Determination of soil permeability coefficient following an updated grading entropy method Open Access

A

relative base grading entropy

B

normalised entropy increment

C₁ to C₄

fitting coefficients for regression models (C ₁ (m/s); C ₂ to C ₄, dimensionless)

C_C

coefficient of curvature

C_U

coefficient of uniformity

D₁₀

effective grain size (mm)

D₃₀, D₅₀, D₆₀

particle sizes corresponding to 30, 50 and 60% dry mass passing, respectively (mm)

e

void ratio

e_compacted

void ratio of compacted specimens

G_C

characteristic gradation parameter (mm)

k

coefficient of permeability (m/s)

k_M

measured permeability coefficient (m/s)

k_P

predicted permeability coefficient (m/s)

m

number of sample subdivisions (fractions)

n

number of data points (observations)

p

p-value of statistical hypothesis testing

R²

coefficient of determination

$Ra2$

adjusted coefficient of determination

x_i

relative frequency of fraction i

The topic of accurate prediction of soil permeability is of importance to geotechnical engineers and researchers. The coefficient of permeability (k), defined as the mean discharge velocity of the fluid flow through a porous medium for a unit hydraulic gradient, is principally dependent on the size and connectivity of the pore voids, which are related to the size distribution and shape of the solid particles, the degree of saturation, soil structure and the pore fluid’s viscosity. As customary, reported laboratory k values typically correspond to the soil in its saturated state and for a standardised permeant temperature of 20°C since fluid viscosity is temperature dependent.

Direct measurement using well pumping, borehole slug and tracer tests and laboratory constant- and falling-head permeameter approaches typically involve relatively high economic cost, are time-consuming and require specialised facilities and expertise (O’Kelly, 2008, 2009, 2016), such that they are often not considered viable options for many projects. The same often applies for indirect measurement approaches, including the calculation of laboratory k values from consolidation parameters determined from oedometer, consolidometer and hydraulic consolidation testing (O’Kelly, 2005, 2006).

At the feasibility stage of large projects or for the design of drainage, soak wells, septic tanks and so on, geotechnical engineers are often needed to provide estimated (mainly for budget constraints) soil permeability. Such assessments are often made by way of empirical index-property-based formulae – for example, considering soil gradation, volumetric characteristics, bulk density, clay content and organic content. Dozens of empirical correlations (models) have been proposed over the decades in order to correlate the k value of a soil sample with some representative particle size from its gradation curve, usually taken as the effective grain size (D₁₀) or mean particle size (D₅₀) index values – that is, the particle sizes corresponding to 10 and 50% dry mass passing, respectively. The void ratio (e) is long recognised as a primary predictor of soil permeability-coefficient values and is included in many empirical and semi-empirical models (e.g. Carman, 1937, 1939; Carrier, 2003; Chapuis, 2004, 2012; Chapuis and Aubertin, 2003; Kozeny, 1927). Some of the most popular correlations, which are primarily based on some representative particle size(s), gradation parameter, void ratio and/or specific surface of the solids, are presented in the papers by Arshad et al. (2019) and Chapuis (2012), along with their intended applications in terms of soil types, gradation ranges and density states.

Recently, approaches based on the grading entropy framework (attributed to Lörincz’s doctoral thesis (Lörincz, 1986; Lörincz et al., 2017; Singh, 2014)) and the characteristic gradation parameter (Arshad et al., 2019) have been proposed for permeability assessments of coarse-grained soils. The primary focus of the present investigation is on the grading entropy approach. The characteristic gradation parameter G_C (Equation 1) approach is mentioned at this stage for completeness and has been employed in the model given by Equation 2 for predictions of the saturated permeability-coefficient (k_P) value for sand and silty sands.

where D₁₀, D₃₀, D₅₀ and D₆₀ are the particle sizes, given in millimetres, for 10, 30, 50 and 60% dry mass of material passing, respectively.

The grading entropy approach applies entropy theory (Shannon, 1948) to the particle-size distribution (PSD) of a granular mixture, thereby establishing a vectorial depiction of gradation variation (McDougall et al., 2013) arising from an ongoing change in the PSD (because of grain crushing or soil erosion) or for materials with different PSDs – that is, each grading curve can be interpreted and plotted as a discrete point in the grading entropy chart, whose coordinates A (relative base grading entropy) and B (normalised entropy increment) are computed according to the following equations

where m is the number of fractions (sample subdivisions) and x_i is the relative frequency of fraction i.

Imre et al. (2012) explained that the variation in fraction number m for soils with different PSDs can induce a discontinuity in the normalised entropy path. As described by Feng et al. (2019a), in order to remove this discontinuity, ‘zero’ fractions (i.e. coarser or finer fractions with zero particle frequency), which leave the non-normalised grading entropy unaffected, are introduced in the analysis (see the paper by Imre et al. (2012)). Based on the normalised grading entropy equations, for a certain fraction number m, the maximum normalised entropy increment (B) curve is fixed (see the calculation process in the book by Singh (2014)); however, with different fraction numbers m, the variation of the maximum B curve is rather unnoticeable (cf. the paper by Imre et al. (2008)).

As demonstrated in Figure 1, variations in different materials’ gradations can be conveniently recorded using this approach in terms of discrete points presented in the one chart, rather than their usual presentation as a family of PSD curves. As explained in the paper by McDougall et al. (2013) and also evident from examination of the various PSD plots presented in this figure, the coordinate A reflects the skewness (symmetry) of a PSD curve, while the coordinate B measures its peakiness (kurtosis). For instance, with coordinate A decreasing from unity to 0·5 and increasing values of coordinate B, the proportion of smaller-sized particles in the soil mass increases, which is associated with lower k_M values.

A perceived advantage of the grading entropy permeability-prediction models is that through the coordinates A and B, they account for the entire soil gradation, rather than considering some representative particle size (e.g. D₁₀ or D₅₀ value) of the grading curve, as employed in many of the other empirical index-property-based formulae. The model given by Equation 2 also has this advantage since the philosophy of the characteristic gradation parameter G_C is to embody the entirety of the soil gradation curve.

Recent studies by Feng et al. (2018a, 2018b, 2019a) on various gravel mixtures and Arshad et al. (2019) on various sands and silty sands investigated the grading entropy coordinates A and B for predictions of the k_P value. The premise of the studies by Feng et al. (2018a, 2018b, 2019a) was that pavement engineers often specify gradation curve envelopes when designing gravel mixtures, such that a method of estimating the permeability-coefficient value straight from the gradation curve is useful. Specifically, they performed multiple linear regression analyses, producing Equations 5–7 from data sets of measured permeability coefficient (k_M) for asphalt concrete (Feng et al., 2018a), various saturated 10 mm crushed basalt–gritstone gravel mixtures (Feng et al., 2019a) and a database of 164 hydraulic conductivity tests for various sands and gravels compiled from ten earlier publications summarised in the paper by Feng et al. (2018b).

where n is the number of data points (observations); p refers to the p-value of the statistical hypothesis testing; and R² is the coefficient of determination.

As per Equations 5–7, Feng et al. (2018a, 2018b, 2019a) found that compared with other possible formulations, power functions of the form $kP=C1AC2BC3$ are statistically superior; where C₁, C₂ and C₃ are the fitting coefficients. Note that C₁ is expressed in the same units as k_P, whereas C₂ and C₃ are dimensionless. This avoids the mathematical and physical inconsistencies discussed by Castillo et al. (2014a, 2014b).

The study presented by Arshad et al. (2019) employed essentially the same approach, investigating saturated standard Proctor (SP)-compacted sand and silty sand samples. Based on their published data, the authors derived the following equation as part of the present investigation

As described earlier, the k_M value is principally controlled by the size and connectivity of the pore voids, which are dependent on the shape and size distribution of the constituent solids, whether well graded or poorly graded, the void ratio and the soil structure. As such, models of the form given by Equations 5–8 have potential shortcomings, most notably since they do not account for the soil densification level (packing state) or particle shape – that is, although dependent on gradation (and hence coordinates A and B) and to a lesser extent on the particle shape, the placement void ratio is also strongly dependent on the densification level (packing state). For a given soil material, an equation of the form $kP=C1AC2BC3$ produces the same k_P value for loose, medium and dense packing states and for particle shape classes as diverse as irregular to rounded, which are clearly not the cases. As such, relationships of the form $kP=fn(A,B,e)$ should prove statistically superior and extend the model’s scope and reliability (Feng et al., 2019b). This is analogous to the approach employed for Equation 2, where the G_C and e parameters account for gradation characteristics and packing state, respectively. The packing state is controlled to some degree by the particle shape, such that the controlling effect of the latter on the permeability-coefficient magnitude is partially accounted for by the inclusion of the e parameter in the model.

Further, from inspection of Equations 6–8 for sand and gravel materials, the values of the coefficients C₁ to C₃ are significantly different between equations. Feng et al. (2018b) explained that this may be due to the diversity of gradation ranges and mineralogical compositions of the various sand and gravel materials considered in their study, compared with the 10 mm basalt–gritstone gravel mixtures investigated in another paper by the same authors (Feng et al., 2019a), which is undoubtedly the case. In other words, the fitting-coefficient values in Equations 5–8 pertain to the particle shape and the gradation and densification (compaction) ranges of the materials investigated for the permeability tests, consistent with the fact that the different samples (data sets) considered in the papers by Feng et al. (2018a, 2018b, 2019a) and Arshad et al. (2019) did not represent the same statistical population – that is, like any empirical correlation, Equations 5–8 cannot be applied with confidence for materials having different physical characteristics, prepared at other densification levels and (or) outside their calibration gradation ranges.

As evident from Figure 2, compared with the gravel and sand materials for Equations 6 and 7, the power function given by Equation 8 for the sand and silty sand materials predicts substantially lower k_P values. In this figure, the domain for each presented model is based on its reported range of k_M or coordinate A and B values, as listed in Table 1. The spatial trend given by these three models over their calibration permeability-coefficient ranges is consistent with the expected reduction in k values for reducing particle size from gravel to sand to silty sand materials. On this basis, it would appear a step too far for a single regression correlation of the form $kP=C1AC2BC3$ to give a satisfactory assessment of k values for a broad spectrum of coarse- and (or) fine-grained soils. This is undoubtedly the case when different densification levels are considered.

The present investigation provides in-depth evaluation of the grading entropy approach for permeability-coefficient assessments. To this aim, two previously published data sets are combined and analysed – namely, for 30 saturated compacted basalt–gritstone gravel mixtures (k_M = 4·2 × 10⁻³ to 5·6 × 10⁻¹ m/s) presented in the paper by Feng et al. (2019a) and for 20 saturated SP-compacted fine and coarse silica sand and silty sand samples (k_M = 7·2 × 10⁻⁷ to 5·6 × 10⁻⁴ m/s) presented in the paper by Arshad et al. (2019). Hereafter, these sources are referred to as data sets X and Y, respectively. From a review of the pertinent literature, these two papers are presently the only ones that report values of the coordinates A and B along with pertinent gradation parameter, void ratio and measured permeability-coefficient values (determined from standard laboratory constant-head permeameter testing) for the investigated soils.

In particular, the dependence of the grading entropy coordinates A and B on the coefficient of uniformity (C_U), coefficient of curvature (C_C), D₁₀, D₅₀ and compacted void ratio (e_compacted) is closely examined. From a practical application point of view, relative density is usually estimated for subsurface soil strata by various means of field tests (i.e. cone penetration test, standard penetration test etc.) and also compaction specifications for earthwork verification are generally defined by a dry density ratio of 95 or 98% of maximum dry density (modified compactive effort). Neither of the data sets X and Y reported the maximum and minimum void ratio values for the various materials examined – that is, only e_compacted values were given, such that it was not possible to perform analysis and evaluations based on the relative density and (or) at a particular dry density ratio, which would have been preferable.

None of the previous grading entropy investigations has explicitly considered the importance of soil gradation in terms of whether the materials were well graded or poorly graded and its impact on deduced correlations. Presumably, it was assumed that this aspect was incorporated in the coordinates A and B values. However, it is the authors’ opinion that correlations with greater statistical significance would be obtained for considering well-graded and poorly graded materials as separate groupings. This aspect is investigated for each of the parameters C_U, C_C, D₁₀, D₅₀ and e_compacted.

Based on new insights gleaned from the detailed analysis of the combined X and Y data sets, improved grading entropy soil permeability-prediction models are proposed and evaluated. Further, whereas the studies reported in the papers by Feng et al. (2018a, 2018b, 2019a) considered sand and gravel materials as well as asphalt concrete, one of the novelties of this paper is the extension of the grading entropy approach for permeability assessments of silty sand materials.

The widely different gradations of 30 gravel mixtures (D ₁₀ = 0·72–7·02 mm and D ₅₀ = 2·17–9·93 mm) and 17 sandy soil samples (D ₁₀ = 0·01–0·42 mm and D ₅₀ = 0·02–0·56 mm) comprising data sets X and Y, respectively, are shown using the grading entropy chart in Figure 3. Note that from the authors’ preliminary analysis of the n = 20 data entries comprising data set Y, unresolvable inconsistencies were found for the data entries corresponding to the three soil samples listed as A, B and A₅₀ + C₅₀ in the paper by Arshad et al. (2019). For this reason, the authors considered it prudent to omit these three soils from statistical analysis reported in the present investigation.

The A coordinate values for all investigated soils ranged from 0·44 to 0·94 – that is, the grading entropy coordinate pairs fall approximately within the right-hand half of the semi-elliptical domain identified in Figure 1. In presenting the data points in Figure 3, the authors have distinguished between well-graded and poorly graded materials to allow for a more critical assessment of the data sets – that is, for gravel soils with C _C = 1–3, C _U ≥ 4 → well graded and C _U < 4 → poorly graded (ASTM, 2017), whereas C _C ≠ 1–3 → gap-graded soils. For sand soils, C _U ≥ 6 and < 6 with C _C = 1–3 → well graded and poorly graded, respectively, whereas C _C ≠ 1–3 → gap graded (ASTM, 2017).

As evident from Figure 3 and noted previously by Feng et al. (2019b), the gradation characteristics of the gravel mixtures comprising data set X were such that their B values negatively correlate with A (R ² = 0·50), whereas only a weak correlation (R ² = 0·16) occurs for data set Y – that is, the soil materials comprising data set X were mostly well graded, whereas those comprising data set Y were predominantly poorly graded.

The C _U and C _C values for the sandy soil samples (data set Y) ranged 1·60–12·50 and 0·73–5·12, respectively, with C _U = 1·51–7·29 and C _C = 0·62–3·5 for the gravel mixtures (data set X). Feng et al. (2019a) reported that for data set X, C _U inversely correlated with coordinate A and directly correlated with coordinate B. As evident from Figure 4, however, these trends are not obvious when considering the combined data sets X and Y.

For data set X, 18 out of the 30 gravel mixtures investigated classify as well graded (C _U ≥ 4), compared with only two of the 17 sandy soil samples (C _U ≥ 6) comprising data set Y. It would appear that the above trends deduced by Feng et al. (2019a) for data set X arose on account of the data points associated with its 18 well-graded gravels. As evident from Figures 4(c) and 4(d), the 20 well-graded soils for the combined data sets X and Y exhibit reasonably strong correlations between C _U and the coordinates A and B (R ² ≈ 0·64), whereas C _U was independent of coordinates A and B for the 27 poorly graded sand and gravel materials.

In relation to C _C, Feng et al. (2019a) observed that for data set X, higher values of C _C peaked around A = 0·82 and B = 0·78, which is generally evident for the combined data sets X and Y presented in Figure 5. This was also the case when the well-graded and poorly graded subsets of the combined data sets were investigated. It is concluded, however, that apart from C _U for the well-graded soils, neither C _U nor C _C is a candidate for useful correlations with the coordinates A and (or) B.

Data sets X and Y reported D ₁₀ and D ₅₀ values for the investigated materials (n = 47), but only data set Y also reports D ₃₀ and D ₆₀. Considering the small sample size for the latter parameters (n = 17), only the dependences of D ₁₀ and D ₅₀ on the coordinates A and B are investigated in the present study. For the D ₁₀–A, D ₁₀–B, D ₅₀–A and D ₅₀–B relationships, exponential fitting of the combined data sets (Equations 9–14) was found to produce marginally higher R ² values compared with other forms of their relationships – that is, log D ₁₀ and log D ₅₀ both positively correlate with coordinate A and negatively correlate with coordinate B for these soils (Figure 6). Further, referring to Figures 6(a)–6(d), both log D ₁₀ and log D ₅₀ correlate more closely with coordinate A (R ² = 0·83 for both) than coordinate B (R ² = 0·71 and 0·67, respectively).

Referring to Figures 6(e) and 6(f), the 20 well-graded soils for the combined data sets X and Y exhibit a very strong correlation between log D ₁₀ and the coordinate A (Equation 13: R ² = 0·94), compared with the moderately strong correlation obtained for the 27 poorly graded soils. Further, compared with the well-graded soils (Equation 14), the correlation between log D ₁₀ and coordinate B was stronger for the poorly graded soils, possibly reflecting its measure of peakiness (kurtosis) of the PSD.

On first viewing of Figures 6(a)–6(f), there appears to be significant scatter in the data points for D ₁₀ < ∼1 mm, but this follows directly from the logarithmic scale employed for the y-axis which always makes the points below 1·0 more dispersed (in the vertical axis), whereas it makes the points above 1·0 more concentrated (in the vertical axis). What is interesting is that the type of curve obtained will predict in a more accurate manner for small values of D ₁₀ (D ₅₀), increasing the prediction interval with larger D ₁₀ (D ₅₀). In the case of D ₁₀ (D ₅₀)–A, this will occur for smaller values of coordinate A (0·4–0·7), whereas in the case of D ₁₀ (D ₅₀)–B, it will happen for large values of coordinate B (0·8–1·4) – that is, in the case of the data that the authors have and referring to the ranges of coordinates A and B for the two data sets shown in Figure 3, the exponential D ₁₀ (D ₅₀)–A and D ₁₀ (D ₅₀)–B models will provide more accurate prediction for data set Y (sand and silty sand materials) compared with data set X (gravels).

Figure 7 presents the measured permeability-coefficient (k _M) values plotted against the coordinate A and B values for the combined data sets X and Y. As evident from this figure, log k _M directly correlates with log A and inversely correlates with log B. The k _P–A and k _P–B models for the combined data sets are presented as the following equations

From Figure 7, it could again be construed that considerably greater scatter in the data points occurs for data set Y (17 sandy soils), although this also arises from the representation, as explained earlier for Figures 6(a)–6(f). Note that here, as the scale is log–log, the effect applies to both x- and y-axes. Given the range of definition of coordinates A and B, the dispersion effect is not so obvious in the x-axis. Similar to the D ₁₀ (D ₅₀) against A and B correlations small values of k will be predicted with more precision. In the case of A–k, this will happen for smaller values of coordinate A. In the case of B–k, this will happen for larger values of coordinate B – that is, again, data set Y (sand and silty sand materials) will be better predicted than data set X by this type of power equation.

Based on the regression results for the combined data sets (Figures 7(a) and 7(b)), compared with log B (R ² = 0·68), log A correlates more closely with log k _M (0·83). However, when the well-graded and poorly graded subsets were analysed separately (Figures 7(c) and 7(d)), the strongest correlations were found for well-graded soils between log k _M and log A and for poorly graded soils between log k _M and log B (see Equations 17 and 18, respectively). Again, this follows from the fact that coordinate A reflects the skewness (symmetry) of the PSD, whereas coordinate B measures the peakiness (kurtosis) of the distribution.

Hence, it should follow that for well-graded coarse soils, a k _P–A power model will provide a more accurate prediction, whereas k _P–B or preferably $kP=fn(A,B)$ power models are more appropriate for their poorly graded counterparts.

The values of the grading entropy coordinate pairs for the 30 gravel and 17 sandy soil samples comprising data sets X and Y are plotted in Figure 8, with the associated order of magnitude of their k _M values indicated. With the values of the coordinate A approximately ranging from 0·5 to unity, from the identified trends presented in this figure, the value of k _M reduces overall for decreasing A and increasing B (as per Equations 15 and 16). For A < 0·5, one could also anticipate that the value of k _M would reduce overall with A decreasing and B increasing in value. As suggested by Feng et al. (2019a), the form of presentation of the experimental data given in Figure 8 may be useful as a design chart for engineers to make deductive assessments of the tendency of coarse-grained soils to exhibit high or low k values.

In the case of data set Y, the 60 mm dia. × ∼150 mm high permeameter specimens were densified to achieve 94 ± 3% of the SP-compaction maximum dry density value determined for the major constituting parent soil element. In contrast, for data set X, the 90 mm dia. × ∼320 mm long permeameter specimens were comprised of four layers, each layer compacted manually by applying 70 blows using a sliding cylindrical tamper of 50 mm dia. and 2·5 kg self-weight to produce a compacted dry density of ∼1·572 Mg/m³. Since maximum and minimum void ratio values were not reported for the various materials investigated, it was not possible to perform analysis and evaluations based on the relative density and (or) at a particular dry density ratio, rather only in terms of the reported e _compacted values.

Compared with the 17 compacted sandy soil samples (0·34–0·56), the void ratio values of the 30 compacted gravel mixtures (e _compacted = 0·51–0·85) were significantly greater. Using the combined data sets X and Y, the linear, exponential and power correlations were investigated for the A–e _compacted (R ² = 0·77, 0·75 and 0·72, respectively) and B–e _compacted (R ² = 0·63, 0·57 and 0·57, respectively) relationships – that is, the normal A–e _compacted and B–e _compacted plots produced the strongest (linear) correlations, with e _compacted positively correlating with A and negatively correlating with B (Figures 9(a) and 9(b) and Equations 19 and 20).

Similar to their dependences on k _M and D ₁₀, when considering the combined data sets, coordinate A (R ² = 0·77) appears to correlate more closely with e _compacted than coordinate B (0·63). When the well-graded and poorly graded subsets were analysed separately, as expected, the fitted e _compacted–A correlations indicated that, overall, the compacted well-graded soils had a marginally lower void ratio: see trend-line equations given in Figure 9(c). Further, compared with the well-graded soils (R ² = 0·47), a stronger correlation was obtained between e _compacted and coordinate B (0·67) for the poorly graded soils, reflecting again the fact that coordinate B measures the peakiness (kurtosis) of the distribution.

This section proposes updated grading entropy permeability-prediction models, incorporating the new insights gleaned from the detailed analyses of the published data sets presented in the previous sections. Specifically, the updated model should incorporate the void ratio parameter and consideration should be given to separate evaluations for well-graded and poorly graded soil materials. Following from the k_P–A and k_P–B power correlations deduced for well-graded and poorly graded coarse-grained soils given by Equations 17 and 18, respectively, it was postulated in the section headed ‘Correlations with permeability coefficient’ that for well-graded soils, a k_P–A power model may be adequate, whereas k_P–B or preferably $kP=fn(A,B)$ power models are more appropriate for their poorly graded counterparts. In general, a correlation of the form $kP=fn(A,B,e)$ should prove statistically superior and extend the model’s scope and reliability, at least for assessments of a given material type with different densification levels. Based on various statistical analyses, the following improved grading entropy permeability-prediction model in which the void ratio is considered as an additional variable was proposed by the authors in the discussion paper by Feng et al. (2019) 

where C₁, C₂, C₃ and C₄ are the fitting coefficients, with C₁ expressed in the same units as k_P, whereas C₂ to C₄ are dimensionless.

The various grading entropy models described earlier were tested using the combined data sets X and Y, and also considering their well-graded and poorly graded soil subsets, with the results presented in Table 2. In all cases, including a second and a third variable into the regression model improves the quality of the model, as shown by the adjusted R² (i.e. $Ra2$⁠), which penalises the number of variables employed in the model. However, it could be argued that, overall, the fitting of the three-variable model (A, B, e) is surprisingly good (R² = 0·99). However, it could be argued that, overall, adding B and the void ratio does not improve the quality of the prediction significantly, as the simple model (i.e. $kP=C1AC2$⁠) already provides a good fit (R² = 0·93); whereas in the case of poorly graded soils, the three-variable model significantly improves the quality of the prediction (R² increasing from 0·81 to 0·95).

The grading entropy framework provides a convenient means of presenting and interpreting the gradation characteristics of various soils. Further, the grading entropy approach provides a powerful tool for permeability-coefficient assessments of coarse-grained soils. From examination of the two published data sets comprising silty sand, sand and gravel materials, the compacted void ratio, log D₁₀ and log D₅₀ were found to correlate positively with coordinate A and negatively correlate with coordinate B. Further, apparently stronger correlations occur for A compared with B. For the well-graded soils subset, strong correlations were found between C_U and coordinates A and B, and also between log D₁₀ (log D₅₀) and coordinate A. Consistent with these findings, log k_M was found to correlate inversely with log B and correlate directly with log A and e (log e), such that for the investigated data sets, power functions of the form $kP=C1AC2BC3$ and particularly $kP=C1AC2BC3eC4$ proved statistically superior compared with other possible formulations.

Compared with the two-variable model (A, B), the three-variable model (i.e. with the e parameter included) gave a better fit to the experimental k_M values and has broader scope in terms of a wider application range of densification (compaction) states. Consideration should be given to separate evaluations for well-graded and poorly graded soil materials. Note that the values of the fitting coefficients C₁ to C₄ determined from regression analysis are specific to the soil gradations and compaction levels investigated for the permeability tests – that is, the deduced correlations cannot be applied with confidence for dissimilar materials.

Further studies investigating a much larger database composed of more diverse sand and silty sand materials are recommended towards increasing the predictive power of the associated three-variable (A, B, e) regression correlation. The inclusion of a particle shape factor and specific surface parameters in the upgraded three-variable model may possibly produce further performance enhancements.