain curve in Fig. 2a. Note that this relationship, along with the other three correlations outlined in Fig. 2a, conform to the well-established “path of optimums” correlation framework reported in numerous earlier studies (e.g., Gurtug and Sridharan 2004; Sivrikaya et al. 2008; Gurtug et al. 2018; Di Matteo and Spagnoli 2023; Soltani et al. 2023). Regarding the CEL, the compaction data cover RSP, SP, RMP and MP efforts (i.e., CEL range of E = 202.0–2723.5 kJ/m3), comprising 31, 70, 42 and 63 cases, respectively (see Fig. 2b). It should be reiterated that the database assembled in the present investigation is entirely different from the one previously employed by the authors (Soltani et al. 2023) in developing the DC-based EC modeling framework given by Eqs. 1–3.

Fig. 2
figure 2

Compaction characteristics of the database soils: a variations of MDUW against OMC; and b detailed analysis of the compaction test results. Note: ZAV denotes the zero-air-voids saturation line (or S = 100%) for an assumed specific gravity value of Gs = 2.90

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3 Results and Discussion

3.1 Converting SP to Other Rational CELs

Figure 3a illustrates the variations of converted against measured OMC values for the compiled database, with the converted RSP, RMP and MP OMCs obtained from the measured SP by Eq. 1. The converted and measured values are strongly correlated with each other, exhibiting a very high R2 value of 0.966 (for NC = 136 conversions). The average prediction error associated with these OMC conversions was quantified using the normalized root-mean-squared error (NRMSE) and mean absolute percentage error (MAPE) parameters, both being dimensionless fit-measure indices, expressed as percentage, through the following relationships (Soltani et al. 2018; Soltani and O’Kelly 2022):

$${\text{NRMSE}}=\frac{{\text{RMSE}}}{{y}_{{\text{M}}\left({\text{max}}\right)}-{y}_{{\text{M}}\left({\text{min}}\right)}}\times 100\%\ni {\text{RMSE}}=\sqrt{\frac{1}{{N}_{{\text{C}}}} \sum_{n=1}^{{N}_{{\text{C}}}}{\left[{y}_{{\text{M}}\left(n\right)}-{y}_{{\text{P}}\left(n\right)}\right]}^{2}}$$
(4)
$${\text{MAPE}}=\frac{1}{{N}_{{\text{C}}}} \sum_{n=1}^{{N}_{{\text{C}}}}\left|\frac{{y}_{{\text{M}}\left(n\right)}-{y}_{{\text{P}}\left(n\right)}}{{y}_{{\text{M}}\left(n\right)}}\right|\times 100\%$$
(5)

where RMSE is the root-mean-squared error (in %MC for OMC or kN/m3 for MDUW); yM and yP are the measured and converted (or predicted) OMC or MDUW data, respectively; yM(max) and yM(min) are the maximum and minimum of the measured OMC or MDUW data, respectively; NC is the number of compaction conversions (or predictions); and n is the index of summation.

Fig. 3
figure 3

a Correlation and b NAPE scatter plots illustrating the level of agreement between the SP-converted (by Eq. 1) and measured OMC values for the investigated RSP, RMP and MP CELs

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The NRMSE and MAPE parameters for the OMC conversions made by Eq. 1 were calculated as 6.0% and 3.8%, respectively, both being acceptable in view of the typically allowable 5–10% reference threshold. Although the high R2 and low NRMSE or MAPE values obtained for the OMC conversions depicted in Fig. 3a would normally warrant accepting the predictive capability of Eq. 1, these fit-measure indices alone are generally not able to elucidate the practical implications of employing Eq. 1 in routine practice (Soltani and O’Kelly 2021a). That is, in addition to conventional statistical fit-measure indices (e.g., R2, RMSE, NRMSE and MAPE), it is also imperative to critically scrutinize the prediction residual values (Bland and Altman 1986; Soltani and O’Kelly 2021b). In the present investigation, this was achieved by the normalized absolute percentage error (NAPE) parameter, which, for a given conversion, can be calculated as follows (AS 1289.5.1.1 2017; AS 1289.5.2.1 2017):

$${\text{NAPE}}=\left|\frac{{y}_{{\text{M}}}-{y}_{{\text{P}}}}{0.5\left({y}_{{\text{M}}}+{y}_{{\text{P}}}\right)}\right|\times 100\%$$
(6)

where yM and yP are the measured and converted (or predicted) OMC or MDUW data, respectively.

According to the Australian AS 1289.5.1.1 (2017) and AS 1289.5.2.1 (2017) standards for SP and MP laboratory compaction testing, acceptable NAPE values for the OMC and MDUW measurements considering a given fine-grained soil can be as high as 20% and 4%, respectively. Based on laboratory compaction tests performed on a wide variety of soils, AS 1289.5.1.1/AS 1289.5.2.1 advises that “these [NAPE = 20% and 4%] values, in 95% of cases, should not be exceeded,” while acknowledging that “in some cases, such as heavy clays, these [NAPE = 20% and 4%] values may be exceeded.” Accordingly, in the present investigation, NAPE values of 20% and 4% were chosen as reference values for cross-examining the prediction residuals associated with the OMC and MDUW conversions (made by Eqs. 1–3), respectively. Figure 3b illustrates the distribution of the NAPE values for the OMC conversions produced by Eq. 1. Compared to the ≥ 95% requirement suggested in AS 1289.5.1.1/AS 1289.5.2.1, this figure shows that 99% (= 135/136 × 100%) of the NAPE values for the OMC conversions were lower than 20%. In other words, it is implied that 99% of the differences between the converted and measured OMC values shown in Fig. 3 are statistically insignificant, being within the acceptable margin of OMC measurement errors permitted by AS 1289.5.1.1 (2017) and AS 1289.5.2.1.5.2.1 (2017). In view of all the above, Eq. 1 can be used with confidence to obtain reliable ESP → ER (where R = RSP, RMP and MP) conversions of the OMC parameter.

Correlation and NAPE scatter plots illustrating the level of agreement between the SP-converted (by Eq. 2) and measured MDUW values for RSP, RMP and MP CELs are presented in Fig. 4. As depicted in Fig. 4a, the converted versus measured MDUW correlation exhibits excellent fit-measure indices; that is, R2 = 0.957, NRMSE = 2.7% and MAPE = 2.1% for NC = 136. However, compared to the ≥ 95% requirement suggested in AS 1289.5.1.1/AS 1289.5.2.1, only 85% (= 115/136 × 100%) of the NAPEs associated with the conversions produced by Eq. 2 were lower than the allowable 4% limit (see Fig. 4b). This implies that Eq. 2, albeit useful for preliminary estimation purposes, should generally not be used in place of actual compaction test results. This was the driver for Soltani et al. (2023) to introduce Eq. 3 (for use instead of Eq. 2) in obtaining more accurate MDUW conversions. Nevertheless, having already demonstrated that an accurate conversion of the OMC parameter can be established using Eq. 1, one can therefore directly measure its corresponding MDUW value by performing a more practical single-point compaction test.

Fig. 4
figure 4

a Correlation and b NAPE scatter plots illustrating the level of agreement between the SP-converted (by Eq. 2) and measured MDUW values for the investigated RSP, RMP and MP CELs

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Moreover, as described earlier, in some instances where the converted MDUW and/or OMC values are overestimates, the theoretically deduced degree of saturation value for the conversion (calculated using \({\gamma }_{\text{dmax}}^{\text{R}}\) and \({{w}}_{\text{opt}}^{\text{R}}\)) may surpass the physically limiting ZAV line boundary, thereby producing a materially meaningless compaction state (Di Matteo et al. 2022; Soltani et al. 2023). In the present study, a total of 15 conversions (out of NC = 136) obtained based on Eqs. 1 and 2 were found to produce degree of saturation values greater than 100% (i.e., giving \({{S}}_{\text{opt}}^{\text{R}}\) = 101.0–120.0%, with a median value of 102.7%). This inconsistency (of \({{S}}_{\text{opt}}^{\text{R}}\) ≥ 100%) was addressed by Soltani et al. (2023) with the introduction of Eq. 3 (in place of Eq. 2), on the premise that the value of Sopt for a given fine-grained soil does not change significantly across different CELs. The validity of this assertion, also reported in numerous previous investigations (e.g., Blotz et al. 1998; Gurtug and Sridharan 2004; Horpibulsuk et al. 2008, 2009; Shimobe et al. 2021), is demonstrated in Fig. 5 for the compiled database of NT = 206 compaction test results. As depicted in this figure, the measured \({{S}}_{\text{opt}}^{\text{SP}}\)-to-\({{S}}_{\text{opt}}^{\text{R}}\) ratio (where R = RSP, RMP and MP) ranged between 0.79 and 1.21 (with mean, median and standard deviation values of 1.00, 0.99 and 0.06, respectively), reaffirming the proposition of \({{S}}_{\text{opt}}^{\text{SP}}\) ≈ \({{S}}_{\text{opt}}^{\text{R}}\). Note that some of the original sources presented their compaction test results without accompanying specific gravity measurements (see Table 1). For these cases, the highest reported Gs value (of 2.83) in consideration of the entire assembled database was employed in the calculations.

Fig. 5
figure 5

Variations of the measured \({{S}}_{\text{opt}}^{\text{SP}}\)-to-\({{S}}_{\text{opt}}^{\text{R}}\) ratio against the corresponding (\({\gamma }_{\text{dmax}}^{\text{SP}}\) + \({\gamma }_{\text{dmax}}^{\text{R}}\))/2 value (where R = RSP, RMP and MP) for the compiled database of NT = 206 compaction test results. Note: UB and LB denote the upper and lower \({{S}}_{\text{opt}}^{\text{SP}}\)-to-\({{S}}_{\text{opt}}^{\text{R}}\) ratio, respectively

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Correlation and NAPE scatter plots depicting the level of agreement between the SP-converted (by Eq. 3) and measured MDUW values for RSP, RMP and MP CELs are presented in Fig. 6. The new MDUW conversions, besides producing physically meaningful values of \({{S}}_{\text{opt}}^{\text{R}}\) < 100% (when employed in conjunction with Eq. 1), were associated with notably improved R2, NRMSE and MAPE values of 0.976, 2.0% and 1.6%, respectively (see Fig. 6a). More importantly, 96% (= 131/136 × 100%) of the NAPE values (i.e., satisfying the ≥ 95% requirement suggested in AS 1289.5.1.1/AS 1289.5.2.1) were found to be lower than 4% (see Fig. 6b). In other words, this indicates that 96% of the differences between the converted and measured MDUW values are within the allowable margin of MDUW measurement errors permitted by AS 1289.5.1.1 (2017) and AS 1289.5.2.1 (2017). In view of these results, when the specific gravity value is available (either via direct measurement or based on a reliable assumption), Eq. 3 should be adopted (instead of Eq. 2) for ESP → ER (where R = RSP, RMP and MP) conversions of the MDUW parameter. This is further validation, based on an entirely independent database, of the authors’ previous findings and recommendations documented in Soltani et al. (2023).

Fig. 6
figure 6

a Correlation and b NAPE scatter plots illustrating the level of agreement between the SP-converted (by Eq. 3) and measured MDUW values for the investigated RSP, RMP and MP CELs

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3.2 Converting RSP to Other Rational CELs

Given the prevalent usage of the SP compaction test compared to other higher-CEL test methods (e.g., MP and West African RMP), its adoption as an input to estimate the OMC and MDUW parameters for lower and higher CELs is certainly merited. Understandably, however, there may still exist an impetus to reduce the time and labor demands associated with SP compaction testing. To this end, it would be beneficial if one could employ measured compaction data from any user-defined RSP test (with its CEL being, for instance, as low as ∼200 kJ/m3) to produce OMC and MDUW conversions for higher CELs. Rewriting the sub- and super-scripts of “SP” as “RSP” in Eqs. 1–3, the following ERSP → ER relationships are obtained (Soltani et al. 2023):

$${w}_{{\text{opt}}}^{{\text{R}}}={w}_{{\text{opt}}}^{{\text{RSP}}} {\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{RSP}}}}\right)}^{-0.178}\ni {E}_{{\text{R}}}>{E}_{{\text{RSP}}}$$
(7)
$${\gamma }_{{\text{dmax}}}^{{\text{R}}}={\gamma }_{{\text{dmax}}}^{{\text{RSP}}} {\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{RSP}}}}\right)}^{+0.068}\ni {E}_{{\text{R}}}>{E}_{{\text{RSP}}}$$
(8)
$${\gamma }_{{\text{dmax}}}^{{\text{R}}}=\frac{{G}_{{\text{s}}}{\gamma }_{{\text{w}}}}{1+\left(\frac{{G}_{{\text{s}}}{\gamma }_{{\text{w}}}}{{\gamma }_{{\text{dmax}}}^{{\text{RSP}}}}-1\right){\left(\frac{{E}_{{\text{R}}}}{{E}_{{\text{RSP}}}}\right)}^{-0.178}}\ni {E}_{{\text{R}}}>{E}_{{\text{RSP}}}$$
(9)

where \({{w}}_{\text{opt}}^{\text{RSP}}\) and \({\gamma }_{\text{dmax}}^{\text{RSP}}\) are OMC and MDUW obtained for RSP compaction effort (here defined as any custom compaction test with ERSP < ESP), respectively.

Omitting those database soils not including compaction test results for ERSP < ESP (see Table 1), the measured OMC and MDUW values corresponding to the lowest CEL for each remaining soil (all being lower than SP) were applied as \({{w}}_{\text{opt}}^{\text{RSP}}\) and \({\gamma }_{\text{dmax}}^{\text{RSP}}\) in Eqs. 7–9 to produce conversions for higher CELs. The predictive performance metrics for the ERSP → ER (here ER > ERSP) conversions are presented in Table 2. In terms of NAPE distribution, 99% (= 88/89 × 100%) and 93% (= 83/89 × 100%) of the NAPE values (compared to the ≥ 95% requirement suggested in AS 1289.5.1.1 (2017) and AS 1289.5.2.1 (2017)) obtained for the OMC and MDUW conversions produced by Eqs. 7 and 9 were found to be lower than 20% and 4%, respectively. These favorable results indicate that the differences between the converted and measured optimum compaction parameters are predominantly within the acceptable margins of laboratory measurement errors permitted by the Australian AS 1289.5.1.1 and AS 1289.5.2.1 standards. Note that much like Eqs. 2 and 3 compared earlier in Section 3.1, direct MDUW conversions produced by Eq. 8 were associated with inferior NAPE values compared to those produced by Eq. 9. That is, compared to the ≥ 95% requirement, only 75% of the MDUW conversions produced by Eq. 8 gave NAPE values lower than the 4% limit. Given the favorable performance metrics obtained for Eqs. 7 and 9 (see Table 2), all being comparable to those noted for Eqs. 1 and 3, it can be concluded that the authors’ proposed EC-based modeling framework (Soltani et al. 2023) has been further confirmed to achieve reliable ERSP → ER conversions. Overall, therefore, the results presented in this study reaffirm the authors’ viewpoints that they originally presented in the discussion paper of Di Matteo et al. (2022) and later substantiated in Soltani et al. (2023), namely:

Table 2 Summary of the predictive performance metrics for the ESP → ER (Eqs. 1–3) and ERSP → ER (Eqs. 7–9) conversions
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For fine-grained soils, the optimum compaction parameters (i.e., OMC and MDUW) across rational CELs are somewhat uniquely related.

The effects of fine-grained soil attributes (e.g., gradation, plasticity and clay mineralogy) on soil compactability are adequately captured and explained by the measured OMC and MDUW values that are employed as the conversion inputs/predictors.

4 Summary and Conclusions

Among the many proposals for estimating the optimum compaction properties of fine-grained soils across rational CELs, EC-based models seem to be gaining increased attention. These models work on the premise of employing measured OMC and MDUW values obtained for a rational CEL (usually SP) to predict the same for other lower and/or higher compactive efforts. Towards fostering greater confidence for its adoption in routine geoengineering practice, the present investigation revisited the authors’ recently proposed EC-based modeling framework for converting the OMC and MDUW parameters of fine-grained soils across rational CELs (i.e., Eqs. 1–3 and 7–9). Specific aims were to critically examine its asserted accuracy and hence identify its true potentials and/or limitations. These aims were achieved by performing comprehensive statistical analyses on a newly compiled database of 206 compaction test results, entailing 70 different fine-grained soils (covering LL = 19–365 %MC and PI = 5–333 %MC) investigated for CELs ranging between 202.0 and 2723.5 kJ/m3.

It was demonstrated that 99% and 96% (compared to the ≥ 95% requirement recommended by the Australian AS 1289.5.1.1 and AS 1289.5.2.1 standards for SP and MP laboratory compaction testing) of the differences between the SP-converted (i.e., obtained by Eqs. 1 and 3) and measured OMC and MDUW values, respectively, fall within the allowable margins of routine moisture content and dry unit weight measurement errors permitted by the said standards. Equally favorable results were also noted for the RSP-based conversions (i.e., employing Eqs. 7 and 9). These findings reaffirm the authors’ viewpoint that the optimum compaction parameters across rational CELs are somewhat uniquely related. Furthermore, they demonstrate that the effects of fine-grained soil properties (e.g., gradation, plasticity and clay mineralogical composition) on soil compactability are adequately captured and explained by the measured OMC and MDUW values that are employed as the conversion inputs/predictors. In summary, the examined EC-based modeling framework (i.e., given by Eqs. 1–3 and 7–9) proposes a practical approach to reliably achieve “direct” conversions of the optimum compaction parameters of fine-grained soils across different CELs, only requiring the energy ratio parameter ER/ESP or ER/ERSP. In respect of MDUW, superior prediction accuracy is achieved using Eqs. 3 and 9 over Eqs. 2 and 8, respectively, but with the added input requirement of the specific gravity value (either obtained via direct measurement or based on a reliable assumption). Based on the presented experimental evidence, the user-friendly framework can be adopted with confidence to eliminate the time and labor demands associated with performing additional compaction tests that would otherwise be required to cover a desired/wide CEL range.