Abstract
Permeability is one of the most important engineering properties of clayey soil. However, the traditional method for determining the permeability coefficient is time-consuming. To establish a simple and accurate predictive method to obtain the permeability coefficient of artificial clayey soil based on the double cut-off transverse relaxation times (T2 cut-offs) using low-field nuclear magnetic resonance (NMR) technology, eight kinds of artificial clayey soil with different mineralogical compositions were prepared in the laboratory. Evaporation tests at 40°C were carried out on the saturated artificial clayey soil samples in an oven. During the evaporation process, NMR tests were also performed on the artificial clayey soil every hour. The results showed that the evaporation process could be divided into three stages according to different evaporation rates: the constant rate stage (CRS), the falling rate stage (FRS), and the residual stage (RS). The water evaporated in the CRS and FRS was defined as the absolute movable water and the partially movable water, respectively. The water that could not evaporate in the RS was defined as the immovable water. Based on the cumulative signal amplitudes in the T2 spectrum corresponding to different kinds of water, the double T2 cut-offs were defined. On the basis of the double T2 cut-offs and T2 spectrum of the saturated sample, an improved Timur–Coates (TC) model was established. The prediction capability of the improved model was evaluated by finding the determination coefficient (R2), mean absolute error (MAE), and root-mean-square error (RMSE). Compared with the typical TC model, the prediction accuracy of the improved model was much higher. In addition, the relationships between the double T2 cut-offs and fractal dimension (D) of the T2 spectrum of saturated artificial clayey soil were also identified.
1. Introduction
In recent years, artificial clayey soil has been widely used in various engineering constructions [1, 2]. As one of the most important engineering parameters, the permeability coefficient is closely related to the consolidation deformation of soil, seepage of groundwater, and diffusion of pollutants in environmental soil [1–3]. The permeability coefficient of artificial clayey soil needs to be determined before the design and construction of civil engineering, highway engineering, dam engineering, and refuse treatment engineering projects. The most commonly used method for measuring the permeability coefficient of an artificial clayey soil is variable head permeability testing. However, variable head permeability testing is time- and effort-consuming [4]. Many previous studies have proposed models for predicting the permeability coefficient of clayey soil [1, 3–9]. However, these models need other parameters which are hard to obtain, and the errors between the measured and predicted values of permeability coefficients cannot be ignored. Therefore, it is necessary to build a simple and accurate predictive model.
In recent years, low-field NMR has been widely used to characterize the pore size distributions of porous media such as coal, shale, soil, and sandstone based on the T2 spectra [10–13]. Based on the pore size distributions of porous media, many studies have been performed to predict the permeability of these porous media. Yao et al. established a permeability model for coal from the Schlumberger Doll Research (SDR) equation and Timur–Coates (TC) equation by using NMR relaxometry [14]. Further, Li et al. classified the pore types of shale using the T2 spectrum and proposed a permeability model for shale based on the regional TC model [15]. Rezaee et al. also developed a predictive model for sandstone on the basis of the NMR parameters from the T2 spectrum [16]. However, few studies have focused on predictive models based on NMR for clayey soil. Therefore, to simplify the determination of the permeability coefficient of clayey soil, we previously proposed a novel method for predicting the permeability coefficients of artificial clayey soil based on the TC model by using low-field NMR [17]. This novel method was proved to have sufficient prediction capability. In this previous study [17], the water in the soil was classified into two kinds: free water and bound water. Based on this classification, a TC model for artificial clayey soil was proposed. The TC model was also called the free fluid model. In theory, the greater the detail in which the water in artificial clayey soil is classified, the higher the prediction accuracy of the model.
On the basis of our previous study [17], we continued to research on this prediction. The aim of this paper was to establish an improved model for predicting the permeability coefficients of clayey soil. Instead of the classification of water used in the previous study, the water in artificial clayey soil was divided into three kinds—absolute movable water, partially movable water, and immovable water—in this study. Based on this novel classification, an improved TC model was developed. The determination coefficient (R2), mean absolute error (MAE), and root-mean-square error (RMSE) were used to evaluate the prediction accuracy of the improved model. Compared with the model proposed in our previous study, the prediction reliability of the novel model is better. The double T2 cut-offs were defined to distinguish cumulative signal amplitudes in the T2 spectrum corresponding to the weights of different kinds of water in artificial clayey soil. Similar to our previous study, the relationships between the double T2 cut-offs and fractal dimension (D) were also established to remove evaporation testing from the determination process of the permeability coefficient. Using this method, the permeability coefficient of artificial clayey soil can be determined simply, quickly, and accurately.
2. Materials and Methods
2.1. Materials
Two clay mineral (montmorillonite and kaolinite) powders and one sand powder were used in this study to prepare artificial clayey soil. The sand powder was purchased from Guangzhou Jialiang Mineral Products Co., Ltd. (Guangzhou, China). The montmorillonite and kaolinite powders were provided by Inner Mongolia Ningcheng Tianyu Chemical Co., Ltd. (Chifeng, China) and Guangzhou Yifeng Chemical Technology Co., Ltd. (Guangzhou, China), respectively. A Mastersizer 2000 laser particle analyzer was used to determine the grain size distributions of these three mineral powders in this study. The grain size distributions are depicted in Figure 1. The Atterberg limits, specific gravity values, and classifications of the three kinds of powders were determined following the methods in ASTM D4318, ASTM D2487, and ASTM D2478, respectively. The X-ray fluorescence was also used to determine the chemical compositions of the powders. The basic properties of the three powders, including their Atterberg limits, specific gravity, and chemical composition, are summarized in Table 1.
2.2. Methods
2.2.1. NMR Theory
As a nondestructive, fast, and accurate testing technology, nuclear magnetic resonance technology is widely used to evaluate the pore characteristics of porous media [10, 12]. The NMR theory is briefly summarized as follows.
NMR relaxation consists of three parts: bulk relaxation, diffusion relaxation, and surface relaxation. The relaxation time can be divided into two categories: longitudinal relaxation time (T1) and transverse relaxation time (T2). The transverse relaxation time is more often used because it is shorter than the longitudinal relaxation time and easier to measure using an NMR analyzer [18]. The transverse relaxation time can be calculated by the following equation [10, 19–21]:where T2B is the transverse bulk relaxation time, T2D is the transverse diffusion relaxation time, and T2S is the transverse surface relaxation time.
T2D can be calculated by the following [22]:where Cd is the diffusion coefficient of the fluid in the pores, G is the gradient magnetic field intensity, and TE is the echo interval time. When the magnetic field is uniform, G equals 0.
T2S can be calculated by the following equation [23]:where S is the surface area of the pore, V is the volume of the pore, and ρ2 is the surface relaxivity of the pore surface. The surface relaxivity is mainly affected by the physicochemical properties of the soil particle surface [24].
The relationship between the surface area of the pore, the volume of the pore, and the diameter of the pore can be expressed by the following equation:where r is the diameter of the pore and Fs is the shape factor of the pore. For planar, columnar, and spherical pores, Fs is equal to 1, 2, and 3, respectively.
Equation (4) can be converted into
When the magnetic field is uniform, G equals 0, so 1/T2D equals 0. When the liquid in the pore is distilled water, T2B is much larger than T2S, so 1/T2B can also be considered negligible [25]. Equation (1) can then be simplified into the following equation:
From the previous equation, there is a linear relationship between T2 and r. The quantity of pores with a specific pore size can then be obtained from the T2 spectra of water-saturated samples. A small T2 corresponds to a small pore and a large T2 corresponds to a large pore [26].
2.2.2. Sample Preparation
Eight artificial clayey soil samples were prepared in the laboratory. All the samples consisted of clay mineral (montmorillonite or kaolinite) and sand. Four samples consisted of montmorillonite and sand, while the other four consisted of kaolinite and sand. These two groups were named the M-group and K-group, respectively. The clay mineral (montmorillonite or kaolinite) contents in each group were increased from 40% to 100% at intervals of 20%. The eight artificial clayey soil samples were named after the clay mineral content and clay mineral type. For example, M80 consisted of montmorillonite and sand, and the clay mineral content was 80%.
All eight artificial clayey soils were prepared under laboratory conditions. The process of sample preparation is briefly summarized as follows. The mineral powders were oven-dried at 250°C for 24 hours to a constant weight. At 250°C, almost all of the water in the artificial clayey soil was removed and the structure of the clay particles was unaffected [27–30]. Next, the clay mineral powder and sand powder in a dry state were thoroughly mixed at fixed proportions. Then, distilled water was added to the mixtures until the water content reached the plastic limit. To make the mixtures reach a full hydration state, the mixtures were stirred using a mortar-mixing apparatus at 60 rounds/min for 5 min and put into self-sealing plastic bags for two days. Following the hydration period, the mixtures were shaped in a hollow cylinder with an inner diameter and height of 31.9 mm and 80 mm, respectively, using a Jack under static pressure to reach the fixed dry density. Finally, the eight artificial clayey soil samples were saturated by the vacuum saturation method.
2.2.3. NMR and Evaporation Tests
A MesoMR12-150H-I NMR analyzer (NIUMAG Co., Ltd., Suzhou, China), shown in Figure 2, was used for NMR tests. The composition of the instrument used in this study was the same as that used in a previous study [21]. The parameters of the NMR tests in this study are shown in Table 2. The pulse sequence of the NMR tests used in this study was the Carr–Purcell–Meiboom–Gill (CPMG) pulse sequence [31]. Because of magnetic field inhomogeneities, the value of T2 measured by a free induction decay (FID) pulse sequence is less than the actual value [10]. The CPMG pulse sequence selected in this study was the same as that in previous studies [20, 23, 26, 32]. The T2 spectra were obtained automatically using the software installed on the NMR analyzer. Inversion software was developed based on inverse Laplace transform [33] and the simultaneous iterative reconstruction technique (SIRT) [34].
The NMR tests were first performed on the eight saturated samples and then carried out on the evaporated samples every hour until the T2 spectrum did not change. The samples were evaporated in an oven, with the drying temperature set to 40°C to simulate the highest temperature of the subgrade under natural conditions. It is noted that the temperature of the sample has dramatic effects on the T2 spectrum. The temperature of the sample needs to be equal to the temperature of the magnetic field (30°C). So, after the sample was taken out from the oven and before each NMR test, the sample needed to be cooled to 30°C. During the cooling process, the sample was wrapped in plastic wrap to prevent further dehydration.
3. Results and Discussion
3.1. Morphology of T2 Spectra during the Evaporation Process
The variations of the T2 spectra during the evaporation process of artificial clayey soils consisting of different clay minerals (kaolinite and montmorillonite) and sand are presented in Figure 3. All the T2 spectra in Figure 3 are unimodal, which suggests that the pores in both saturated and evaporated samples were distributed uniformly and continuously [15, 35]. The T2 spectra of artificial clayey soil in the initial state shifted to the left with increasing clay content. According to equation (6), this might be because the relaxation coefficient increased or the pore diameter decreased with increasing clay content in the artificial clayey soil. All of the T2 spectra of the eight artificial clayey soils shifted to the left with increasing evaporation time. According to NMR theory, the pore size is proportional to T2, so this means that the pore size kept decreasing during the evaporation process. This indicates that decreases in the pore size may be the reason for clayey soil shrinkage during evaporation [36].
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According to our previous study [17], the weight of distilled water (m) in artificial clayey soil is proportional to the T2 total signal amplitude (At), which is equal to the total area under the T2 spectrum curve. The relationship between At and m can be expressed by equation (7); the R2 value of equation (7) was 0.996, as shown in Figure 4. This indicated that the weight of evaporated water can be expressed by the attenuated cumulative T2 signal amplitudes:
The variation of the total T2 signal amplitude with respect to evaporation time is shown in Figure 5. During evaporation, At kept decreasing, while the decay rate was different at various evaporation times. The decay rate of At was defined as Rt, which can be calculated by the following equation:where ΔAt is the attenuated At during evaporation time t.
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The Rt values for each hour during evaporation were calculated and are presented in Figure 6. From Figure 6, it can be seen that Rt was approximately constant firstly and then decreased to 0. According to the Rt values, the evaporation process can be divided into three stages: the constant rate stage (CRS), the falling rate stage (FRS), and the residual stage (RS). The demarcation points of the three stages were defined as Tf and Tr. The demarcation point of CRS and FRS is Tf, and the demarcation point of FRS and RS is Tr. The Tf values of artificial clayey soils which consisted of the same clay mineral were found to be the same. The Tf value of artificial clayey soil consisting of kaolinite and sand was 20 h, while the Tf value of artificial clayey soil consisting of montmorillonite and sand was 6 h. The results indicated that the Tr values of artificial clayey soils with different clay mineral types and contents varied greatly.
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3.2. The Improved TC Permeability Prediction Model
The two most commonly used permeability prediction models of porous media based on NMR technology are the SDR model and the TC model. According to our previous study [17], the SDR model was not suitable for predicting the permeability coefficients of clayey soils that consist of different minerals. This can be explained by the fact that the relaxation coefficients of clayey soils with different mineralogical compositions vary greatly. The geometric mean transverse relaxation time (T2gm) of clayey soils with different mineralogical compositions cannot be used to compare the average pore size directly.
The TC model can be described by the following equation:where FFV and BFV are the volumes of free fluid and bound fluid, respectively, in a porous medium. The three empirical constants (a, m, n) are related to the properties of porous media. φNMR is the total porosity of artificial clayey soil as measured by NMR tests. For saturated artificial clayey soil, φNMR can be determined by the following equation:where is the density of distilled water and V is the volume of artificial clayey soil.
In our previous study [17], it was proved that the TC model can be used to determine the permeability coefficient of artificial clayey soil. In addition, the TC model is also called the free fluid model. The key to determining the permeability coefficient via the TC model is to calculate the volumes of two kinds of fluids (FFV and BFV) accurately. The FFV and BFV were defined as the water evaporated in the constant and other rate stages, respectively. However, it is more reasonable to divide the evaporation process into three stages according to Figure 6. This is because the greater the detail in which the water is classified, the higher the prediction accuracy of the model. In this study, the water evaporated in the CRS and FRS was defined as the absolute movable water and the partially movable water, respectively. The water that could not evaporate in the RS was defined as the immovable water. According to NMR theory, the weights of water evaporated in different stages correspond to different parts of the cumulative signal amplitudes in the T2 spectra of saturated artificial clayey soils. The T2 cut-off was used to discriminate the cumulative signal amplitudes corresponding to the different kinds of water. In this study, double T2 cut-offs (T2cf and T2cr) were defined to discriminate the three kinds of water. As shown in Figure 7, the cumulative signal amplitudes between T2min and T2cr (Aimw) correspond to the immovable water. Similarly, the cumulative signal amplitudes between T2cr and T2cf (Apmw) correspond to the partially movable water, and the cumulative signal amplitudes between T2cf and T2max correspond to the absolute movable water (Aamw). At, φNMR, Aimw, Apmw, and Aamw values of the eight artificial clayey soils are summarized in Table 3.
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To establish the improved TC permeability coefficient prediction model, the permeability coefficients (KTC) of the eight artificial clayey soils were also measured by variable head permeability tests. It is noted that the permeability coefficients are not the same as the permeability (kTC) in the TC model. According to previous studies [17, 37], KTC is proportional to kTC. The TC model can thus be converted into the following equation:
From the previous equation, there are relationships between KTC and φNMR and the volume ratios of different kinds of water. The relationships between KTC and φNMR, Apmw/Aamw, and Aimw/Aamw are shown in Figure 8. As presented in Figure 8, KTC decreased with increasing Apmw/Aamw and Aimw/Aamw. It is noted that KTC decreased with increasing φNMR mainly because there were differences in the mineralogical compositions of the eight artificial clayey soils. For clayey soils consisting of the same mineral, KTC decreased with decreasing φNMR.
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As shown in Figure 8, the relationships between KTC and φNMR, Apmw/Aamw, and Aimw/Aamw can be expressed by the following equations:
The R2 values of equations (12)–(14) were 0.927, 0.984, and 0.983, respectively. On the basis of equations (11)–(14), the improved TC model could be further established. This model can be expressed by the following equation:
Figure 9 presents a comparison of the predicted and measured permeability coefficients. All points are very close to the line y = x. This indicates that the improved TC model can be used to reliably predict the permeability coefficient of artificial clayey soil.
R2, mean absolute error (MAE), and root-mean-square error (RMSE) are often used to evaluate the prediction capability of a predictive model. These parameters can be determined by the following equations:where Xprei, Xmeai, and symbolize the predicted, measured, and mean measured values, respectively.
This improved TC model was proposed based on double T2 cut-offs. The typical TC model based on a single T2 cut-off for artificial clayey soil was established in our previous study [17]. The R2, MAE, and RMSE values of the two models are listed in Table 4 and shown in Figure 10. As shown in Table 4, the R2, MAE, and RMSE values of the improved TC model are 0.999, 1.050, and 1.370, respectively. However, the corresponding values of the typical TC model are 0.957, 3.096, and 3.828, respectively. It can be seen that the R2 value of the improved TC model is bigger than that of the typical TC model. This suggests that the connection between the measured K and predicted K in the improved TC model is stronger than that in the typical TC model. In addition, the MAE and RMSE values of the improved TC model are much smaller than those of the typical TC model. This indicates that the error between the measured K and predicted K of the improved TC model is much smaller than that of the typical TC model. The prediction capability of the improved TC model is thus much higher than that of the typical TC model.
3.3. Determination of the Double T2 Cut-Offs Based on Fractal Analysis
Before applying the improved TC model, it is necessary to determine the double T2 cut-offs to classify the cumulative signal amplitudes of T2 spectra corresponding to different kinds of water in artificial clayey soil. It takes a long time and great effort to determine the double T2 cut-offs by evaporation tests. According to our previous study [17], the T2 cut-off can be determined by fractal analysis. There is a linear relationship between the T2 cut-off and the fractal dimension (D) of the T2 spectrum of saturated artificial clayey soil. The D value can then be used to determine the double T2 cut-offs. It is noteworthy that the fractal dimensions were determined based on the T2 spectra obtained from the results of the NMR tests. The pore size of an artificial clayey soil is proportional to T2, so, actually, the fractal dimensions were determined according to the pore size distributions of the artificial clayey soil. So, the fractal dimensions in the manuscript were treated for space. The D, T2cr, and T2cf values of the eight artificial clayey soils are listed in Table 5. As shown in Table 5, a smaller fractal dimension was obtained for samples with higher clay content. Further, the fractal dimensions of the M series were larger than those of the K series. According to the relevant literature [38, 39], the probability density function of pore size can be written as follows:where c is constant. It can be concluded from equation (19) that the fractal dimension of artificial clayey soil decreases with increasing uniformity of the pore distribution. This indicates that the uniformity of artificial clayey soil with higher clay content is higher. For artificial clayey soil with the same clay content, the uniformity of the M series was worse than that of the K series.
The relationships between the double T2 cut-offs and fractal dimensions are shown in Figure 11. As shown in Figure 11(a), the relationships between the double T2 cut-offs and fractal dimensions of artificial clayey soil in the M-group can be expressed by the following equations:
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As presented in Figure 11(b), the relationships between the double T2 cut-offs and fractal dimensions of artificial clayey soil in the K-group can be expressed by the following equations:
The predicted double T2 cut-offs of artificial clayey soil can be calculated by equations (20)–(23). A comparison between the predicted and measured double T2 cut-offs is presented in Figure 12. The points in Figures 12(a) and 12(b) are very close to the line y = x; this indicates that the error between the predicted and measured double T2 cut-offs is small.
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The R2, MAE, and RMSE values were also determined to evaluate the prediction accuracy of equations (20)–(23), and these values for the predictive equations are shown in Table 6. From Figure 12 and Table 6, it can be seen that the prediction capability of the predictive equations for the double T2 cut-offs is sufficient.
4. Conclusions
Evaporation and NMR tests were carried out on eight artificial clayey soils with different mineral compositions. On the basis of the results of tests and a discussion, the main conclusions are as follows:(1)The T2 total signal amplitude is proportional to the weight of distilled water in artificial clayey soil. During the evaporation process, the T2 total signal amplitude decreased until it was stable. The decay rate of the T2 total signal amplitude was constant at first and then decreased gradually to zero. Based on the different decay rates, the evaporation process can be divided into three stages: the constant rate stage, the falling rate stage, and the residual stage.(2)Based on the three stages of the evaporation process, the water in artificial clayey soil can be classified into three categories: absolute movable water, partially movable water, and immovable water. Double T2 cut-offs were also defined to distinguish the cumulative signal amplitudes in the T2 spectrum corresponding to the different kinds of water.(3)An improved TC model for predicting the permeability coefficient of artificial clayey soil was proposed. The improved and typical models were evaluated by determining their R2, MAE, and RMSE values. The prediction accuracy of the improved model is much higher than that of the typical one.(4)To simplify the prediction process, the relationships between the double T2 cut-offs and the fractal dimension of the T2 spectrum of saturated clayey soil were also established.
Data Availability
The data used to support the findings of this study have not been made available because the data also form part of an ongoing study (National Key Research and Development Program of China).
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This research was funded by the National Key Research and Development Program of China (Grant no. 2018YFB1600200).