Improving Hydraulic Conductivity Estimation for Soft Clayey Soils, Sediments, or Tailings Using Predictors Measured at High-Void Ratio

Babaoglu, Y.; Simms, P.

doi:10.1061/(ASCE)GT.1943-5606.0002344

Open access

Technical Notes

Jul 23, 2020

Improving Hydraulic Conductivity Estimation for Soft Clayey Soils, Sediments, or Tailings Using Predictors Measured at High-Void Ratio

Authors: Y. Babaoglu https://orcid.org/0000-0002-6513-9375 [email protected] and P. Simms https://orcid.org/0000-0002-9766-1336Author Affiliations

Publication: Journal of Geotechnical and Geoenvironmental Engineering

Volume 146, Issue 10

https://doi.org/10.1061/(ASCE)GT.1943-5606.0002344

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Abstract

Consolidation parameters are required to support the disposal management of soft soils or mine tailings. The estimation of these parameters from simple correlations using easily measured properties can be advantageous, and considerable work has been done on this topic. This paper proposes two innovations that advance this work: (1) hydraulic conductivity–void ratio (

k - e

) estimation can be substantially improved by using a single measured value at a high-void ratio, and (2) the compressibility curve itself can be a useful predictor of

k - e

. Using these findings, general equations are derived that describe

k - e

using a power law, where the power is either 4 or 5. Examining 79

k - e

data sets from clays, clayey tailings, and dredged materials, 94% of all predicted

k

values are within an order of magnitude of measured

k - e

values. This level of accuracy, coupled with the advantage of anchoring the

k - e

function by a measured value at a high-void ratio, is shown to result in robust predictions of settlement in large strain consolidation analyses.

Introduction

The geotechnical behavior of soft soils or tailings is very dependent on their hydraulic conductivity–void ratio functions (henceforth,

k - e

), which are highly variable, and can be time consuming and/or expensive to obtain from experimental tests: for soft soils these tests can range from months to years. Estimating these parameters from commonly measured geotechnical properties, such as Atterberg limits or grain size curves, can therefore be advantageous for practitioners, especially when testing candidate amendments to improve the

k - e

of dredged sediments or tailings. These correlations have been much studied, and recent review papers in the geotechnical literature include Pandian et al. (1995), Berilgen et al. (2006), Sridharan and Nagaraj (2000), and Chandra Paul and Azam (2013). However, the accuracy of these correlations remains insufficient for most types of design. The accuracy is typically worse where the initial water contents are substantially above the liquid limit, where the fabric of the soil will be influenced by various chemical and physical processes (for example, the influence of water chemistry on flocculation, water-movement induced shear during deposition, and its effect on floc size). However, determining the Atterberg limits requires a reduction of water content, which will substantially alter the fabric and potentially remove any signal in the Atterberg values that reflects the actual state of the soil or tailings at a high water content.

Readers should be reminded that the consolidation behavior of soft materials in this range is complex. Imai (1981) described three stages during the sedimentation of dilute clay-water mixtures: flocculation, sedimentation, and consolidation. The transition between sedimentation and consolidation has been studied by various researchers (Edil and Fox 2000; Kynch 1952; McRoberts and Nixon 1976; Pane and Schiffman 1997). The rate of consolidation is also affected by electrochemical interactions between pore water and clay particles (Azam 2011 Caughill et al. 1993; Demoz and Mikula 2012), which alters pore size and interconnectedness through a result in changes in the microstructure (Mitchell and Soga 2005). Time dependent processes, such as creep and structuration, may bear on a long term consolidation analysis (Burland 1990). However, the scope of this paper deals only with the rapid prediction of the

k - e

function.

This paper proceeds by analyzing an existing data set of 79

k - e

functions obtained from the literature on soft soils or clayey mine tailings (listed in Table 1), with 17

k - e

correlations to the Atterberg limits (see the “Appendix” section). Based on the form of the best-fitting equations, improved correlations are proposed based on two predictors providing information at a high-void ratio. The two predictors are the compressibility curve itself and a single measured value of

k

.

Table 1. Reference list for all data used in this study

Soil type	References
Clay	Mesri and Olson (1971), Znidarčić et al. (1986), Nagaraj et al. (1993), Pane and Schiffman (1997), Tanaka et al. (2001), Sivapullaiah et al. (2000), Sridharan and Nagaraj (2000), and Khan and Azam (2016)
	Compressibility: 28 individual data sets, 230 data points
	Hydraulic conductivity: 22 data sets, 175 data points
Dredged soils	Cargill (1985), Shields (1988), Poindexter (1988a, b), and Berilgen et al. (2006)
	Compressibility: 22 individual data sets, 353 data points
	Hydraulic conductivity: 21 data sets, 409 data points
Oil sands tailings	Suthaker (1995), Pollock (1988), Jeeravipoolvarn (2005, 2010), Miller (2010), Beier (2015), Owolagba (2013), Farkish and Fall (2013), Yao (2016), and Wilson et al. (2018)
	Compressibility: 31 individual data sets, 290 data points
	Hydraulic conductivity: 36 data sets, 253 data points

Existing Hydraulic Conductivity Correlations

Most of the correlation relationships presented in the literature are listed in the “Appendix.” The majority of the equations relate

e

to

k

through log-linear expressions or power expressions, where the characteristic parameters are liquid limit (LL), plastic limit (PL), or liquidity index (LI) (and hence, the initial void ratio). Several equations are derived from the Kozeny-Carman equations (Carman 1939), and these equations require a determination of the specific surface of the soil—for these, it is often required to estimate the specific surface from the Atterberg limits.

Each method was tested against each class of soils or tailings, and the closest comparisons are shown in the Figs. S1–S4 in the “Supplemental Material” section. The results are presented in terms of a cumulative distribution function of the log of ratios between measured and predicted hydraulic conductivity data. The ratio of measured to predicted values are demonstrated in the horizontal line in logarithmic form, and the probability is presented in the vertical line, which falls between 0 and 1. A perfect match would be a straight vertical line at 0 on the

x

-axis (indicating that the predicted values are equal to the measured data). Methods that would provide a good fit on average would be centered around the same vertical line.

A number of predictive equations showed relatively good agreement with the measured data. It should be acknowledged that the specific surface methods may have apparently performed less well, only due to a lack of direct specific surface measurements for most data sets. The best agreement was shown by the Samarasinghe et al. (1982) and the Carrier and Beckman (1984) equations. The parameters of these equations [such as the power and modifier of the plasticity index (PI) in the Samarasinghe equation] were optimized for each of the three data sets, using a nonlinear least squares regression algorithm in MATLAB version R2018a, and are presented in Fig. 1. The optimized equations show high

R^{2}

values (

> 0.96

). These optimized equations are presented as follows:

•

Clays

k (m / s) = 1 \times 10^{- 10} {(P I)}^{0.6245} (\frac{e^{5}}{1 + e}) R^{2} = 0.97

(1)

•

Dredged materials

k (m / s) = 1.756 \times 10^{- 11} {(PI)}^{0.486} [\frac{e^{5}}{1 + e}] R^{2} = 0.96

(2)

LI = 18.67 {(k (1 + e))}^{0.043} - 7.353 R^{2} = 0.9884

(3)

•

Oil sands tailings

k (m / s) = 2 \times 10^{- 12} (PI) (\frac{e^{5}}{1 + e}) R^{2} = 0.97

(4)

Fig. 1. Best-fit $k - e$ models optimized to the data sets of (a) clay; (b) dredged material; and (c) oil sands tailings where colored dots denote the predicted values.

Suitability of Optimized Equations for Design

However, the relatively high

R^{2}

is somewhat misleading in terms of the utility for design of these

k - e

estimates. Even the optimized methods [Eqs. (1)–(4)] exhibit a relatively poor predictability at water contents above the liquid limit. Above this limit, an orders of magnitude difference between predicted and measured hydraulic conductivities at a given void ratio is expected for 20% of the dredged data and 20.6% for the oil sands data. This range will demonstrate very different results in a large strain consolidation analysis. This is illustrated in Fig. 2, which shows the settlements predicted from a hypothetical analysis for a 10-m deposit using the large-strain consolidation model UNSATCON (Qi et al. 2017). Predictions are made using either the measured

k - e

relationship,

k = 7.53 \times 10^{- 11} e^{3.8}

, as provided in Jeeravipoolvarn (2010), and using the prediction of Eq. (4),

k = 5 \times 10^{- 11} e^{5}

. The initial void ratio is 5.0, and the compressibility equation is

σ^{'} = 49.33 e^{- 3.6}

. The difference in the predictions is substantial.

Fig. 2. Large strain consolidation analysis of a hypothetical fluid fine tailings deposited at an original thickness and void ratio of $10 m$ and 4, respectively.

Predicting $k - e$ Using the Compressibility Curve

All data sets with both compressibility and

k - e

data were used to investigate if compressibility itself could be a useful predictor of

k - e

(17 for clay, 21 for dredged materials, and 28 for oil sands tailings). As shown in Fig. 3, many data sets displayed a very strong similarity between

k - e

and

e - σ_{v}^{'}

curves (where

k - e

data points are presented as dark grey dots and

e - σ_{v}^{'}

data is displayed in a light gray color), which could be defined

\log k = \frac{\log e}{A} - B

(5)

Fig. 3. Relationship between the hydraulic conductivity-void ratio-effective stress for untreated oil sands tailings.

If optimized parameters

A

and

B

are determined for each soil using a linear regression algorithm in MATLAB version R2018a and rearranged in a power form, then the soil specific equations are:

•

Clays

\log k = \frac{\log e}{0.2} - 9.436 \to k (m / s) = 3.66 \times 10^{- 10} e^{5}

(6a)

•

Dredged soils

\log k = \frac{\log e}{0.2} - 10.795 \to k (m / s) = 1.60 \times 10^{- 11} e^{5}

(6b)

•

Oil sands tailings

\log k = \frac{\log e}{0.2} - 10.5 \to k (m / s) = 3.16 \times 10^{- 11} e^{5}

(6c)

The equations are presented in terms of the most optimized integer value of power. The common parameter in all three equations is

e^{5}

: this is the optimal value for every half-power (that is, powers of 5.5 or 4.5 give poorer results). Fig. 4 displays different examples of fits to the linear relationship provided in Eq. (5) for various samples of fine-grained soils, where

A = 0.2

and parameter

B

varies for different types of soils. In almost all data sets, a good fit can be obtained by changing only the

B

parameter (the offset). However, using Eq. (6) only results in a marginally better agreement than using the optimized equations, as shown in Table 2.

Fig. 4. Improving the predictions of hydraulic conductivity when using a single measured data at a high-void ratio for both clays and dredged soils, also oil sands tailings (non-amended and amended) respectively.

Table 2. Sensitivity analysis for proposed optimized

k - e

relationships

Materials	Percent error less than order of magnitude
Materials	Optimized $k - e$ using Atterberg limits (%)	Optimized $k - e$ using compressibility curve [Eq. (6)] (%)	Eq. (7) (%)	Eq. (8) (%)
Clay	63	73.2	98	99
Dredged materials	80	78.4	94	92
Oil sands tailings	79.4	77.3	95	92

Predicting Void Ratio – Saturated Hydraulic Conductivity ( $e - k$ ) Relationship Using Atterberg Limits/Compressibility Curves with a Single Measured Data Point $k_{sat}$ at Higher Void Ratios

As all the methods examined show the poorest performance at high-void ratios, the authors have examined how the methods can be improved using only a single measurement of

k

at a high-void ratio. When either the Atterberg limit-based equations or the compressibility equations are combined with a single measured

k

at a known and high-void ratio, this results in the following equations:

k = k_{measured e_{0}} \frac{e^{5}}{e_{0}^{5}}

(7)

k = k_{measured e_{0}} \frac{e^{5}}{e_{0}^{5}} \frac{1 + e_{0}}{1 + e}

(8)

As shown in Fig. 4, the agreement is much improved. For Eqs. (7) and (8), the difference between the measured and estimated

k

values are less than an order of magnitude for 90% and 88% of the data for oil sands tailings, 98% and 99% of the data for clay soils, and 90% and 83% of the data for dredged soils, respectively. However, the

k

values at the very highest void ratios are probably the most susceptible to error, and it is more practical to target a given effective stress rather than a given void ratio when performing a measurement. Reanalyzing the data sets and picking the measured

k

value at the nearest 10, 5, 2, 1, 0.5, and 0.1 kPa, it is found that the most accurate predictions are made when the

k_{measured}

value is selected to be the measurement at an effective stress between 1 and 2 kPa, where there are multiple measurements between 1 and 2 kPa, and the one closest to 2 kPa is used. This improves the agreement for the dredged soil and tailings to 94% and 95% for Eqs. (7), respectively, and to 92% for both data sets for Eq. (8). An example of the agreement for a subset (9

k - e

functions) of oil sands tailings is shown in Fig. 5.

Fig. 5. Performance of Eq. (7) using $k$ measured for points between 1 and 2 kPa for amended oil sands tailings.

To show the utility afforded by the relatively good accuracy of Eqs. (7) and (8), predictions of a hypothetical soft soil deposit, instantaneously deposited at

e = 4

for a 10-m height is shown in Fig. 6. Each case is simulated using the same compressibility, as for Fig. 6, and the following three

k - e

functions:

•

Base case:

k (m / s) = 3 \times 10^{- 9} e^{5}

•

Higher

k

:

k (m / s) = 3 \times 10^{- 8} e^{3.34}

•

Lower

k

:

k (m / s) = 3 \times 10^{- 10} e^{6.66}

Fig. 6. Predictions of $10 m$ deposit $e_{0} = 4$ , assuming an order of magnitude difference of $k$ at $e = 1$ between the base case and either the higher or lower case.

These functions are chosen to give

k

values less than an order of magnitude apart at the initial void ratio but

k

values two orders of magnitude apart at

e < 1

. This would be a very conservative estimate of the error due to Eqs. (7) and (8). Even with this very conservative estimate of the error, the spread of predictions is relatively narrow. This is because the hydraulic conductivity values at a high-void ratio strongly influence the results of the consolidation modeling (

k - e

values at lower void ratios exert less and less influence), and these values are anchored to the measured value at a high-void ratio.

Limitations

The proposed method substantially abbreviates the time required to obtain a full

k - e

curve and provides estimations that are at least accurate enough to employ the method as a screening tool, say to examine the effect of different proposed amendments (polymer or coagulant type, dose, and mixing methods) on the consolidation efficiency of dredged soil or tailings deposits. However, for a final design, it would still be prudent to measure the full

k - e

curve. The proposed method relies on an accurate measurement of

k

at a high-void ratio, which can be a challenging task for many materials.

Summary and Conclusions

This paper analyzes correlations for

k - e

applied to soft soils: clays, clay-based tailings (oil sands), and dredged soils. The data set comprises 79 data sets of

k - e

. Modified versions of the best performing

k - e

correlations were obtained and optimized for the analyzed data on clays, oil sands tailings, and dredged soils. Despite high

R^{2}

coefficients, these optimized correlations would still not be sufficiently accurate for even a preliminary design for many applications, as demonstrated using a hypothetical large-strain consolidation analysis. Using the compressibility function as a predictor somewhat improved agreement. However, the agreement between measured and predicted

k - e

substantially improved when hydraulic conductivity values measured at effective stresses between 1 and 2 kPa were themselves used as predictors. This approach generated functions that depended only on one value of

k_{measured}

and assumed either powers of 4 or 5 for the void ratio. Although these powers emerged from the statistical analysis of a large data set, fundamental studies have long suggested a power between 4 and 5 (Richardson and Zaki 1997; Marshall 1958). A further discussion of fundamentals is beyond the scope of this paper, although we conclude with a cautionary comment that the hydraulic equation applicable to the sedimentation stage appears to be the same as the consolidation for some soils (Pane and Schiffman 1997) but not for others (Winterwerp 2002).

Notation

The following symbols are used in this paper:

$A$ , $B$ , $C$ , $D$ , $Z$: material coefficient;
$C_{K C}$ , $C_{p}$: Kozeny-Carman coefficient;
$D$: material coefficient;
$D_{R}$: specific weight;
$E$: permeability parameter;
$e$: void ratio;
$G$: gravitational acceleration;
$k$: hydraulic conductivity;
LI: liquidity index;
LL: liquid limit;
$P_{a t m}$: atmospheric pressure;
PI: plasticity index;
PL: plastic limit;
$p$: weight portion of clay minerals in soil;
SG: specific gravity;
SI: shrinkage index;
$S_{s}$: specific surface;
$x$: permeability parameter;
$γ_{w}$: specific weight of water;
$μ_{W}$: dynamic viscosity;
$ρ_{w}$: density of water;
$σ^{'}$: effective stress; and
$υ_{s}$: velocity of tailings-water interface.

Supplemental Materials

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Appendix. Summary of Correlation Relationships for $k - e$ Relation

Correlation relationship	Materials	References
$\log k = \frac{e}{0.01 (P I) + 0.05} - 10$	Clays	Nishida and Nakagawa (1969)
$k = (0.00104 P I^{- 5.2}) [\frac{e^{x}}{1 + e}]$	Normally consolidated remolded clay	Samarasinghe et al. (1982)
$L I = 95.21 {(k (1 + e))}^{0.233} - 0.242$	Clay and Slurried mineral waste	Carrier et al. (1983)
$k = \frac{0.0174 {(\frac{e - 0.027 [(P L) - 0.242 (P I)]}{P I})}^{4.29}}{(1 + e)}$	Remolded clay	Carrier and Beckman (1984)
$\frac{e}{e_{L}} = 2.38 + 0.233 \log k$	Normally consolidated clays	Nagaraj et al. (1993)
$\frac{e}{e_{L}} = 2.162 + 0.195 \log k$	Over-consolidated clay	Nagaraj et al. (1994)
$\log k = \frac{e - 0.0535 (L L) - 5.286}{0.0063 (L L) + 0.2516}$	Mine tailings	Sivapullaiah et al. (2000)
$\frac{e}{e_{L}} = 29.80 {(k (1 + e))}^{0.177} - 0.09527$	Dredged materials	Morris et al. (2000)
$\frac{e}{e_{L}} = 2.23 + 0.204 \log k$	Normally consolidated clays	Sridharan and Prakash (2002)
$k (c m / s) = C_{p} \frac{γ_{w}}{μ_{w}} \frac{e^{3 + x}}{1 + e} \frac{1}{ρ_{s}^{2} {(L L)}^{2 χ}}$ where $x = 7.7 {(L L)}^{- 0.15} - 3$	Plastic soils	Mbonimpa et al. (2002)
$\log [k_{pre d i c t e d}] = 0.5 + \log [\frac{e^{3}}{D_{R}^{2} S_{S}^{2} (1 + e)}]$ where $\frac{1}{S_{S}} = 1.3513 (\frac{1}{L L}) - 0.0089$	Clay	Chapuis and Aubertin (2003)
$\log [\frac{k}{1 m / s}] = 1.5 {0.5 + \log [\frac{e^{3}}{D_{R}^{2} S_{S}^{2} (1 + e)}]} + 2$	Mine tailings	Chapuis and Aubertin (2003)
$\frac{e}{e_{L}} = 12.55 {(k (1 + e))}^{0.109} - 0.372$	Dredged materials	Morris (2003)
$k = (2.5 \times 10^{- 4} {(S I)}^{- 3.69}) [\frac{e^{3.79 + 0.044 (S I)}}{1 + e}]$	Remolded clays	Sridharan and Nagaraj (2005)
$k = \exp [- 5.51 - 4 ln (P L)] {(e)}^{7.52 \exp [- 0.25 (L I)]}$	Dredged materials	Berilgen et al. (2006)
$k = \frac{6.31 \times 10^{- 7}}{{(P I - 8.74 p)}^{3.03}} e^{2.66 {(P I - 8.47 p)}^{0.234}}$ $\begin{matrix} A_{s} = (L L - 31.91 p) / 0.81 \\ A_{s} = (P L - 23.16 p) / 0.27 \\ A_{s} = (P I - 8.47 p) / 0.54 \end{matrix}$	Clay	Dolinar (2009)
$k = [1.06 \times 10^{- 5} {(P I)}^{- 2.24}] {(e)}^{16.23 {(P I)}^{- 0.4}}$	Fine-grained soils	Chandra Paul and Azam (2013)
$\begin{array}{l} k (m / s) = 10^{- 7} {(\frac{S_{s}}{m^{2} / g})}^{- 2} e^{β} \\ β = 5 \pm 1 and S_{s} = 1.8 (L L) - 34 \end{array}$	Fine-grained soils	Ren and Santamarina (2018)

Data Availability Statement

Some or all data, models, or code generated or used during the study are available in a repository or online in accordance with funder data retention policies. (URL: https://paulsimms0.wixsite.com/tailings-carleton/research).

Acknowledgments

The research was supported by funding from the Natural Sciences and Engineering Research Council (NSERC) and Canada’s Oil Sands Innovation Alliances (COSIA).

References

Azam, S. 2011. “Large strain settling behavior of polymer-amended laterite slurries.” Int. J. Geomech. 11 (2): 105–112. https://doi.org/10.1061/(ASCE)GM.1943-5622.0000031.

Abstract

Introduction

Existing Hydraulic Conductivity Correlations

Suitability of Optimized Equations for Design

Predicting k-e Using the Compressibility Curve

Predicting Void Ratio – Saturated Hydraulic Conductivity (e-k) Relationship Using Atterberg Limits/Compressibility Curves with a Single Measured Data Point ksat at Higher Void Ratios

Limitations

Summary and Conclusions

Notation

Supplemental Materials

Appendix. Summary of Correlation Relationships for k-e Relation

Data Availability Statement

Acknowledgments

References

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Predicting $k - e$ Using the Compressibility Curve

Predicting Void Ratio – Saturated Hydraulic Conductivity ( $e - k$ ) Relationship Using Atterberg Limits/Compressibility Curves with a Single Measured Data Point $k_{sat}$ at Higher Void Ratios

Appendix. Summary of Correlation Relationships for $k - e$ Relation