A New Model for Evaluating the Behaviour of Swelling Soils

Abstract

In many areas around the world, there are clayey soils that have the potential to change their volume caused by the variation in their water content. Increasing or decreasing the water content caused the clayey soils to swell or shrink, respectively. This phenomenon may cause the uplifting and settlement of structures, which may lead to considerable financial damages. The estimation of swelling displacement without addressing the swelling rate have been published by several research works. This drawback leads to the development of a new model that takes into account the swelling behavior of soils with time. The model, which consists of two hyperbolic curves, was compared with swelling test results performed on soil samples taken from several locations in Israel. Data test results were used to compare the newly introduced model with other existing mathematical models found in the literature. This analysis shows that the new model represents more accurately the behavior with time of the swelling clayey soils measured in laboratory test results than the existing hyperbolic models.

Introduction

In many areas worldwide, there are clayey soils that have the potential to change their volume due to the variation of their water content. Increasing or decreasing the water content caused the clay soils to swell or shrink, respectively. The phenomenon of swelling-shrinking may cause the uplifting and settlement of structures, which may lead to considerable financial damages. In semi-arid countries, the penetration of water can reach several meters below the ground surface and may cause vertical and horizontal pressure on the structures. The amplitude of the swelling–shrinking phenomenon depends, among other things, on the initial water contents, the liquid limit, the initial density of the clayey soils, and the type of clay. The type of clay soils may consist of several minerals such as Montmorillonite, Illite, and Kaolinite. The clay soils that contain Montmorillonite have the most potential for swelling and shrinking. Several research works have been published regarding the estimation of the swelling displacement and the developed pressure of the clayey soils without addressing the swelling rate that developed with time [1,2,3,4]. This drawback leads to the development of a new kinetic model considering the adsorption behavior of swelling clay with time when submerged in distilled water.

Existing Models of Adsorption

The use of kinetics models to describe and estimate the adsorption of a liquid into an absorbent material has been studied since the end of the nineteenth century [5]. These studies were performed in order to evaluate the ending value of the adsorption process (isotherm) and the rate and shape of the adsorption kinetic models. The submersion of clayey soils in water creates an adsorption process which causes an augmentation of soil volume. Clay soils have a layered mineralogical structure. The expansion of clayey soils is due to the attraction of negative ions present at the surface of the clay layers with positive ions of the dipolar water molecules (adsorption). That induces the penetration of water molecules between the clay mineralogical layers and thus an augmentation of the volume of the clayey soil [6, 7]. This augmentation of volume may last several weeks until it reaches a stage of stabilization or a relatively small variation of volume with time. At this stage, the soil is almost saturated and there is extremely small additional attraction between anions present at the surface of the clay sheets and cations present in the water. The kinetics of the swelling of clayey soils include the following three stages: (i) A relatively fast rate of swelling. At this stage, water penetrates into the soil, and its water content increases relatively very rapidly. (ii) An intermediary stage when the swelling begins to stabilize, (iii) A third stage when the swelling rate is relatively very low. The following existing models of adsorption liquid on materials were first analyzed in order to develop a new model for evaluating the adsorption behavior of swelling clayey soils with time:

Pseudo-First-Order (PFO) Kinetic Model

General Formulation (Langmuir [5] and Lagergren [8])

This model is based on the following differential equation [9,10,11,12,13,14,15,16,17]:

$$\frac{dS(t)}{dt}={k}_{1}({S}_{f}-S(t))$$

(1)

where $S(t)$: The swelling fraction at time t, ${S}_{f}$: The swelling fraction at the end of the experiment, ${k}_{1}$: A first-order constant or first-order swelling rate [mm⁻¹].

By integration, Eq. (1) can be written as follows:

$$S(t)={S}_{f}(1-exp(-{k}_{1}t))$$

(2)

Maghrabi Model

The Maghrabi Model which uses the following equation is based on the Pseudo-First-Order Kinetic Model: [18,19,20]

$$S(t)={S}_{f}\left(1-\frac{1}{{e}^{kt}+C\sqrt{t}}\right)$$

(3)

Where $k$: is a constant [min⁻¹]$,C$: is a loss parameter $\left[ {{\text{min}}^{ - 1/2} } \right]$.

According to the Maghrabi Model, the parameter $\text{k}$ is reliant on the viscosity of the water-based mud system. Maghrabi added a filtrate loss parameter denoted as $C$ assigned to a value between zero and one.

Pseudo-Second-order (PSO) Kinetic Model

General Formulation (Blanchard et al. [18])

This model uses the following differential equation: [12,13,14, 16, 17, 21,22,23,24,25]

$$\frac{dS(t)}{dt}={k}_{2}({S}_{f}-S(t){)}^{2}$$

(4)

where: ${k}_{2}$: is a pseudo-second-order constant.

By integration, Eq. (4) can be written in three different forms as follows:

$$S(t)=\frac{{S}_{f}^{2}{k}_{2}t}{1+{S}_{f}{k}_{2}t}=\frac{t}{\frac{1}{{S}_{f}^{2}{k}_{2}}+\frac{1}{{S}_{f}}t}=\frac{{S}_{f}t}{\frac{1}{{S}_{f}{k}_{2}}+t}$$

(5)

The hyperbola equation (5) can be rewritten:

$$\frac{t}{S(t)}=\frac{t}{{S}_{f}}+\frac{1}{{k}_{2}{S}_{f}^{2}}$$

(6)

Equation (6) is linear in the coordinates $t/S(t)$ versus $t$.

Dakshanamurthy and Peleg Model

Dakshanamurthy [26] and Peleg [27] developed the following empirical equation for estimating the swelling of clayey soils and for the adsorption of milk powder and rice, respectively, $S(t)$:

$$S(t)=\frac{t}{a+bt}$$

(7)

where a and b are parameters of the hyperbola. Compared to Eq. (5), coefficients a and b can be written as follows:

$$\text{a}=\frac{1}{{S}_{f}^{2}{k}_{2}} b=\frac{1}{{S}_{f}}$$

(8)

Vayssade Model

Vayssade [28] proposed the following equation for estimating the relative displacement of clayey soils $S(t)$ at time $t$:

$$S(t)=\frac{t{S}_{f}}{B+t}$$

(9)

where: $B$: A constant depending on the clayey soil.

In the coordinates $S(t)$ versus $S(t)/t$, Eq. (9) is linear ($S(t)={S}_{f}-B(S(t)/t$) and the coefficient $B$ can be obtained by measuring the slope of this line. ${S}_{f}$ can be obtained by extrapolation of the line $S(t)/t$ equal to zero. According to Vayssade [25] model:

$$B={t}_{0.5}=t\left(\frac{1}{2}{S}_{f}\right)$$

(10)

where: ${t}_{0.5}$: The time at half stabilization swelling ${S}_{f}$.

Equation (9) can be written as follows:

$$S(t)=\frac{{S}_{f}t}{{t}_{0.5}+t}$$

(11)

Parcevaux Model

Parcevaux [29] proposed the following equation for estimating the swelling of clayey soils $S(t)$:

$$S(t)=G\frac{t}{B+t}$$

(12)

where: $G$ and $B$: Constants of the hyperbola depending on the clayey soil properties.

Equation (12) can be rewritten:

$$\frac{t}{S(t)}=\left(\frac{1}{G}t+\frac{B}{G}\right)$$

(13)

The parameters $G$ and $B$ can also be obtained by drawing the linear Eq. (12) in the coordinates $t/S(t))$ versus $t$. $G$ is the inverse of the slope of the linear line and $B/G$ is obtained when the time is equal to zero.

The parameters ${k}_{1}$, $k$, $C$, ${k}_{2}$, $a$ and $B$ in Eqs. (2), (3), (4), (7), (9), (12) are constants which influence the rate of the curvature of the functions mentioned above.

Development of a New Model for Evaluating the Adsorption Behaviour of Swelling Soils

Formulation of the Splitting Pseudo-Second-Order Model

The main objective of this study is to find a new model that estimates more accurately the behavior of adsorption of water in soils that cause swelling of clayey soils. One of the drawbacks of the above-mentioned models is that they do not include time-dependent parameters. Their constant parameter does not allow to estimate accurately the behavior of the swelling of soil development with time. That leads to the development of a new empirical model that includes time-dependent parameters allowing the description of this behavior. The following new developed pseudo-second-order model equation, which includes a time-dependent rate function $C(t)$, describes the behavior of soil swelling, $S(t)$. The new model includes a hyperbolic function $C(t)$ that changes with time. This model can be written as follows:

$$S(t)=t\frac{{S}_{f}(1+\frac{C({t}_{end})}{{t}_{end}})}{C(t)+t}$$

(14)

where ${S}_{f}$: The stabilized swelling fraction at the end of the experiment, ${t}_{end}$: Elapsed time at the stabilization, $C(t)$: Curvature coefficient function dependent on time, $C({t}_{end})$: Curvature coefficient at the end of the test.

In order to describe the adsorption behavior of the swelling process Eq. (14) is divided into two sections, which are defined by the two following equations ${S}_{1}(t)$ and ${S}_{2}(t)$. The hyperbolic function ${S}_{1}(t)$ starts at the beginning of the test until the displacement reaches a value slightly greater than ${S}_{f}/2$. The function ${S}_{2}(t)$ starts at the displacement ${S}_{f}/2$ until the swelling is stabilized at the end of the test (Appendix A).

$${S}_{1}(t)=\frac{t{S}_{f}(1+{C}_{1}{e}_{c})}{{C}_{1}+t} t\in [0,\approx {t}_{{s}_{f}/2}+]$$

(15)

and

$${S}_{2}(t)={S}_{f}\frac{{C}_{2}+t(1+{C}_{2}{e}_{c})}{2{C}_{2}+t} t\in [\approx {t}_{{s}_{f}/2}+,{t}_{end}]$$

(16)

where ${C}_{1}$: Curvature coefficient of the hyperbola ${S}_{1}(t)$, ${C}_{2}$: Curvature coefficient of the hyperbola ${S}_{2}(t)$. ${e}_{c}$: Ending correcting factor equals $1/{t}_{end}$. ${t}_{{S}_{f}/2}+$: Slightly greater than the time at half swelling.

Computation of ${{\varvec{C}}}_{1}$ and ${{\varvec{C}}}_{2}$

${C}_{1}$ And ${C}_{2}$ can be evaluated by using a least-square method or by adjustment of ${S}_{1}(t)$ and ${S}_{2}(t)$ equations to the swelling test results. Another way to evaluate ${C}_{1}$ is by substituting ${t}_{{S}_{f}/2}$ (time of half swelling) into Eq. (15) (Appendix C). ${C}_{2}$ can also be evaluated by substituting ${t}_{inf{l}_{s}}$ into Eq. (16), where ${t}_{inf{l}_{s}}$ is the time of inflection point which defined as the beginning of the secondary swelling (Appendix C). ${t}_{int}$ is the time where Eqs. (15) and (16) intersect. It can be approximately evaluated as ${t}_{int}\approx \frac{{C}_{1}{C}_{2}}{({C}_{2}-{C}_{1})}$ (Appendix B).

Test Results Performed on Clay Soils

Experiment Process for Obtaining Test Results

Swelling tests were performed according to ASTM 4546-96 [30] in order to evaluate the accuracy of the new model. Controls 26-WF31E20/SW consolidometer apparatus was used for measuring the vertical displacement. The consolidometer has two sensors, one sensor for measuring the vertical displacement and one sensor for maintaining a constant vertical pressure. A fixed constant vertical pressure of one kPa was applied to the dry sample. The initial pressure remained on the specimen until the displacement was stabilized. Petro Tech distilled water was used to submerge the soil sample in the consolidation ring. The consolidometer recorded the vertical swelling displacement of the soil sample which was developed during 24 h.

Description of the Soil Sample

The soil sample was taken from an Ariel site at a depth of 50 cm from the ground surface. The samples were classified according to ASTM D4318-17 [31] as CH (High plasticity Clay).

Preparation and Characteristics of Soil Sample

The soil sample was first washed out through a sieve ASTM # 200 (opening of 75 microns) into a large container to eliminate sand and pebbles. The large container stayed still for at least 24 h. The clear limpid water was taken out using a siphon. The mixture of silt and clay lying on the bottom was dried out in an oven at 60 °C for 3 days. The dried mixture was crushed with a cylindrical plastic hammer and passed through a sieve ASTM# 40 (opening of 425 microns) to get powder soil. The fine particles of the powder soil sample were classified according to ASTM D7928-21e1 [32]. The hydrometer analysis shows that 58% of all particles are less than 0.002 mm (Fig. 1). This dry powder soil sample was poured into the consolidometer mold of 20 ${\text{cm}}^{2}$ area. The upper surface was levelled using 300 g cylinder which caused 1.5 kPa pressure on the soil sample. A porous stone and a cap causing extra pressure of 0.5 kPa were laid on the top of the specimen. Table 1 shows the characteristics of the soil specimen at the beginning of the test.

Fig. 1

Hydrometer analysis of ariel soil

Table 1 Characteristics of the Ariel soil specimen at the beginning and the end of the test

Swelling Test Results

Table 1 summarizes the characteristics of the Ariel soil sample at the beginning of the test and after the swelling was stabilized at the end of the test. Figure 1 shows the Hydrometer Analysis of Ariel Soil.

Figures 2 and 3 show the test result of swelling [%] versus time [minutes] in linear and semi-logarithmic scales respectively.

Fig. 2

Test results at constant vertical pressure of 1kPa (linear scale)

Fig. 3

Test results at constant vertical pressure of 1kPa (semi-logarithmic scale)

The test was ended at a displacement rate of 0.003 mm per hour which was obtained after 24 h. The following observations can be derived from the test results:

• 1. The swelling begins after a relatively short time, about 5 s, after pouring the water into the consolidation ring.

• 2. The rate of swelling increases very sharply after about 5 s and stabilizes at the end of the test.

• 3. The moisture content of the soil sample (ω) reaches 53% at the end of the experiment.

• 4. The total unit weight of the soil sample (${\gamma }_{t}$) increases during the saturation process.

• 5. The dry unit weight of the soil sample (${\gamma }_{d}$) decreases during the saturation process.

• 6. The void ratio of the soil sample ($e$) significantly increases during the saturation process.

• 7. The three phases of swelling, initial, primary, and secondary, can be observed in the semi-logarithmic swelling graph (Fig. 3).

• 8. The inflection point at the primary swelling phase is relatively easily visible.

• 9. The primary swelling started after 30 s from the beginning of the test which is 4% of the total swelling.

• 10. The secondary swelling started after 406 min from the beginning of the test which is 97% of the total swelling.

• 11. An inflection change of rate during the primary swelling phase starts at 5 min from the beginning of the test which is 54% of the total swelling.

Comparison Between Swelling Test Results and the New Model

The coefficients ${C}_{1}$ and ${C}_{2}$ were obtained using the three methods mentioned above; least-square, adjustment, and substituting ${t}_{{S}_{f}/2}$ and ${t}_{inf{l}_{s}}$ into Eqs. (15) and (16) respectively (Table 2). Figures 4 and 5 show the relationship between the swelling versus the time for the coefficients ${C}_{1}$ and ${C}_{2}$ which were obtained using the adjustment method.

Table 2 Values of the coefficients ${C}_{1}$, ${C}_{2}$, and the intersection time of the two hyperbolae (${S}_{1}(t)$ and ${S}_{2}(t)$) of the new model using three different methods

Fig. 4

Comparison between the test results and the new model using the coefficient ${C}_{1}$ (semi-logarithmic scale)

Fig. 5

Comparison between the test results and the new model using the coefficient ${C}_{2}$ (semi-logarithmic scale)

Figures 6 and 7 shows the relationship between the swelling versus the time for the test results and the new model, on a linear and semi-logarithmic scale, respectively. These figures show the good adjustment between the new model and the test results.

Fig. 6

Comparison between the behavior of free swelling of the test results on ariel soil and the new model (linear scale)

Fig. 7

Comparison between the behavior of free swelling of the test results on ariel soil and the new model (semi-logarithmic scale)

The adjustment of the new model and the swelling test results are evaluated using the following statistical equations:

$$RMSE=[\frac{1}{N}\sum_{i=1}^{N}({d}_{i}-{S}_{i}{)}^{2}{]}^{1/2}$$

(17)

where $RMSE$: Root Mean Square Error, $N$: Number of time samples, ${d}_{i}$: Data at time i, ${S}_{i}$: Model at time i.

$${R}^{2}=1-\frac{\sum_{i=1}^{N}({d}_{i}-{S}_{i}{)}^{2}}{\sum_{i=1}^{N}({d}_{i}-\widehat{d}{)}^{2}} \,with: \widehat{d}=\frac{1}{N}\sum_{i=1}^{N}{d}_{i}$$

(18)

where ${R}^{2}$: Root Mean Square Error, $\widehat{d}$: The average Data value.

$$KS=\,\stackrel{N}{\underset{i=1}{\text{max}}}(|CS{D}_{i}-CS{S}_{i}|)$$

(19)

with:

$$\begin{array}{ll} {CSD_{i} = \sum\limits_{i = 1}^{i} {d_{i} } } \\ {CSS_{i} = \sum\limits_{i = 1}^{i} {S_{i} } } \\ \end{array}$$

(20)

where $KS$: The Kolmogorov–Smirnov parameter, $CS{D}_{i}$: Cumulative sum of experiment data at time i, $CS{S}_{i}$: The cumulative sum of model data at time i.

The criteria for adjustment evaluation of the new model to the test results for the statistical Eqs. (17), (18), and (19) are the following: The more the ${R}^{2}$ parameter is close to one, the better the adjustment. The more the $RMSE$ parameter is close to zero, the better the adjustment. The more the $KS$ parameter is small, the better the adjustment. The computed parameters ${R}^{2}$, $RMSE$ and $KS$ obtained for the new model are 0.995, 0.4038, and 10.98 respectively (Table 3). These parameters show a good agreement between the suggested new model and the swelling test results on Ariel soil.

Table 3 Statistical parameters obtained from analyses for the new model and the existing models using test results on Ariel soil

Comparison Between Swelling Test Results, the New Model, and the Existing Swelling Models

Figures 8 and 9 shows the comparison between the swelling test results for the Ariel soil and the swelling behavior obtained using different models. It can seen that the best agreement between the test results is for the new suggested model. The statistical parameters calculated for the different models using Eqs. (17), (18), and (19) show that the new presented model has the best adjustment with the swelling test results (Table 3).

Fig. 8

Comparison between different (PFO) model analyses with test data results (semi-logarithmic scale)

Fig. 9

Comparison between different (PSO) model analyses with test data results (semi-logarithmic scale)

Application of the New Model for Different Clayey Soils

The new model was applied to estimate the swelling behavior of six different types of clayey soils. Table 4 summarizes the characteristics of these soils obtained by tests performed according to the ASTM as described above. The tests were performed according to the process described in paragraph [4.3]. Figures 10, 11, 12, 13, 14, and 15 show the results of the analyses using the new suggested model and the existing PSO models. Tables 5 and 6 show the ${C}_{1}$ and ${C}_{2}$ parameters (Eqs. 15 and 16) obtained for the new model and Table 7 shows B parameters (Eq. 9) obtained for the PSO model. According to these analyses, it can be deduced that the new proposed model estimates more accurately the behavior of the swelling soils than the existing PSO model (Table 8).

Table 4 Geotechnical characteristics of tested clay soils

Fig. 10

Comparison between the swelling of highway 4 soil test results with the new and the existing PSO models

Fig. 11

Comparison between the swelling of Petach Tikva soil test results with the new and the existing PSO models

Fig. 12

Comparison between the swelling of highway 6 soil test results with the new and the existing PSO models

Fig. 13

Comparison between the swelling of North Valley soil test results with the new and the existing PSO models

Fig. 14

Comparison between the swelling of Haifa soil test results with the new and the existing PSO models

Fig. 15

Comparison between the swelling of Montmorillonite clay soil test results with the new and the existing PSO models

Table 5 Values of ${C}_{1}$ and ${C}_{2}$ parameters, intersection time (least square method), and the maximum swelling for different soils obtained for the new model

Table 6 Values of ${C}_{1}$ and ${C}_{2}$ parameters, intersection time (adjustment method), and the maximum swelling for different soils obtained for the new model

Table 7 Values of B parameter (Eq. 9) for the existing PSO model obtained for several soils

Table 8 Comparison of statistical analysis results between the new model and the existing PSO model for several soils

Conclusions

• 1. The statistical analysis shows that the new proposed model can estimate in a relatively accurate manner, the behavior of swelling clayey soils caused by changing their water contents.

• 2. The new introduced swelling model allows estimating the behavior of the swelling process of clayey soils during the laboratory test more accurately than the estimation obtained using the existing PSO models.

• 3. The statistical analyses show that the new proposed model can estimate, in a relatively accurate manner, the swelling behavior of the clayey soils tested in this research work.

Appendices

Appendix A

Demonstration of Eq. (16) from Eq. (14)

$$S(t)=t\frac{{S}_{f}(1+\frac{C({t}_{end})}{{t}_{end}})}{C(t)+t}$$

(A1)

Assumptions:

$${C}_{2}=C({t}_{end})$$

$$C(t)=\frac{{C}_{2}t}{{C}_{2}+t}$$

$${e}_{c}=\frac{1}{{t}_{end}}$$

$${C}_{2}(1+{e}_{c})\approx {C}_{2}$$

Equation (14) becomes Eq. (16):

$$S(t)={S}_{f}\frac{({C}_{2}+t(1+{C}_{2}{e}_{c}))}{2{C}_{2}+t}$$

(A2)

with:

$$S(0)=\frac{{S}_{f}}{2}$$

$$S({t}_{end})={S}_{f}$$

Appendix B

Demonstration of ${{\varvec{t}}}_{{\varvec{i}}{\varvec{n}}{\varvec{t}}}\approx \frac{{{\varvec{C}}}_{1}{{\varvec{C}}}_{2}}{{{\varvec{C}}}_{2}-{{\varvec{C}}}_{1})}$

Equating Eq. (15) to (16):

$${S}_{1}(t)={S}_{2}(t)$$

(B1)

Approximations:

$${C}_{1}{e}_{c}\approx {C}_{2}{e}_{c}\approx 0$$

$$\frac{t}{{C}_{1}+t}\approx \frac{{C}_{2}+t}{2{C}_{2}+t}$$

Time of intersection of the two curves:

$${t}_{int}\approx \frac{{C}_{1}{C}_{2}}{{C}_{2}-{C}_{1}}$$

(B2)

Appendix C

Computation of ${{\varvec{C}}}_{1}$ and ${{\varvec{C}}}_{2}$ by Substitution

From Eq. (15):

$${S}_{1}(t)=\frac{t{S}_{f}(1+{C}_{1}{e}_{c})}{{C}_{1}+t}$$

(C1)

$${S}_{1}({t}_{{s}_{f}/2})=\frac{{S}_{f}}{2}$$

where ${t}_{{s}_{f}/2}$ the time when half the swelling occurs.

$$\frac{{S}_{f}}{2}=\frac{{t}_{{s}_{f}/2}{S}_{f}(1+{C}_{1}{e}_{c})}{{C}_{1}+{t}_{{s}_{f}/2}}$$

Approximations:

$${C}_{1}{e}_{c}\approx 0$$

${C}_{1}$ can be expressed:

$${C}_{1}\approx {t}_{{s}_{f}/2}$$

(C2)

From Eq. (16):

$${S}_{2}\left(t\right)={S}_{f}\frac{{C}_{2}+t\left(1+{C}_{2}{e}_{c}\right)}{2{C}_{2}+t}$$

(C3)

Observe the time and the swelling at the secondary swelling inflection: ${t}_{inf{l}_{s}}$ and ${S}_{2}({t}_{inf{l}_{s}})$ (Fig. 3). The Eq. (16) becomes:

$${S}_{2}({t}_{inf{l}_{s}})={S}_{f}\frac{({C}_{2}+{t}_{inf{l}_{s}}(1+{C}_{2}{e}_{c})}{2{C}_{2}+{t}_{inf{l}_{s}}}$$

Approximations:

$${C}_{2}{e}_{c}\approx 0$$

$${C}_{2}\approx {t}_{inf{l}_{s}}\frac{{S}_{f}-{S}_{2}({t}_{inf{l}_{s}})}{2{S}_{2}({t}_{inf{l}_{s}})-{S}_{f}}$$

(C4)