Quantitative characterization of soil micropore structure and pore water content using nuclear magnetic resonance: Challenges and calibration methods

Abstract

Though nuclear magnetic resonance (NMR) has been applied in soil science over several decades, the quantitative relation between NMR signals and soil pore-water distribution is complex and still covered by some cloud. The major debates include: 1) the quantitative relation between transverse relaxation time (T₂) and pore radius varies in different studies; 2) Is the relation between NMR signals and soil–water contents unique? To clarify the aforementioned issues over the application of NMR in soil science, a comprehensive study was carried out. The results demonstrate that: 1) a unique linear relationship between peak area of NMR signals and soil volumetric water content (θ) exists, independent of the soil’s initial molding conditions, such as molding dry density (ρ_d) and molding water content (w_ini); 2) the ratio between T₂ and pore radius, defined as the pore structure coefficient (C_r) of NMR, varies with pore water morphologies and soil types; 3) three methods were proposed to determine the value of C_r and can help to provide insights for better understanding of the NMR results in soil science.

Graphical abstract

Keywords

nuclear magnetic resonance / unsaturated soil / pore water distribution / soil–water characteristic curve / calibration method

1 Introduction

The micropore structure of soils gives essential information regarding the soil hydro-mechanical characteristics [1–4]. In addition, prediction models for soil macroscopic properties based on measurements of microstructures have also advanced rapidly [5]. With technological advancements, the laboratory measurement of soil pore structure has become increasingly mature. Measurements of soil micropore structures can be categorized into qualitative and quantitative methods. Qualitative methods encompass scanning electron microscopy (SEM) [6], optical microscopy (OM) [7], and X-ray scanning. Quantitative methods encompass mercury intrusion porosimetry (MIP) [8], nuclear magnetic resonance (NMR), and nitrogen adsorption (NA) [9]. Among these, NMR offers a rapid, eco-friendly, and non-invasive approach to measure the pore structure of porous media [10].

The NMR technique has been extensively utilized to measure the pore size and porosity of rocks and soils [11,12]. Furthermore, some scholars have used the spin-spin (T₂) relaxation time distribution curve of NMR to predict the mechanical properties of geotechnical materials [13,14]. For example, Tian et al. [15] analyzed the microstructure mechanism of pore water distribution and movement in sandy clay during the drying process. Ma et al. [16] studied the evolution of the microstructure of bentonite during multiple drying-wetting cycles employing the saturated salt solution vapor equilibrium method and NMR technique. Kong et al. [17] explored the affect of multiple drying-wetting cycles on the soil–water characteristic curve (SWCC) and soil pore size distribution, and proposed a modified van Genuchten model to obtain the best-fit pore size distribution curves. Li et al. [18] utilized the NMR technique to obtain the soil freezing characteristic curve of clayey soils and validated a SWCC prediction formula based on the Clapeyron model using the collected data. These studies demonstrate the wide application of the NMR technique in analyzing soil drying-wetting cycles, changes in moisture distribution caused by temperature variations, internal structure evolution, and damage characteristics [19–22]. Past studies have provided strong support from a microscopic level for the analysis of the macroscopic mechanical properties of soils [23].

However, there are still some unresolved issues with the NMR technique in the quantitative characterization of soil pore size distribution and pore water content. First, while many scholars have established linear relationships between peak area under T₂ spectra and different types of water content, such as Tian et al. [15] suggesting a linear relation between the peak area and the degree of saturation (S_r), and Liang et al. [24] proposing a positive correlation with the gravimetric water content (w), it should be noted that the NMR apparatus only probes the hydrogen atom in water. This implies a proportional relationship between peak area and the amount of water. Thus, it may be more reasonable to establish a linear relationship between T₂ peak area and volumetric water content (θ). Furthermore, the differences in functional expressions between the three commonly used water contents (i.e., S_r, w, and θ) and peak area, under varying initial sample preparation conditions (initial molding dry density and initial molding water content), need to be analyzed and compared through measured data. Secondly, the quantitative relation between T₂ relaxation time (T₂) and pore radius (r) is still a puzzle. Assuming that the NMR relaxation of water-saturated soils occurs in the conditions of a fast-diffusion regime [25], there is a direct proportionality relation between T₂ and r. However, this conversion requires the pore structure coefficient (C_r) of NMR, which is related to the particular composition of mineral grain and pore fluid. Currently, in geotechnical engineering, the C_r value is typically determined by comparing NMR relaxation data with surface-volume measurements obtained from methods such as MIP [26] or NA [27]. However, due to the different working principles of these methods (e.g., MIP measures pore throat diameter distribution; NA measures the total internal surface area of porous media), this comparison does not guarantee consistent C_r [28,29]. So far, no convenient methods have been proposed to determine C_r, which leads to variations in the quantitative relation between T₂ and r in past studies. Furthermore, different types of soil components (or pore water morphologies) may result in variations in C_r, implying that C_r is not a fixed parameter for different soils (e.g., sand, silt, clay). Therefore, when converting T₂ to r, it is necessary to calibrate the C_r value for each type of test soil, which significantly limits the application of the NMR technique in the quantitative analysis of pore characteristics of porous media.

This paper aims to address the aforementioned issues by conducting NMR tests and soil–water retention characteristics tests on different types of soils (sand, silty clay, and clay). This research analyzes the relationship between the peak area and three common types of soil–water content (S_r, w, and θ). Additionally, we explored the connection between T₂ and r. Based on results and findings, three methods, namely the empirical value method, the T_2max–ψ method, and the saturated soil cumulative pore size distribution curve (CPSD) method, were proposed to quickly obtain the pore structure coefficient C_r between T₂ and r. These research results are expected to provide more profound support and guidance for the application of NMR technique in geotechnical engineering.

2 Testing materials and methods

2.1 Testing material

A series of experiments were conducted to explore the relationship between T₂ and r, and the correlation between the peak area and soil–water content. The experiments involved different types of soils, containing coarse-grained and fine-grained soils, and were conducted in both SWCC and NMR tests. The coarse-grained soils included three types of sand: coarse sand, international standard (ISO) sand, and silty sand. The fine-grained soils included silty clay from Yanqing District, Beijing, China, referred to as Yanqing (YQ) silty clay, and bentonite from Gaomiaozi town, Inner Mongolia Autonomous Region, China, referred to as Gaomiaozi (GMZ) bentonite. The grain size distribution curves of tested soils are illustrated in Fig.1. The physical properties of the fine-grained soils are presented in Tab.1. The physical properties of the coarse-grained soils are detailed in Tab.2.

2.2 Sample preparation

The soil was initially oven-dried and subsequently stored in plastic bags. The soil mass and water contents were determined based on the given molding condition. Then, the homogeneous wet soil was compressed into the tailor-made rings to obtain compacted samples. The prepared samples were saturated by using the degassed water and subsequently air-dried to achieve the desired gravimetric water contents (w). To ensure moisture equilibrium, all unsaturated samples were sealed for three days. The procedure of sample preparation is referred to as the saturation-air drying method. Given the volume shrinkage of fine-grained soils during drying, the void ratio (e) of the unsaturated soil sample was measured by utilizing the fluid volume displacement method [30]. According to w and e of the unsaturated sample, the degree of saturation (S_r) of each sample can be calculated. It is worth noting that, to eliminate interference from iron elements on the NMR signals, the polytetrafluoroethylene (PTFE) rings were used instead of the steel rings. Tab.3 presents the initial molding states for sample preparation.

The entire experimental procedure is illustrated in Fig.2. The prepared samples are sequentially subjected to SWCC measurements, NMR tests, and volume measurement experiments. To avoid errors introduced by parallel samples, all three tests are conducted by using the same soil sample.

2.3 Nuclear magnetic resonance tests

The NMR technique relies on the behavior of proton spins within a uniform magnetic field, which are stimulated by a brief pulse of radio frequency (RF) energy. Using a Carr-Purcell-Meiboom-Gill (CPMG) sequence, the T₂ time of the pore fluid is measured. Subsequently, a Fourier transformation is applied to obtain a T₂ curve. The dimensionless peak area under the T₂ curve corresponds to the moisture content within the respective T₂ time, facilitating the determination of soil moisture content.

The NMR tests were performed utilizing a low-field NMR analyzer (model MesoMR23-060H-I) manufactured by Niumag Corporation, China. The equipment features a permanent magnetic field of 0.5 Tesla and a probe diameter of 60 mm.

2.4 Soil–water retention tests

The SWCC of coarse-grained soils (coarse sand, ISO sand, and silty sand) was measured by using the pressure plate technique, whereas a combination of the pressure plate, contact filter paper, and WP4C was adopted for fine-grained soils (YQ silty clay and GMZ bentonite). A detailed description of the combined method is as follows.

1) Low suction range: According to a given air pressure path, saturated samples were subjected to air pressure using a pressure plate equipment. Each air pressure level was sustained for two weeks. Additional air pressures of 250 and 400 kPa were applied for bentonite due to its higher air-entry value.

2) Medium suction range: each unsaturated sample was sandwiched between three layers of filter paper, with a middle layer of Whatman No. 42 filter paper (40 mm in diameter) and ordinary filter paper (50 mm in diameter) on the top and bottom layers. To ensure moisture equilibrium, the samples were sealed for ten days [31]. Matric suction was obtained by measuring w of the Whatman No. 42 filter paper, and the calibration formulas are shown as follows [32]:

(1)

(2)\lg \psi =-0.0112w_{\mathrm{f}}+2.4423,w_{\mathrm{f}}⩾59.5\mathrm{\%},

where w_f is the gravimetric water content of the Whatman No. 42 filter paper in percentage (%); ψ is the matric suction (kPa).

3) High suction range: the suctions of soil samples were obtained by utilizing the WP4C produced by Decagon, USA. During the measurement, the sample equilibrated with the water vapor in the testing space. This is an indirect method for measuring soil–water potential.

4) After the suction measurements were completed, the w of each soil sample was obtained by using the oven dry method.

3 Results and analyses

3.1 The definition of physical parameters in T₂ distribution curves

To clarify the physical parameters employed in the T₂ distribution curves in this study, each parameter is explicitly defined, as shown in Fig.3(a). The “T₂ at peak” (dominant T₂) refers to the T₂ value corresponding to the maximum signal amplitude, indicating the optimum pore radius filled with water. For convenience, the area under the T₂ curve is referred to as the “peak area” (P_A). The “maximum T₂” (T_2max) is defined as the maximum T₂ value in the T₂ curve at which the NMR signal is nonzero.

3.2 Typical examples of T₂ distribution curves

3.2.1 T₂ distribution curves for various types of soils

The T₂ curves for various types of soils under saturated conditions are shown in Fig.3(b). The curves exhibit a left-to-right shift as the grain size of the soil increases, with the soil types ranging from bentonite, silty clay, silty sand, ISO sand, to coarse sand. This shift in the T₂ distribution curves indicates that larger pore sizes correspond to larger T₂ values. Thus, the NMR can be utilized to qualitatively compare the pore size distribution characteristics of different types of soils.

3.2.2 T₂ distribution curves for samples with various degrees of saturation

For a given initial state, Fig.4 shows the T₂ curves for silty clay (ρ_d = 1.44g/cm³, w_ini = 16.5%) and bentonite (ρ_d = 1.2g/cm³, w_ini = 18.0%) with different S_r. The accompanying table in T_2max values for each S_r point of the soil samples.

It can be observed that the T₂ value corresponding to the maximum NMR signal gradually decreases as S_r decreases, and the “maximum T₂” shifts to the left, that is to say, both values of dominant T₂ and T_2max continuously decrease. The peak area decreases progressively with the decrease of S_r. The larger pores complete the drainage process first, then the water in the smaller pores begins to be drained.

3.3 The relationship between peak area and S_r, w, and θ

The relationship between peak area and S_r, w, and θ for silty clay with different ρ_d is shown in Fig.5, respectively. The measured data show well-defined linear relationships between the peak area and S_r, as well as peak area and w, with variations in intercepts and slopes. However, when θ is plotted on the horizontal axis, the peak area data points of soil samples with different ρ_d overlap, indicating that the functional relationship between peak area and θ remains unaffected by the variation in soil ρ_d. The main reason is that NMR instrument only probes the hydrogen atoms in water (i.e., the amount of water), while water content (i.e., w or S_r) represents the relative amount of water within the soil. Hence, there is a unique linear relationship between θ (i.e., the amount of water) and peak area, whereas S_r or w, which are not entirely equivalent to “the amount of water,” show a divergent linear relationship with peak area.

The functional relationship between the peak area and θ is independent of the w_ini of the soil, as shown in Fig.6(a). When ρ_d remains constant, the functional relationship between the peak area and θ does not change with variations in initial water content.

Combining all data points and conducting regression analysis, the study established a unique expression for the peak area of YQ silty clay in relation to θ, as shown in Eq. (3). The linear curves fit with a 95% confidence interval and a fit correlation coefficient R² = 0.99, as shown in Fig.6(b).

(3)P_{\mathrm{A}}=1556.97\theta -3746.34,

where P_A is the peak area; θ is the volumetric water content (%).

Regarding GMZ bentonite, the relationship between peak area and S_r, w, and θ exhibits the same pattern. Specifically, only the relationship between peak area and θ can be fitted by using a unique linear formula, as shown in Fig.7. These research results further confirm that the linear relationship between peak area and θ remains consistent regardless of initial molding states. Consequently, the NMR is a reliable approach for detecting the quantity and distribution of protons (hydrogen nuclei) per unit volume. After a regression analysis is conducted on all available data, a unique expression (Eq. (4)) that describes the relationship between peak area and θ of GMZ bentonite can be established.

(4)P_{\mathrm{A}}=3854.25\theta -25628.16.

3.4 Theoretical relationship between T₂ relaxation time and pore radius

3.4.1 The definitions of equivalent pore radius r_c and pore structure coefficient C_r

Assuming that the fast diffusion condition is fulfilled, the T₂ relaxation observed in this research is primarily due to surface relaxation, occurring at the interface between water and soil. Surface relaxation is dependent on the surface-to-volume ratio of the pore water:

(5)\frac{1}{T_2}={\rho }_2\frac{S}{V},

where T₂ is the transverse relaxation time; S/V is the surface-to-volume ratio, which is related to the pore size; and ρ₂ is a constant representing the surface relaxation coefficient (μm/ms).

Assuming the pores in the soil have regular shapes, Eq. (5) can be expressed as:

(6)\frac{1}{T_2}={\rho }_2\frac{\alpha }{{\mathit{r}}_{\mathrm{c}}}=\frac{{\mathit{C}}_{\mathrm{r}}}{{\mathit{r}}_{\mathrm{c}}},

where α is the correction factor of pore geometry; r_c is the equivalent pore radius; and C_r is the pore structure coefficient, i.e., C_r = α*ρ₂. The definitions of equivalent pore radius r_c and pore structure coefficient C_r are as follows, respectively.

The definition of the equivalent pore radius r_c is important. In this study, r_c is defined similarly to the pore radius used in MIP and SWCC experiments. This assumption considers that r_c applies to the entire pore size range, including capillary and adsorption pores, as shown in Eqs. (7) and (8).

(7)\mathrm{M}\mathrm{I}\mathrm{P}:r_{\mathrm{c}}=\frac{2T_{\mathrm{s}}\cos \beta }{P},

(8)\mathrm{S}\mathrm{W}\mathrm{C}\mathrm{C}:r_{\mathrm{c}}=\frac{2T_{\mathrm{s}}\cos \beta }{\psi },

where T_s is the surface tension of soil and water; β is the contact angle between the water molecule and the pore surface; P is the intrusion pressure; ψ is the suction.

In Eqs. (7) and (8), the pores are assumed to be cylindrical, which does not represent the actual geometric morphology. A given P (or ψ) value corresponds to a specific value of pore radius, where the pore radius is an equivalent concept representing the free energy in the adsorption. The definition of the equivalent radius r_c in this study follows the same principle. For regularly shaped and saturated pores, such as spherical, cylindrical, and interlayer water, the equivalent radius r_c corresponds to the actual pore radius. However, for irregularly shaped and unsaturated pores, such as water lens and film water, the equivalent radius r_c represents the free energy associated with the suction ψ, rather than the physical pore size (the equivalent pore radius for different geometric morphologies of pore water will be derived in five cases in Subsubsection 3.4.2).

From Eq. (6), it is evident that the conversion of T₂ to r_c necessitates determining the values of two parameters, α and ρ₂. The parameter α varies with different pore geometries, while ρ₂ varies with different materials. It proves challenging to ascertain the values of α and ρ₂ independently. Therefore, by multiplying α and ρ₂, we introduce a pore structure coefficient C_r as a conversion factor between T₂ and r_c. The influence of the parameter C_r on the SWCC prediction is shown in Fig.8. As the C_r value increases, the CPSD shifts to the right, indicating a growth in pore size for the same water content. Specifically, larger C_r values correspond to larger pore sizes within the soil, implying a transformation of the soil type from fine-grained to coarse-grained. By determining the optimal C_r value, it is possible to acquire true information about the soil pore size distribution by analyzing the T₂ curve of saturated samples. The key issue addressed in this study is how to interpret the value of C_r.

3.4.2 The derivation of the equivalent pore radius for pore water with different geometric morphologies

The pore water of different geometric morphologies can be classified into saturated pore water (e.g., spherical pore water, cylindrical pore water, and interlayer water) and unsaturated pore water (e.g., water lens, and film water). With regard to saturated pore water, r_c represents the physical pore radius, which varies with the pore shapes. With regard to unsaturated pore water, r_c mainly represents the free energy associated with ψ. The expressions for the equivalent pore radius of pore water with different geometric morphologies are derived as follows, and the corresponding values are summarized in Tab.4.

1) Case 1: Spherical pore water

For sand, the pore shape is spherical, and the morphology of pore water is gravitational water, as shown in Fig.9(a):

S_{\text{spherical}}=4\text{π}{r}_{\text{c}}^2,

V_{\text{spherical}}=\frac{4}{3}\text{π}{r}_{\text{c}}^3,

\alpha =\frac{S_{\text{spherical}}}{V_{\text{spherical}}}{r}_{\mathrm{c}}=3,

(9)r_{\mathrm{c}}=r_{\text{p-s}}.

The α for the spherical pore water is 3, and r_c is the physical sphere radius r_p-s.

Note: The r_p-s is the physical sphere radius; the r_p-c is the physical cylinder radius; the r_p-h is the half of the physical distance between two parallel plates; the r_c represents the free energy associated with water lens and film water.

2) Case 2: Cylindrical pore water

For silt and silty clay, the pore shape is cylindrical, and the morphology of pore water is capillary water, as demonstrated in Fig.9(b):

S_{\text{cylindrical}}=2\text{π}{r}_{\text{c}}L,

V_{\text{cylindrical}}=\text{π}{r}_{\text{c}}^{2}L,

\alpha =\frac{S_{\text{cylindrical}}}{V_{\text{cylindrical}}}{r}_{\mathrm{c}}=2,

(10)r_{\mathrm{c}}=r_{\text{p-c}},

where L is the cylinder height.

The α for the cylindrical pore water is 2, and r_c is the physical cylinder radius r_p-c.

3) Case 3: Interlayer water

For clay, the shape of soil particles is platy, and the morphology of pore water is interlayer water, as shown in Fig.9(c):

S_{\text{interlayer}}=2LH,

V_{\text{interlayer}}=2LH{r}_{\mathrm{c}},

\alpha =\frac{S_{\text{interlayer}}}{V_{\text{interlayer}}}{r}_{\mathrm{c}}=1,

(11)r_{\mathrm{c}}=r_{\mathrm{p}-\mathrm{h}},

where L is the plate length; and H is the plate width.

α for the interlayer water is 1, and r_c is the half of the physical distance between two parallel plates r_p-h.

4) Case 4: Water lens

When the S_r of the granular soils is low, the morphology of pore water is interparticle pore water menisci (or lens) between two contacting spherical particles, as shown in Fig.9(d). The radius of the particles is R. Two additional spherical radii, r₁ and r₂, are introduced to describe the water lens [33]. The radius r₁ is rotated about an axis perpendicular to the cross-section. Conversely, the radius r₂ is rotated about an axis parallel to the cross-section. The filling angle δ, connects the center of each soil particle to the center of the circle defined by r₁. The three-phase interface involving solid particles, pore air, and pore water is characterized by an interfacial contact angle β [34].

In the case of water lens, the equivalent pore radius r_c represents the free energy associated with ψ. The parameters α and r_c for the water lens can be expressed as follows (see the detailed derivation in Electronic Supplementary Materials):

\alpha =\frac{S_{\text{lens}}}{V_{\text{lens}}}{r}_{\mathrm{c}}=\frac{2S_{\text{lens}}{T}_{\mathrm{s}}}{V_{\text{lens}}\psi },

(12)r_{\mathrm{c}}=\frac{2T_{\mathrm{s}}}{\psi },

where S_lens is the surface area of the lens; V_lens is the volume of the lens; T_s is the surface tension, with a value of 73.05mN/m when the laboratory temperature is 18 °C [35]; ψ is the matric suction.

According to the expression equation for α, a) the parameters are fixed at δ = 30°, β = 0°, r₁ = 10⁻⁷ m, r₂ = 2.76 × 10⁻⁷ m, R = 6.5 × 10⁻⁷ m, T_s = 73.05 mN/m, ψ = 462 kPa. The α for the water lens is 14.7; b) the parameters are fixed at δ = 10°, β = 0°, r₁ = 10⁻⁸ m, r₂ = 10⁻⁷ m, R = 6.5 × 10⁻⁷ m, T_s = 73.05 mN/m, ψ = 6574 kPa. The α for the water lens is 2.8. From the two calculation results above, as the content of the water lens decreases (ψ increases), the α value for the water lens decreases.

5) Case 5: Film water

When the water molecules are adsorbed in the interlamellar space of clay minerals or intensively adsorbed on the particle surface, the matric potential is low and dominated by the adsorptive forces between soil particles and water molecules. The predominant morphology of pore water is called film water. To characterize the energy state of pore water between two parallel plates, Lu and Zhang [36] proposed the soil sorptive potential theory. Soil sorptive potential is defined as free energy change per unit volume of soil resulting from the interaction of water molecules with external components of the system. The semi-infinite parallel plate represents typical clay particles in Fig.9(e). For the film water within semi-infinite parallel plates, the van der Waals force primarily dominates the soil sorptive potential, which can be expressed as:

(13)\psi (h)=-\frac{A_{\text{H}}}{6\text{π}{h}^3},

where A_H is the Hamaker constant (J), i.e., −6 × 10⁻²⁰ J [37]; h is the thickness of water film (nm).

In the case of film water, the equivalent pore radius r_c represents the free energy associated with ψ. Substituting Eq. (13) into Eq. (8):

(14)r_{\text{c}}=\frac{12T_{\text{s}}\text{π}{h}^3}{A_{\text{H}}}.

The expression for α in the film water is as follows:

S_{\mathrm{film}}=2a,

V_{\text{film}}=ah,

(15)\alpha =\frac{S_{\text{film}}}{V_{\text{film}}}{r}_{\text{c}}=\frac{24T_{\text{s}}\text{π}{h}^2}{A_{\text{H}}},

where a is the surface area of the single-sided water film; h is the water film thickness.

According to Eq. (15), 1) the layer number (n) of water molecule adsorbed on the surface of soil particles is 2, with the water film thickness h = 0.6 nm, corresponding to the ψ = 15 MPa. The α for the film water is 33. 2) the parameters are fixed at n = 1, h = 0.3nm, ψ = 118 MPa. The α for the film water is 8. From the two calculation results above, it is evident that as the content of the film water decreases (h decreases), the α value for the film water decreases.

3.5 Determination of the pore structure coefficient C_r

According to Eq. (6), it can be observed that the key to solving the relationship between T₂ and r lies in accurately determining the pore structure coefficient C_r. This paper proposes three methods based on empirical or theoretical approaches for obtaining the C_r values of different soil types.

3.5.1 Empirical judgment method

The T₂ curve of the saturated soil sample indicates the pore size distribution. On the T₂ curve, there is a specific T₂ threshold value that separates the curve into two distinct ranges [38,39]. When the T₂ value of the pore fluid exceeds this threshold, the fluid is considered mobile (capillary water or free water). Conversely, when the T₂ value falls below this threshold, the fluid is confined within the pores (adsorbed water). This threshold relaxation time is defined as the T₂ cutoff value (T_2c). Two main methods are used to determine T_2c: 1) The centrifuge calibration method: a saturated sample is subjected to laboratory centrifugal testing, and the T₂ spectrum of the centrifuged sample is compared with that of the saturated sample to obtain T_2c; 2) The empirical judgment method [40,41]: Based on the results of centrifuge calibration, a classification analysis is conducted to summarize the morphologies of different T₂ spectra, thereby establishing an empirical method suitable for quickly determining the T_2c value. Although the centrifuge calibration method is an accurate method to determine T_2c, it often encounters significant challenges due to the long measurement time or the absence of instruments. Therefore, the empirical judgment method is frequently employed to determine T_2c in NMR tests. It should be noted that using the empirical judgment method to determine T_2c is relatively arbitrary. Therefore, this method is employed in this study as a reference. Tab.5 shows the selection of T_2c for different T₂ spectra.

The T₂ curves of saturated silty clay samples under different molding states are shown in Fig.10. In Fig.10(a), it is evident that T₂ distribution curves for different ρ_d samples exhibit a bimodal characteristic. The longer T₂ time of the right peak is attributed to the presence of mobile fluids (capillary and free water), while the shorter relaxation time of the left peak represents bound fluids (adsorbed water) attributed to the presence of clay particles. The T_2c is selected near the trough between the two peaks, i.e., T_2c = 10 ms. In Fig.10(b), the evolution of T₂ curves from unimodal to bimodal as the w_ini increases. For the sample with w_ini = 12.5%, the T₂ curve is unimodal, with the dominant peak (T₂ at peak) being less than 10ms. The T_2c is determined by using the half-amplitude method [42,43], typically located near the right half-amplitude width of the dominant peak, i.e., T_2c = 10 ms. For samples with w_ini = 16.5% and 20.5%, the NMR T₂ curves exhibit a bimodal characteristic, and the T_2c is also chosen near the trough between the two peaks, i.e., T_2c = 10ms. In summary, based on the empirical judgment, the T_2c value for the YQ silty clay is approximately 10 ms. Li and Zhang [1] used MIP and SEM techniques to investigate the micropore structure of clayey soils during drying processes. The test results revealed the existence of a dual-porosity structure within compacted soil samples, consisting of intra-aggregate pores and inter-aggregate pores. The characteristic pore size (r_c), approximately 1μm, was divided into these two types of pores, consistent with the observations of other researchers [27].

In Fig.10 and Fig.11, the S_r values labeled in the legend represent the actual S_r values of the saturated soil samples obtained by using the vacuum saturation method.

Substituting r_c = 1 μm and T_2c = 10 ms into Eq. (6):

(16)C_{\mathrm{r}}=0.10\text{μ}\mathrm{m}/\mathrm{m}\mathrm{s}.

Fig.11 presents the T₂ curves of saturated bentonite samples under different initial molding states. The T₂ curves exhibit a single-peak feature, and the dominant T₂ value is less than 10 ms. According to the half-amplitude method, the T_2c is located near the right half-amplitude width of the dominant T₂, i.e., T_2c = 3.41 ms. Some researchers [44–46] have studied the micropore structure of GMZ bentonite under different initial states using MIP and SEM techniques. Based on experimental results, they determined a critical pore radius of approximately 0.075 μm as the boundary between intra-aggregate and inter-aggregate pores in GMZ bentonite. By substituting r_c = 0.075 μm and T_2c = 3.41 ms into Eq. (6), the C_r value for GMZ bentonite can be calculated as follows:

(17)C_{\mathrm{r}}=0.022\text{μ}\mathrm{m}/\mathrm{m}\mathrm{s}.

3.5.2 The method based on the correlation between soil-water characteristic curve and cumulative pore size distribution curve

The SWCC describes the relationship between ψ and soil–water content, while the NMR CPSD of the saturated sample characterizes the distribution of pore volume and size within the soil. Both SWCC and CPSD typically exhibit an “S” shape. Numerous studies have shown that soil–water retention characteristics are closely related to the pore structure. Therefore, it is feasible to predict the SWCC from a microstructure perspective. The C_r value can be determined by combining the SWCC with the CPSD of saturated soil (referred to as the SWCC-CPSD method):

(18)C_{\mathrm{r}}=\frac{2T_{\mathrm{s}}\cos \beta }{\psi T_2}.

According to Eq. (18), the product of T₂ in the NMR curve and the corresponding ψ in the SWCC is a constant value. This implies that the C_r value for the soil can be obtained by combining the CPSD and the SWCC.

The ordinate of the T₂ curve represents signal intensity, which lacks a specific physical unit. Through Eq. (19), the pore water volume for a given pore size can be calculated, allowing for a connection with signal intensity to be established [47]:

(19)V_{\text{i}}=\frac{A_i}{{\displaystyle \sum A_i}}\frac{m_{\text{w}}}{{\rho }_{\text{w}}}=\frac{A_i}{\sum A_i}\frac{m_{\text{s}}-m_{\text{d}}}{{\rho }_{\text{w}}},

where A_i is the NMR signal amplitude; m_w is the mass of pore water; m_s is the total mass of the moist sample; m_d is the mass of the dried soil; ρ_w is the density of water.

For a given pore size, the volumetric water content θ can be calculated:

(20){\theta }_i=V_i\frac{{\rho }_{\text{d}}}{m_{\text{d}}}=\frac{A_i}{{\displaystyle \sum A_i}}\frac{\left(m_{\text{s}}-m_{\text{d}}\right)G_{\text{s}}}{(1+e)m_{\text{d}}}.

The T₂ peak area can characterize the pore water content (θ) within the corresponding T₂ relaxation time. According to Eq. (20), the vertical axes of the CPSD and the SWCC can be unified.

Assuming that the SWCC follows the following equation:

(21)y_1=f_1(\psi ).

The NMR CPSD follows the following equation:

(22)y_2=f_2\left(T_2\right).

Combining Eqs. (21) and (22):

(23)f_1(\psi )=f_2\left(T_2\right)+\mathit{\text{z}},

where z is the intercept, which reflects the difference between the actual θ and the theoretical θ of the saturated sample.

According to Eq. (18), T₂ can be converted to ψ by means of the coefficient C_r:

(24)\psi =g\left(T_2,C_{\mathrm{r}}\right).

Substituting Eq. (24) into Eq. (23):

(25)f_1\left[g\left(T_2,C_r\right)\right]=f_2\left(T_2\right)+\mathit{\text{z}}.

According to Eq. (25), an optimization by using the least squares method (LSM, the detailed calculation process is described in Electronic Supplementary materials B) can be performed to obtain the minimum value of the optimization objective coefficient ε, and the optimal values of parameters C_r and z can be obtained, as shown in Tab.6. The results of the SWCC and CPSD overlapping are depicted in Fig.12 (other results for silty clay and bentonite are shown in Electronic Supplementary Materials Figs. B1 and B2).

(26)\epsilon =\sum {\left(f_1(\psi )-f_2\left(T_2\right)-\mathit{\text{z}}\right)}^2.

The calculation results in Tab.6 indicate that: 1) the C_r values of coarse sand, ISO sand, and silty sand are 0.696, 0.390, and 0.354 μm/ms, respectively; 2) for a given soil, the variation in C_r values under different initial molding states is relatively small. Therefore, the calculated average C_r values for silty clay and bentonite are 0.088 and 0.027 μm/ms, with standard deviations of 0.021 and 0.002, respectively.

3.5.3 T_2max−ψ method

Based on the Young−Laplace theory, the pores within the soil can be considered as parallel capillaries with different diameters. During soil drying, the size of the drainage pore decreases as ψ increases. For a given ψ, the size of the drainage pore is fixed. For unsaturated soils, the maximum pore size in water-filled pores corresponds to the maximum T₂ value (T_2max) in the T₂ curve.

According to Eq. (18), it is theoretically possible to determine the C_r value by calculating the product of one-point T_2max and its corresponding ψ. Therefore, we propose the one-point T_2max−ψ method to assess the feasibility of obtaining the C_r value by just using the information from one unsaturated sample. The relationship between ψ and the corresponding r_c concerning T_2max is shown in Fig.13. As ψ increases (or r_c decreases), T_2max gradually decreases. Since one T_2max value corresponds to one ψ point (Fig.4), the C_r value for each point can be obtained. However, the coefficient of variation C_v for the calculated results obtained by using the one-point T_2max−ψ method is relatively large, with a C_v value of 1.187 for silty clay and 0.433 for bentonite. This indicates significant dispersion in the results. Therefore, a multi-point averaging method is employed to enhance the precision of the estimation. For silty clay and bentonite, the average values of {\overline{C}}_{\mathrm{r}} are 0.076 and 0.021 μm/ms, respectively.

3.5.4 Mercury intrusion porosimetry-nuclear magnetic resonance method

Two CPSDs were measured through NMR and MIP tests on a saturated silty clay sample (initial molding condition: ρ_d = 1.44 g/cm³, w_ini = 16.5%), respectively [14]. It is possible to align the two curves by selecting an optimal C_r value, which is considered as the optimal value. The impact of the C_r value on the CPSD prediction is shown in Fig.14. As the C_r value increases, the NMR CPSD shifts to the right. By means of the LSM, the optimal C_r value for the silty clay is determined to be 0.11 μm/ms.

4 Discussion

4.1 The comparison of C_r values obtained by means of different methods

The results of obtaining the coefficient C_r by utilizing various methods are shown in Fig.15. It can be observed that the values of C_r obtained by using various methods are relatively consistent.

4.2 The empirical values of coefficient C_r from the literature

We conducted a statistical analysis of C_r values for different types of soils from past studies. These values were systematically categorized based on the soil types and particle size distribution range, as shown in Tab.7.

The relationship between coefficient C_r and the average soil particle diameter (d̅) [54,55] is shown in Fig.16. It can be seen that as d̅ increases, the value of C_r also shows an increasing trend. This finding is consistent with the results obtained from Fig.8. Based on the soil types and particle size distribution range, Tab.8 provides suggested ranges and values for the coefficient C_r of different soil types.

5 Recommended calibration methods for determination of C_r

5.1 Empirical C_r value

Empirical C_r values for different types of soils can be determined according to Tab.8.

5.2 T_2max−ψ method

The one-point T_2max−ψ method exhibits significant data dispersion when calculating the C_r value. Although this method can provide a rough estimate of the C_r value, it is recommended to adopt a multi-point measurement method to obtain more accurate results by calculating the average C_r value for soil samples with different S_r.

5.3 Saturated soil cumulative pore size distribution curve method

Three methods can be used to determine the C_r value by using the CPSD of a saturated soil sample: the one-point, two-point, and SWCC−CPSD methods. Since the two-point method significantly improves prediction accuracy compared to the one-point method and requires fewer tested samples than the SWCC−CPSD method (see the comparison results in Electronic Supplementary materials C), we recommend using the two-point method to quickly determine the C_r value. The two-point method is described in detail as follows.

1) First, prepare two unsaturated samples with different S_r, e.g., S_r1 = 80% and S_r2 = 30%, measure the matric suction (ψ_0.8 and ψ_0.3) of two samples by using the filter paper method, and substitute ψ_0.8 and ψ_0.3 into Eq. (8) to calculate the corresponding pore radius, r_c-0.8 and r_c-0.3.

2) Second, use the NMR to measure the T₂ curve of the saturated soil sample, plot the CPSD, and determine the T₂ values (T_2-0.8 and T_2-0.3) corresponding to S_r1 = 80% and S_r2 = 30% on the CPSD, as shown in Fig. C1.

3) Third, substitute r_c-0.8, T_2-0.8, r_c-0.3 and T_2-0.3 into Eq. (18) to calculate the values of C_r-0.8 and C_r-0.3, respectively.

4) Finally, calculate the average value of C_r-0.8 and C_r-0.3.

6 Conclusions

This paper explored the quantitative relation between T₂ peak area and water content by combining NMR with suction measurement techniques. It proposed calibration methods for rapidly determining the pore structure coefficient C_r between T₂ and r_c. The main conclusions are summarized as follows.

1) For a given soil, a unique linear relationship exists between the peak area and θ, and this linear relationship is independent of the molding states of the soil, confirming that the NMR is a method for detecting the quantity and distribution of hydrogen nuclei per unit volume.

2) The pore structure coefficient C_r varies according to the properties of the soil, primarily due to differences in the morphologies of pore water in various types of soils.

3) Based on theoretical research and experimental results, three efficient methods for determining the pore structure coefficient C_r are proposed: the empirical value method, the T_2max−ψ method, and the saturated soil CPSD method. These methods provide valuable references for the rapid, non-destructive, and accurate characterization of the soil pore size distribution by means of the NMR technique.

References

[1] Li X, Zhang L M. Characterization of dual-structure pore-size distribution of soil. Canadian Geotechnical Journal, 2009, 46(2): 129–141

[2] Li X K, Li X, Liu J K. A dynamic soil freezing characteristic curve model for frozen soil. Journal of Rock Mechanics and Geotechnical Engineering, 2024, 16(8): 3339–3352

[3] Liu Y, Wang X, Zhao Y X, Chen L H, Li X. A method for determining yield stress of unsaturated soils from lateral pressure. Geotechnical Testing Journal, 2024, 47(6): 1313–1327

[4] Zhao Y X, Wu L Z, Li X. NMR-based pore water distribution characteristics of silty clay during the soil compaction, saturation, and drying processes. Journal of Hydrology, 2024, 636: 131240

[5] Li X, Zhang L M, Wu L Z. A framework for unifying soil fabric, suction, void ratio, and water content during the dehydration process. Soil Science Society of America Journal, 2014, 78(2): 387–399

[6] Collins K, McGown A. The form and function of microfabric features in a variety of natural soils. Geotechnique, 1974, 24(2): 223–254

[7] Cousin I, Issa O M, Le Bissonnais Y. Microgeometrical characterisation and percolation threshold evolution of a soil crust under rainfall. Catena, 2005, 62(2): 173–188

[8] Penumadu D, Dean J. Compressibility effect in evaluating the pore-size distribution of kaolin clay using mercury intrusion porosimetry. Canadian Geotechnical Journal, 2000, 37(2): 393–405

[9] Prost R, Koutit T, Benchara A, Huard E. State and location of water adsorbed on clay minerals: Consequences of the hydration and swelling-shrinkage phenomena. Clays and Clay Minerals, 1998, 46(2): 117–131

[10] Tian H H, Wei C F, Yan R T. Thermal and saline effect on mineral-water interactions in compacted clays: A NMR-based study. Applied Clay Science, 2019, 170: 106–113

[11] Tian H H, Wei C F. Characterization and quantification of pore water in clays during drying process with low-field NMR. Water Resources Research, 2020, 56(10): e2020WR027537

[12] Novotny E H, de Godoy G, Consalter D M, Cooper M. Determination of soil pore size distribution and water retention curve by internal magnetic field modulation at low field 1H NMR. Geoderma, 2023, 431: 116363

[13] Zhou J, Meng X, Wei C, Pei W. Unified soil freezing characteristic for variably-saturated saline soils. Water Resources Research, 2020, 56(7): e2019WR026648

[14] Tao G L, Ouyan Q, Lei D, Chen Q, Nimbalkar S, Bai L, Zhu Z. NMR-based measurement of AWRC and prediction of shear strength of unsaturated soils. International Journal of Geomechanics, 2022, 22(9): 04022150

[15] Tian H H, Wei C F, Wei H Z, Yan R T, Chen P. An NMR-based analysis of soil–water characteristics. Applied Magnetic Resonance, 2014, 45(1): 49–61

[16] Ma T T, Wei C F, Yao C Q, Yi P P. Microstructural evolution of expansive clay during drying–wetting cycle. Acta Geotechnica, 2020, 15(8): 2355–2366

[17] Kong L W, Sayem H M, Tian H H. Influence of drying–wetting cycles on soil−water characteristic curve of undisturbed granite residual soils and microstructure mechanism by nuclear magnetic resonance (NMR) spin-spin relaxation time (T₂) relaxometry. Canadian Geotechnical Journal, 2018, 55(2): 208–216

[18] Li X, Zheng S F, Wang M, Liu A Q. The prediction of the soil freezing characteristic curve using the soil–water characteristic curve. Cold Regions Science and Technology, 2023, 212: 103880

[19] Tian H H, Wei C F, Wei H Z, Zhou J Z. Freezing and thawing characteristics of frozen soils: Bound water content and hysteresis phenomenon. Cold Regions Science and Technology, 2014, 103: 74–81

[20] Ma T T, Wei C F, Xia X L, Zhou J A, Chen P. Soil freezing and soil–water retention characteristics: Connection and solute effects. Journal of Performance of Constructed Facilities, 2017, 31(1): D4015001

[21] Yao C Q, Wei C F, Ma T T, Chen P, Tian H H. Experimental investigation on the influence of thermochemical effect on the pore–water status in expansive soil. International Journal of Geomechanics, 2021, 21(6): 04021080

[22] Zhou J Z, Liang W P, Meng X C, Wei C F. Comparison of freezing and hydration characteristics for porous media. Permafrost and Periglacial Processes, 2021, 32(4): 702–713

[23] Zhou J Z, Zhou Y, Yang Z J, Wei C F, Wei H Z, Yan R T. Dissociation-induced deformation of hydrate-bearing silty sand during depressurization under constant effective stress. Geophysical Research Letters, 2021, 48(14): e2021GL092860

[24] Liang W Y, Yan R T, Xu Y F, Zhang Q, Tian H H, Wei C F. Swelling pressure of compacted expansive soil over a wide suction range. Applied Clay Science, 2021, 203: 106018

[25] Brownstein K R, Tarr C E. Importance of classical diffusion in NMR studies of water in biological cells. Physical Review A: General Physics, 1979, 19(6): 2446–2453

[26] Yao Y B, Liu D M, Che Y, Tang D Z, Tang D Z, Tang S H, Huang W H. Petrophysical characterization of coals by low-field nuclear magnetic resonance (NMR). Fuel, 2010, 89(7): 1371–1380

[27] LowellSShieldsJ E. Powder Surface Area and Porosity. Berlin: Springer Science & Business Media, 1991

[28] Jaeger F, Bowe S, Van As H, Schaumann G E. Evaluation of 1H NMR relaxometry for the assessment of pore-size distribution in soil samples. European Journal of Soil Science, 2009, 60(6): 1052–1064

[29] Saidian M, Prasad M. Effect of mineralogy on nuclear magnetic resonance surface relaxivity: A case study of middle bakken and three forks formations. Fuel, 2015, 161: 197–206

[30] Gapak Y, Das G, Yerramshetty U, Bharat T V. Laboratory determination of volumetric shrinkage behavior of bentonites: A critical appraisal. Applied Clay Science, 2017, 135: 554–566

[31] ASTMD5298-16. Standard Test Method for Measurement of Soil Potential (Suction) Using Filter Paper. West Conshohocken, PA: ASTM, 2016

[32] ZhaoY XZhangL QLiXZhaoH F. Unsaturated shear strength characteristics of coarse-fine mixed soils in a wide range of degree of saturation: Experimental phenomena. Chinese Journal of Geotechnical Engineering, 2023, 45(11): 2278–2288 (in Chinese)

[33] Likos W J, Lu N. Hysteresis of capillary stress in unsaturated granular soil. Journal of Engineering Mechanics, 2004, 130(6): 646–655

[34] Gilpin R R. A model for the prediction of ice lensing and frost heave in soils. Water Resources Research, 1980, 16(5): 918–930

[35] FredlundD GRahardjoH. Soil Mechanics for Unsaturated Soils. Toronto: Wiley, 1993

[36] Lu N, Zhang C. Soil sorptive potential: Concept, theory, and verification. Journal of Geotechnical and Geoenvironmental Engineering, 2019, 145(4): 04019006

[37] Or D, Tuller M. Liquid retention and interfacial area in variably saturated porous media: Upscaling from single-pore to sample-scale model. Water Resources Research, 1999, 35(12): 3591–3605

[38] Yao Y B, Liu D M, Cai Y D, Li J Q. Advanced characterization of pores and fractures in coals by nuclear magnetic resonance and X-ray computed tomography. Science China Earth Sciences, 2010, 53(6): 854–862

[39] Yao Y B, Liu D M. Comparison of low-field NMR and mercury intrusion porosimetry in characterizing pore size distributions of coals. Fuel, 2012, 95: 152–158

[40] Morriss C, Rossini D, Straley C, Tutunjian P, Vinegar H. Core analysis by low-field NMR. Log Analyst, 1997, 38(2): 84–93

[41] Li H J, Li X B, Ma S M, Jing J B, Zhang X H, Yuan C J. Conventional reservoir fluid identification method and application of NMR logging technology. Mud Logging Engineering, 2020, 31(2): 72–78

[42] Mitchell J, Howe A M, Clarke A. Real-time oil-saturation monitoring in rock cores with low-field NMR. Journal of Magnetic Resonance, 2015, 256: 34–42

[43] Westphal H, Surholt I, Kiesl C, Thern H F, Kruspe T. NMR measurements in carbonate rocks: Problems and an approach to a solution. Pure and Applied Geophysics, 2005, 162(3): 549–570

[44] Chen Y G, Sun Z, Cui Y J, Ye W M, Liu Q H. Effect of cement solutions on the swelling pressure of compacted GMZ bentonite at different temperatures. Construction and Building Materials, 2019, 229: 116872

[45] Sun D A, Zhang Q Y, Peng F. Effect of aging on shear strength of compacted GMZ bentonite. Engineering Geology, 2022, 302: 106632

[46] Sun Z, Chen Y G, Cui Y J, Ye W M, Chen B. Effect of synthetic Beishan site water and cement solutions on the mineralogy and microstructure of compacted Gaomiaozi (GMZ) bentonite. Soil and Foundation, 2019, 59(6): 2056–2069

[47] An R, Zhang X W, Kong L W, Liu X Y, Chen C. Drying−wetting impacts on granite residual soil: A multi-scale study from macroscopic to microscopic investigations. Bulletin of Engineering Geology and the Environment, 2022, 81(10): 447

[48] Keating K, Knight R. A laboratory study to determine the effect of iron oxides on proton NMR measurements. Geophysics, 2007, 72(1): 27–32

[49] Stingaciu L R, Weihermüller L, Haber-Pohlmeier S, Stapf S, Vereecken H, Pohlmeier A. Determination of pore size distribution and hydraulic properties using nuclear magnetic resonance relaxometry: A comparative study of laboratory methods. Water Resources Research, 2010, 46(11): 2009WR008686

[50] Bian X, Zhang W, Li X Z, Shi X S, Deng Y F, Peng J. Changes in strength, hydraulic conductivity and microstructure of superabsorbent polymer stabilized soil subjected to wetting−drying cycles. Acta Geotechnica, 2022, 17(11): 5043–5057

[51] Yang G S, You Z Y, Wu D, Zhao L Q. Experimental study on the relation of undisturbed loess pore size distribution and mechanical property under freezing-thawing environment. Coastal Engineering, 2019, 51(3): 107–112

[52] Matteson A, Tomanic J P, Herron M M, Allen D F, Kenyon W E. NMR relaxation of clay/brine mixtures. SPE Reservoir Evaluation & Engineering, 2000, 3(5): 408–413

[53] LiZ MZengW XGaoM L. Nuclear magnetic resonance test and analysis on water phase of the ultra-soft soil under different load level and rate. Acta Physica Sinica, 2014, 63(1): 018202 (in Chinese)

[54] Zhou R, Bai B, Chen L, Zong Y C, Wu N. A granular thermodynamic constitutive model considering THMC coupling effect for hydrate-bearing sediment. Ocean Engineering, 2024, 310: 118689

[55] Song Z Y, Zhang Z H. Shear strength equation of soils in a wide suction range under various initial void ratios. Vadose Zone Journal, 2024, 23(5): e20368

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