Soil water content detection based on acoustic method and improved Brutsaert’s model

Geoderma 359 (2020) 114003

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Soil water content detection based on acoustic method and improved Brutsaert’s model Yan Xua, Jun Lia,b, Jieli Duana,b, Shuaishuai Songa, Rui Jianga, Zhou Yanga,b, a b

T

⁎

College of Engineering, South China Agricultural University, Guangzhou 510641, China Key Laboratory of Key Technology on Agricultural Machine and Equipment, Ministry of Education, Guangzhou 510642, China

A R T I C LE I N FO

A B S T R A C T

Handling Editor: L.S. Morgan Cristine

The traditional Brutsaert’s model does not consider the influence of fixed Poisson’s ratio on the accuracy of the model. Therefore, we extend Brutsaert’s model by introducing Poisson’s ratio dynamic solution model, and obtain the improved Brutsaert’s model which considers the change of soil Poisson’s ratio. The primary objective of this work was to explore whether improved Brutsaert’s model (Poisson’s ratio dynamic solution model is introduced) can more accurately predict soil volumetric water content (SVWC) than traditional Brutsaert’s model. Acoustic velocities (the first compressional wave and shear wave) of natural silica sand (sand) and three different textures of farming soils (loam, sandy loam and clay loam) were measured using pulsed acoustic waves with frequencies less than 900 Hz under the conditions of wet swelling and drying shrinkage, respectively. The traditional Brutsaert’s model and improved Brutsaert’s model are validated by the measured acoustic velocities. We compare the result calculated by improved Brustsaert’s model with that calculcated by traditional Brustsaert’s model. The compared results show that the improved Brustsaert’s model is more suitable for measuring the relationship between the velocity and the soil water content (SWC) than traditional Brustsaert’s model in four different textures of soil samples. In order to verify the feasibility of the improved Brustsaert’s model for predicting soil water content (SWC), field experiments were carried out on three kinds farming soil (loam, sandy loam and clay loam) sampling sites. The results show that the detected precision measured by improved Brustsaert’s model was around 5%.

Keywords: Soil volumetric water content Poisson Sratio Improved Brutsaert Smodel

1. Introduction Soil water content (SWC) is a physical metric that measures the degree of soil dryness and wetness. It is also an important parameter for irrigation management, regional hydrological research and water balance calculation in river basins (Han et al., 2016; Lee et al., 2017; Sigouin et al., 2016). With the global water resources crisis increasing, the idea of precise management of crop irrigation arises, and SWC plays an important role in water-saving agriculture research (Mubarak et al., 2016; Huan et al., 2016; Sugimoto et al., 2013). Therefore, precisely measuring the SWC and understanding the water content changing rule is an important factor for making the suitable irrigation plan. Several methods have been proposed for the measurement of SWC, however, their accuracy and efficiency need to be significantly improved in practice. The traditional drying and weighing method is the only way to directly measure SWC at different soil depths, but it need extra cost on time and labor (Ma et al., 2016). The neutron scattering method is suitable for long-term monitoring of soil moisture in the field, it has

⁎

Corresponding author. E-mail address: [email protected] (Z. Yang).

https://doi.org/10.1016/j.geoderma.2019.114003 Received 22 May 2019; Accepted 5 October 2019 0016-7061/ © 2019 Elsevier B.V. All rights reserved.

potential radiation hazards (Bogena et al., 2015). Other measurement methods have been developed such as Time Domain Reflectometry (TDR), and it is most commonly used method to detect SWC in some cases, because of fast, safe and simple measurement. It relies on measuring the dielectric constant of soil to measure the SWC (Campbell, 2010; Evett et al., 2005). However, due to the time difference of electromagnetic wave incident reflected by water-bearing soil is limited to 10−10 s (Lin, 2003), TDR instruments are generally expensive. As a one of non-destructive method, the acoustic detection strategy attracts many researchers attention (Chandrappa and Biligiri, 2016; Kapoor et al., 2018). Acoustic wave is a good information carrier. When it interacted with the soil, it carries the information of basic physical parameters related to soil in the receiving wave (Oshima et al., 2015). The amount of water content in soil directly affects the change of parameters of soil (Zhang et al., 2016). Thus, acoustic detection method is suitable for the measurement of SWC. Recently, some researchers have used experimental calibration method to analyzed the relationship between acoustic parameters and SWC, and constructed a functional

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ratio dynamic solution model, and obtain the improved Brutsaert’s model which considers the change of soil Poisson’s ratio. The effects on volumetric water content changing of four types of soil on the first compressional wave velocity, shear wave velocity and Poisson’s ratio of low frequency pulsed acoustic wave (< 900 Hz) were studied. The experimental results were used to validate the Brutsaert’s model and improved Brutsaert’s model. The aims of this study were to: (1) Explore the feasibility of Brutsaert’s model and improved Brutsaert’s model for describing soil volumetric water content (SVWC), and compare with the measured acoustic velocity. (2) Evaluate the feasibility of improved Brutsaert’s model for SWC detection in agricultural land.

relationship between acoustic parameters and SWC (Wei and Huang, 2017; Li et al., 2017; Huang et al., 2013; Oelze et al., 2002). However, this function is from the empirical results rather than the theoretical analysis. We cannot guarantee that it holds the generalizability. In fact, Brutsaert’s theory (1964) of elastic wave propagation in unsaturated porous granular media with three different phases (i.e., solid and two immiscible fluids) is over 50 years. However, the theory of elastic wave propagation in inhomogeneous granular media was originally developed by Brandt (1955). Biot (1956) established a theoretical model of acoustic wave propagation by studying a continuum consisted of a rigid frame and a pore which contains a compressible viscous fluid. Brutsaert and Luthin (1964) introduced the Lagrangian equation to the Biot model. It introduces the effective fluid density and the bulk modulus as the appropriate weighted average values of density and volume of two incompatible viscous compressible fluids to construct a model, which can reflect the relation between the unsaturated SWC and the velocity of acoustic wave. From Brutsaert equation, we can see that the inertial coupling between constituent phases is not considered. Therefore, the low-frequency acoustic signal should be used in the propagation of acoustic waves in three-phase granular media. Adamo et al. (2004) studied the feasibility of Brutsaert’s model to detect SWC, they obtained the relationship between acoustic velocity and saturation under different soil types, and analyzed the validity of the Brutsaert’s model. They use a specific sand which has known the physical properties to validate their hypothesis. The results show that Brutsaert’s model is effective for calculating the water content in sand (Adamo et al., 2007; Adamo et al., 2009). Attivissimo et al. (2010) measured the acoustic velocity of sandy soil through the seismic wave propagation-based methods, they took the water content of sandy soil measured by TDR method as reference value, and compared the theoretical value of acoustic velocity deduced by Brutsaert’s model with the measured value. The experimental results show the effectiviness of Brutsaert’s model. Sharma and Gupta (2010) measured the acoustic velocity of different textures sandy loam soil samples with frequency less than 900 Hz in the laboratory. Brutsaert’s model was used to calculate the theoretical values of the acoustic velocity of soil samples with different textures. By comparing the measured values with the theoretical values, the error between the theoretical values of acoustic velocity and the measured values was relatively small, but there is a distortion of the theoretical values in the lower stage of soil saturation, and the distortion range becomes larger with the increasing of soil clay content. The previous studies mainly focused on the detection of mineral media such as porous rocks and sand. However, Biot theoretical model is based on Brutsaert’s model, and the porous medium is rigid skeleton at the initial stage when Biot model has been established. Therefore, Biot model can better detect the acoustic characteristics of porous mineral media (Berryman, 1999; Carcione et al., 2000; Leclaire et al., 1994). For Brutsaert’s model, the precondition for its application in agricultural soils is to understand the influence of swelling from contraction of clay. In fact, we can see that Brutsaert’s model is mainly applicable to the prediction of water content in sandy soil. In the experiments of literature (Sharma and Gupta, 2010), the samples with the highest viscosity have only 11% clay content. However, when the soil saturation (S ) is less than 0.1, the deviation between the theoretical value and the measured value is serious. Therefore, it remains challenge in which one uses Brutsaert’s model to calculate water content in agricultural soils. In addition, some researchers neglected the specific value of Poisson’s ratio (0.20) in Brutsaert’s model (Adamo et al., 2004; Adamo et al., 2009; Adamo et al., 2010; Attivissimo et al., 2010; Sharma and Gupta, 2010; Lu and Sabatier, 2009; Flammer et al., 2001). In fact, Poisson’s ratio of 0.20 is only an example of Brandt’s model (1955). Brutsaert and Luthin did not consider the impact of fixed Poisson’s ratio on model accuracy in subsequent studies. In this study, we extend Brutsaert’s model by introducing Poisson’s

2. Improved Brutsaert’s model Brutsaert’s model was first derived from the combination of Biot theory model and linear elasticity theory. For the traditional Brutsaert’s model, the Poisson’s ratio (σs ) is fixed at 0.20 regardless of the change of soil texture or soil saturation. For other researchers, the value of Poisson’s ratio of soil is also fixed at 0.20, which does not take into account the impact of fixed Poisson’s ratio of aggregate on test results. However, according to published papers, soil Poisson’s ratio would be changed when the soil saturation changing, and is related to the density of soil samples and confining pressure (Kumar and Madhusudhan, 2012; Fredlund, 2006; Inci et al., 2003; Wang et al., 2002). Therefore, it is unsuitable for using a fixed, independent of saturation Poisson’s ratio for unsaturated porous granular media. In order to solve this problem, a modified Brutsaert’s model is established by replacing the traditional fixed Poisson’s ratio with the dynamic method equation in the measurement of soil Poisson’s ratio to demonstrate the relationship between the acoustic compressional wave velocity and SWC. 2.1. Brutsaert’s model Brutsaert and Luthin (1964) proposed a unsaturated soil elastic wave propagation model in 1964, and proofed that the elastic wave propagation pattern in soil has three compressional waves (called first, second and third compressional waves) and one shear wave. The first compressional wave propagation speed (Vp ) is the fastest at low frequency, and its relationship with soil saturation is as follows: 1/3

1/2

0.306ape Z ⎤ VP = ⎡ ⎢ ρ f b2/3 ⎥ ⎣ tot 0 ⎦

(1)

In Eq. (1), Vp is the first compressional wave velocity; f0 is the porosity of the tested soil; ρtot is the total density of the tested soil and can be defined as ds (1 − f0 ) + ρω ∙f0 ∙S , where ds , ρω and S are assigned for the density of soil particles, water density (1 g/cm3), and the degree of soil liquid saturation, respectively. a and b are empirical parameters determined by fitting experimental data and both of them depend on the elastic properties of a single soil solid particle. The variation range of coefficient a is [0,1], and that of coefficient b is [10–10,10–12]. The effective stress Pe of the tested soil can be defined as:

pe = pt − pa − Spc

(2)

where pt is the total pressure, which defined to be ρtot ∙g∙h , where g and h are assigned for acceleration of gravity (9.8 m/s2) and the distance of sound propagating in soil, respectively. pa is the air pressure on soil particles, which its value can be negligible. According to the conversion formula of three-phase proportionality index of soil (Zhao, 2014), S can be reformulated as:

S=

θv f0 ·ρw

(3)

In Eq. (3), θv indicates the soil volumetric water content (SVWC). pc is soil matrix potential (Whalley et al., 2012), which is caused by the adsorption force and capillary pressure of soil particles. In fact, the 2

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elastic theory model when we set the value of soil Poisson’s ratio (σs ) to 0.20 (Brutsaert and Luthin, 1964). The expressions are as follows:

measurement of soil matrix potential needs to be determined by soil water retention curve, and it needs a lot of experimental data fitting and analysis to calculate (Shao, 2006). Van Genuchten’s model is most commonly used for describing soil water retention curves because its curves are similar to measured data curves and its parameters are commonly used in practice. Therefore, according to Van Genuchten’s model (1980), pc can be expressed as:

⎟

Kb = −Vb

(4)

In Eq. (4), θs indicates the soil saturated water content and θr indicates the soil residual water content.α , m and n are the shape parameter of soil water retention curve (m = 1–1/n, n > 1), and they can be obtained by referring to the empirical values calculated by Carsel and Parrish (1988). In Eq. (1), Z is a dimensionless quantity, which can be reformulated as follows:

{1 + Z= ⎨ ⎩

·b

}5/3 3/2

⎧ 1+

ka·kw ⎤ 46.12 ⎡ ⎣ ka (1 − S ) + kw·S ⎦ pe1/2

·b ⎫

⎬ ⎭

(5)

θ 1 θ −θ 0.306·a·⎧[ds (1 − f0 ) + θv]·g ·h + f v ⎧ α ⎡ θs − θr v r 0⎨ ⎣ ⎨ ⎩ ⎩ [ds (1 − f0 ) + θv]·f0 ·10−8

(

Vp =

1/ m

)

− 1⎤ ⎦

(8)

σS =

Vp 2

( ) −1 ( ) −1

0.5·

In Eq. (5), the added symbols are: ka , k w for bulk moduli of the air and the water respectively (ka ≅ 1.4∙105 Pa; k w ≅ 2∙109 Pa). It can be seen that parameter Z mainly depends on S and pe , and pe is also affected by S , so the parameters Z and S in Eq. (5) can be analyzed by regression. Adamo et al. (2004) has done this work. In his research, no matter how the other regression variables in Eq. (5) are valued, the value of Z is close to 1 in the range of S < [0,0.95], and the smaller the value of parameter b, the closer the value of Z is to 1. Thus, in this paper, the values of Z and b are 1 and 10−12, respectively. Finally, the relationship model between the first compressional wave velocity and soil volumetric water content (SVWC) from Brutsaert’s model can be expressed as follows: 1/ n

0.153ape1/3 Z dpt = dVb f0 b2/3

Only if the granular medium is macro-isotropic and homogeneous, the coefficient is 0.153 in Eq. (8) (Brutsaert, 1964). The calculated coefficient within Eqs. (1) and (6) is from Eqs. (7) and (8), based on the Poisson’s ratio with 0.20. The velocities of first compressional wave (P-wave) and shear wave (S-wave) in the medium are different, because their values are determined by the elastic parameters of the medium. For the soil, the velocity of P-wave (Vp ) and that of S-wave (Vs ) will be changed with the change of SWC. According to the theory of elasticity, we can use the ratio of P-ware and S-wave velocities to calculate the soil continuous dynamic Poisson’s ratio (σs ) can be obtained by Eq. (9) (Liu et al., 2007).

3/2

ka·kw ⎤ 30.75 ⎡ ⎣ ka (1 − S ) + kw·S ⎦ pe1/2

(7)

where Kb is the bulk modulus of soil. According to the physical meaning of bulk modulus and the theoretical deduction of Brandt (1955), Kb can be expressed as:

1 n

1

⎤ 1 ⎡ θ − θr ⎞ m − 1⎥ pc = − ⎢ ⎛ s α ⎝ θv − θr ⎠ ⎦ ⎣ ⎜

1/2

3K (1 − σs ) ⎤ Vp = ⎡ b ⎢ ρ (1 + σs ) ⎥ ⎦ ⎣ tot

Vs Vp 2

(9)

Vs

According to Eq. (9), the soil dynamic Poisson’s ratio is a function of the ratio of P-wave velocity to S-wave velocity. Thus, the soil dynamic Poisson’s ratio can be obtained by calculating the velocity of the P-wave and S-wave. We have improved the traditional Brutsaert’s model which only considers the fixed Poisson’s ratio of soil. By substituting Kb from Eq. (8) and σs from Eq. (9) into Eq. (7), we get improved Brutsaert’s model, which considers the change of soil Poisson’s ratio expressed by Eq. (10). 1

1/3 2 1 ⎡ 0.612ape Z ⎤ [3·(3vp2 − 4vs2)] 2 = ⎢ ⎥ 2/3 ρ f b ⎣ tot 0 ⎦

1/3

⎫⎫ ⎬ ⎭⎬ ⎭

(10)

3. Material and methods

(6) Brutsaert (1964), whose work was the basis of Brutsaert and Lithin (1964), proposed the theoretical model of elastic wave propagation in porous aggregate media is based on the assumption which is no inertial dynamic coupling between gas and liquid in porous media. Although many researchers have noticed this problem (Arora et al., 2008; Lo and Sposito, 2013; Sharma and Kumar, 2011), the inertia-viscosity factor (β) of porous aggregate media is less than 1 according to Biot (1956) study, and the effect of inertia coupling can be neglected. In exploring the applicable conditions of Brutsaert’s model, Adamo et al. (2004) pointed out that the inertia-viscosity factor (β) of the tested soil is less than 1 in all soil types when the detection frequency of acoustic wave is less than 900 Hz. Therefore, despite there are many problems, the Brutsaert’s model can be applicable at sufficiently low frequencies. It is noticed that Brutsaert’s model does not consider the effect of soil temperature on the acoustic velocity propagation in soil, and other parameters shown in Eq. (6) have no significant temperature sensitivity. Therefore, in the process of application we can reduce the detection error caused by environmental temperature difference by adjusting the value of empirical parameter a .

3.1. Soils and other physical data A common silica sand and three representative agricultural soils were selected for this study. According to the Soil Texture Classification Standard of USDA, the selected soil types were sand (Fine silica sand), loam (Ferrallitic soil), sandy loam (Siallitic soil) and clay loam (Hydragric soil), respectively. The basic physical properties of the tested soils are shown in Table 1. Table 2 shows the empirical values (θr , α , m and n) of tested soils calculated by Carsel and Parrish. 3.2. Experimental apparatus overview Our study adopts TH204 (Tiangong Technology Co., Ltd, Xiangtan, China) acoustic parameter detector, which consists of acoustic detector, conductor, soil acoustic emission transducer and receiving transducer. Acoustic time accuracy of this detector is + 0.05 μs, sampling interval is from 0.05 to 200 μs, transmitting pulse voltage can be selected at 500 V and 1000 V, transmitting pulse width is from 2 to 100 μs, and transmitting frequency range of acoustic transducer is between 500 Hz and 20 KHz. The parameter settings of test process detector are shown in Table 3. In order to reduce the influence on measuring soil samples conducted by manual operation, an information acquisition device is

2.2. The problem of Poisson’s ratio The coefficient 0.306 in Eq. (1) and Eq. (6) is calculated by linear 3

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Table 1 Basic physical properties of soils used in this work. The gravity of soil(ds ) was measured by the pycnometer method. Soils

Sampling locations

Sand

22°01′24.96″N 113°34′72.93″E 23°61′47.49″N 113°59′31.41″E 23°16′94.01″N 113°37′66.49″E 23°26′04.71″N 113°76′92.45″E

Loam Sandy loam Clay loam

Sampling depth/cm

Soil particle composition/%

The density of soil particles(ds)

Sand

Silt

Clay

0–10

89.46

6.72

3.82

2.72

0–20

46.72

36.14

17.14

2.55

0–20

54.58

33.75

11.67

2.58

0–20

43.29

24.26

32.45

2.69

used in our experiment had been made. Fig. 2b is the paddy soil sample. Other types of soil samples were also obtained by the same method.

Table 2 Values of θs , θr , α , m and n for soils used in this work. Soils

θs (cm3·cm−3)

θr (cm3·cm−3)

α (cm−1)

m

n

Sand Loam Sandy loam Clay loam

0.432 0.446 0.489 0.440

0.045 0.078 0.065 0.095

0.145 0.036 0.075 0.019

0.63 0.36 0.47 0.24

2.68 1.56 1.89 1.31

3.3.2. Measurement methods The mass water content (i.e., the ratio of water quality to dry soil quality; θm = m w / ms ), total density (i.e., the ratio of total soil quality to soil sample volume; ρtot = (ms + m w )/ V ), bulk density (i.e., the ratio of dry soil quality to soil sample volume; ρd = ms / V ), volumetric water content (i.e., the product of soil sample mass water content and bulk density; θv = θm ∙ρd ), and the porosity (i.e., 1 minus the ratio of bulk density to the density of soil particles; f0 = 1 − ρd / ds ) of the prepared soil samples were calculated, respectively. Then, we gradually spray 150 ml water to the top of the soil sample with a pneumatic spray bottle to moisten the soil. Soil acoustic parameters acquisition device shown in Fig. 1 was used to detect the P-wave and S-wave velocities of soil samples in different time periods after irrigation. It is observed that the measured acoustic velocity of soil samples would be significantly changed after 40 min irrigation. This phenomenon shows that the moisture can lead to the change of soil micro-structure, making the soil structure change from one physical state to another physical state. After 1 h, the measurement results of acoustic velocity gradually stabilized. We then recorde the P-wave and S-wave velocities of the current soil samples, and calculating the relevant physical parameters (mass water content, total density, bulk density, volumetric water content and porosity) of the current sample, according to the scale of the container. We sprinkle 150 ml water in soil sample again with a pneumatic spray bottle. After 1 h, the P-wave velocity, S-wave velocity and the related physical parameters of the soil samples were measured. We continued to conduct the above operations. After adding water 150xNml, some of the water were permeated into the tray, and the soil sample was nearly saturated under such a scenario. In order to ensure that the soil is fully saturated (i.e., there is no gas in the soil pore), water 150 ml were added into the soil sample again. After that, we measured the total volume (Vs ) of the saturated soil sample and the volume of the remaining water (Vr ) in the tray. In addition, soil saturation water content (θs ) was estimated by using Eq. (11). The physical parameters for the soil samples during wet swelling are shown in Table 4.

designed as shown in Fig. 1. The main structure of the device is a slider bracket for linear guides in which two linear guides rail are fixed parallel in the ground through the bearing block, while the sliders are fixed together through the acoustic transducer clamp. Moving the slider in the linear guide, the acoustic transducers are driven by the transducer clamp, which is close to the soil sample mould, to collect the acoustic information. The soil sample mould is made of woods with 5 mm thickness, soundproof cotton, which can reduce the impact of external acoustic waves on the detection results. The acoustic transducer is driven by TH204 acoustic parameter detector, which can real-time detect the compressional and shear acoustic velocities. In order to facilitate data storage and processing, USB communication cable is used to send data information to the external computer to analyze data directly. 3.3. Experiments and methodology 3.3.1. Sample preparation The collected soil (sand, loam, sandy loam and clay loam) was ground and laid in the sunshine until it was completely natural airdried. The soil with ms weight was put into a self-made container with a bottom area of 20 cm*20 cm as shown in Fig. 2a, and the height was marked with scale. Each side centre of the self-made container is drilled a large circular hole with a diameter of 60 mm for convenient soil testing with acoustic emission transducer and receiving transducer. In other regions in the side, we drill several holes with a diameter of 15 mm, which can increase the ventilation surface of the soil. Each hole is sealed by 500 mesh steel wire mesh, which only allows the air to pass through it. The bottom of the self-made container is composed of fixed support plate and 200 mesh steel wire mesh in which 200 mesh steel wire mesh can allow the water to pass through it. In order to reduce the influence on the test results from soil structure difference, uniform humidification treatment was carried out in the self-made container. After soil saturating, we place it into a well-ventilated area until the soil weight in the container was within the range of ms ± 10 g, which can prove that the soil was completely air-dried. After that, the soil samples

θs =

(150·N ) − Vr ms · ms Vs

(11)

Saturated soil sample were placed in a well ventilated area for 24 h and recorded the P-wave velocity, S-wave velocity and related physical parameters of the current soil sample. Soil sample was recorded in a fixed interval, 24 h, until the volumetric water content of soil sample

Table 3 Parameter setting of TH204 acoustic wave detector. Sampling byte/Kb

Sampling interval/μs

Emission voltage/V

Emission pulse width/μs

Transducer frequency/Hz

Trigger type

8

1

1000

5

900

μP

4

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Fig. 1. Structure of the soil acoustic parameter acquisition device. (a) Structure of soil sample (b) Photograph of clay loam sample.

was lower than the residual water content. The physical parameters for the soil samples during drying shrinkage are shown in Table 5.

4. Results and discussion 4.1. Effect of soil water content on P-wave velocity and S-wave velocity We conduct the acoustic experiments on the four soil types, which are sand, loam, sandy loam and clay loam respectively. We focus on analyzing the changing situation of P-wave velocity and S-wave velocity as the water content is changing. It was found that the acoustic Pwave velocity and S-wave velocity were highly affected by the volumetric water content of soil samples. Fig. 3 illustrates the relationship between the SWC and the acoustic velocity (including P-wave and Swave) during the process of soil swelling and drying shrinkage. From Fig. 3, we can see that changing SVWC can cause the corresponding change of pulse acoustic velocity. With the increasement of SVWC, the P-wave and S-wave velocities decrease monotonously. In general, the water within soil particles consists of the bound-water and free-water (Shao et al., 2011). The different water contents have different effects on the soil porosity, directly affecting the structure of the soil (Rabot et al., 2018), resulting therefore in a variation of the acoustic P-wave velocity and S-wave velocity. When the SVWC is lower than the residual water content, the bound-water film does not be formed on the surface of most soil particles. With the water content increment, the bound-water film will be thicker, which enlarges the size of soil particles and the soil porosity. In bound-water, the polar water molecules and cations in aqueous solution are attracted by the electric field within soil particles. Water molecules are arranged around the surface of particles in a directional manner. Bound-water does not have the typical characteristics of water and it is actually approximated to the solid. When the water content is higher than the residual water content, the soil water suction decreases and the bound-water in the soil particles becomes saturated. The water in the soil gradually transits from bound-water to free-water. Free-water exists out of the surface electric field of soil particles, and its properties are close to ordinary water. Free-water can move through the soil pores due to gravitational forces. Furthermore, hydrodynamic pressure increases soil porosity; and soil pores are occupied by air (Zhao, 2014). Therefore, soil porosity affects acoustic velocity during transmission. Under the same soil texture, both P-wave velocity and S-wave velocity decrease with the SVWC increment.

Fig. 2. Structure diagram (a) and photograph (b) of soil sample. (a) Relationship between Soil volumetric water content and acoustic P-wave velocity for four types of soil sample. (b) Relationship between Soil volumetric water content and acoustic S-wave velocity for four types of soil sample.

5

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Table 4 Basic physical parameters for the soil samples during wet swelling. Soil samples

Watering times

θm (g·g−1)

ρtot (g·cm−3)

ρd (g·cm−3)

θ v (m3·m−3)

f0 (%)

θs (m3·m−3)

Sand

0 1 2 3 4 5 6 7 8 9 10 11 12

0.017 0.041 0.066 0.091 0.116 0.141 0.166 0.191 0.216 0.240 0.265 0.290 0.315

1.580 1.619 1.640 1.646 1.642 1.678 1.674 1.710 1.736 1.748 1.770 1.768 1.779

1.554 1.554 1.538 1.508 1.471 1.471 1.436 1.436 1.428 1.409 1.399 1.370 1.370

0.026 0.063 0.102 0.137 0.171 0.207 0.238 0.274 0.308 0.338 0.371 0.397 0.432

0.429 0.429 0.435 0.446 0.459 0.459 0.472 0.472 0.475 0.482 0.486 0.496 0.496

0.432

Loam

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.019 0.047 0.075 0.104 0.132 0.160 0.189 0.217 0.245 0.274 0.302 0.330 0.358 0.377

1.428 1.453 1.518 1.516 1.531 1.538 1.537 1.543 1.542 1.555 1.575 1.595 1.622 1.629

1.402 1.387 1.387 1.373 1.352 1.325 1.293 1.268 1.238 1.221 1.210 1.199 1.199 1.183

0.027 0.065 0.104 0.143 0.178 0.212 0.244 0.275 0.303 0.335 0.365 0.396 0.429 0.446

0.450 0.456 0.456 0.462 0.470 0.480 0.493 0.503 0.515 0.521 0.525 0.530 0.530 0.536

0.446

Sandy loam

0 1 2 3 4 5 6 7 8 9 10 11 12

0.024 0.059 0.095 0.130 0.165 0.201 0.234 0.272 0.307 0.343 0.378 0.390 0.426

1.353 1.383 1.429 1.448 1.467 1.477 1.494 1.503 1.528 1.556 1.566 1.620 1.673

1.322 1.306 1.306 1.282 1.259 1.230 1.209 1.182 1.169 1.159 1.149 1.149 1.149

0.032 0.077 0.124 0.167 0.208 0.247 0.283 0.322 0.359 0.398 0.434 0.448 0.489

0.488 0.494 0.494 0.503 0.512 0.523 0.531 0.542 0.547 0.551 0.555 0.555 0.555

0.489

Clay loam

0 1 2 3 4 5 6 7 8 9 10 11

0.010 0.041 0.072 0.103 0.134 0.165 0.195 0.226 0.257 0.288 0.319 0.334

1.525 1.552 1.579 1.595 1.602 1.608 1.632 1.637 1.660 1.692 1.714 1.754

1.509 1.491 1.473 1.446 1.413 1.381 1.365 1.335 1.321 1.314 1.299 1.299

0.015 0.061 0.106 0.149 0.189 0.228 0.266 0.302 0.339 0.378 0.414 0.434

0.439 0.446 0.452 0.462 0.475 0.487 0.493 0.504 0.509 0.512 0.517 0.517

0.434

gradually higher than the residual water content, the matrix potential of soil samples decreases when the soil water content increasing. Thus, the measured S-wave velocity decreases. In the saturated state of soil samples, the measured S-wave velocity has a large jumping range and distortion. Therefore, the S-wave velocity in the saturated state of soil samples in this paper is removed. In general, the acoustic velocity measured in the process of natural air-drying soil samples is higher than that measured in wetting process. Soil can cause different changes in soil structure during the process of wet swelling and drying shrinkage. The internal resistance of soil to particle rearrangement makes the volume of soil during drying shrinkage larger than that during wetting swelling under a certain suction balance. Thus, the porosity of soil sample during drying shrinkage is larger than during wetting swelling under the same water content. Therefore, the acoustic wave propagation velocity in soil samples during drying shrinkage is faster than that during wet swelling.

Fig. 3b shows a relationship between the pulse acoustic S-wave velocity and SVWC for four types of soil sample. There is a key point in the relationship curve between S-wave velocity and SVWC in the stage of soil wet swelling, which corresponds to the residual water content. At that stage, when the SVWC is lower than the residual water content, the relationship curve between S-wave velocity and SVWC decreases slowly, but the relationship curve decreases rapidly when the SVWC is higher than the residual water content. This phenomenon is related to the matrix potential of soil samples. Some researchers have studied the relationship between soil matrix potential and S-wave velocity, and found that the S-wave velocity increases when the soil matrix potential increasing. When the soil matrix potential increases to a certain value, the S-wave velocity does not increase (Whalley et al., 2012). In our experiments, when the SVWC of soil samples is less than the residual water content, the matrix potential of soil samples is generally large, while the measured S-wave velocity changes slowly. When the SVWC is 6

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Table 5 Basic physical parameters for the soil samples during drying shrinkage. Soil samples

Measurement times

θm (g·g−1)

ρtot (g·cm−3)

ρd (g·cm−3)

θ v (m3·m−3)

f0 (%)

θs (m3·m−3)

Sand

1 2 3 4 5 6 7 8 9 10

0.287 0.264 0.242 0.203 0.165 0.137 0.116 0.089 0.056 0.039

1.774 1.757 1.746 1.722 1.675 1.668 1.647 1.641 1.629 1.597

1.375 1.416 1.430 1.430 1.434 1.457 1.468 1.512 1.536 1.536

0.395 0.374 0.346 0.290 0.237 0.200 0.170 0.135 0.086 0.060

0.494 0.480 0.474 0.474 0.473 0.464 0.460 0.444 0.435 0.435

0.432

Loam

1 2 3 4 5 6 7 8 9 10 11 12

0.362 0.341 0.319 0.293 0.267 0.234 0.203 0.177 0.161 0.129 0.098 0.064

1.625 1.614 1.607 1.605 1.594 1.581 1.563 1.547 1.535 1.528 1.517 1.479

1.192 1.203 1.212 1.224 1.231 1.248 1.261 1.317 1.331 1.365 1.373 1.385

0.432 0.410 0.387 0.359 0.329 0.292 0.256 0.233 0.214 0.176 0.135 0.089

0.533 0.528 0.525 0.520 0.517 0.511 0.505 0.484 0.478 0.465 0.462 0.457

0.446

Sandy loam

1 2 3 4 5 6 7 8 9 10 11 12

0.404 0.381 0.354 0.310 0.289 0.256 0.212 0.187 0.151 0.119 0.086 0.057

1.642 1.618 1.559 1.538 1.517 1.496 1.484 1.471 1.453 1.436 1.421 1.385

1.157 1.162 1.162 1.173 1.184 1.189 1.206 1.240 1.276 1.298 1.306 1.316

0.467 0.443 0.411 0.364 0.342 0.304 0.256 0.232 0.193 0.154 0.112 0.075

0.552 0.550 0.550 0.545 0.541 0.539 0.533 0.519 0.505 0.497 0.494 0.490

0.489

Clay loam

1 2 3 4 5 6 7 8 9 10 11 12 13

0.322 0.304 0.279 0.258 0.227 0.204 0.183 0.165 0.148 0.122 0.095 0.078 0.066

1.736 1.703 1.681 1.661 1.639 1.627 1.616 1.604 1.597 1.592 1.588 1.579 1.574

1.304 1.314 1.321 1.333 1.346 1.368 1.382 1.391 1.403 1.427 1.461 1.475 1.498

0.420 0.399 0.369 0.344 0.306 0.279 0.253 0.230 0.208 0.174 0.139 0.115 0.099

0.515 0.512 0.509 0.504 0.500 0.491 0.486 0.483 0.478 0.470 0.457 0.452 0.443

0.434

was slightly affected by soil type, confining pressure, and porosity. As shown in Fig. 4, the dynamic Poisson’s ratio of four types of soil sample increases when SVWC increasing. In the lower and higher stages for SVWC, the change of Poisson’s ratio is slighter. The Poisson’s ratio of sand sample measured in our experiment is ranged from 0.10 to 0.25; the Poisson’s ratios of loam sample and clay loam sample are ranged from 0.20 to 0.35; and the Poisson’s ratio of sandy loam sample is ranged from 0.23 to 0.38. Published values for Poisson’s ratios for soils are:0.15–0.20 (unsaturated sand), 0.20–0.25 (sandy soil), 0.2–0.3 (Silty soil), 0.25 (Silty clay in hard state), 0.30 (Silty clay in plastic state), 0.35 (Silty clay in soft plastic or flowing state) (Yang, 2007). The Poisson’s ratios measured in the tests are in good agreement with the published data for soils. Because it is difficult to measure the S-wave velocity when soil samples are saturated, the limit Poisson’s ratio of saturated soil samples could not be determined. Inci et al. (2003) have pointed out that saturated soils are basically incompressible, so the corresponding limit Poisson’s ratio should be close to 0.5. As can be seen from Fig. 4, the Poisson’s ratio of soil samples measured in the stage of wet swelling is basically larger than that measured in the stage of drying shrinkage.

It should be mentioned that the experiments that took place in this study showed that acoustic P-wave velocity is related to soil texture (clay loam > loam > sandy loam > sand). When the soil texture is more sticky and the content of clay is relatively higher, the faster the acoustic velocity is, indicating that soil texture is also one of the factors of affecting the acoustic wave propagation velocity. 4.2. Effect of soil water content on Poisson’s ratio It was assumed that the soil medium was isotropic, semi-infinite, and elastic similar to the assumption provided in Richart et al. (1970). The variation of Poisson’s ratio of four different types of soil sample under different water contents was studied. The P-wave velocity and Swave velocity of soil samples under wet swelling and drying shrinkage were measured. Then, the dynamic Poisson’s ratio of soil samples can be calculated by the Poisson’s ratio dynamic solution model. The relationship curve between the dynamic Poisson’s ratio and SVWC for those soil samples is shown in Fig. 4. Pipinato (2016) indicated that Poisson’s ratio depended highly on the degree of saturation and slightly on the plasticity of the soils, and 7

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Fig. 4. Relationship between Soil volumetric water content and Poisson’s ratio for four types of soil sample. (a) Sand sample (b) Loam sample (c) Sandy loam sample (d) Clay loam sample.

4.3. Verification of the theoretical models From the theoretical model discussed in Section 2, it can be seen that the acoustic velocity propagation in soil depends on soil parameters a, α , m , n , θr , θs , θv , ds and f0 (through Eq. (6)). The value of parameter a depends only on the soil type, while the values of parameters θs , θv , ds and f0 depend on the experiment. The results have been obtained in Section experiment. Parametersα , m , n and θr are refers to the empirical values calculated by Carsel and Parrish, and the empirical values of the soil samples studied in this paper are presented in Table 2. The premise of using Brutsaert’s model to study the relationship between the acoustic velocity propagation in soil and SVWC is that each soil parameter except for θv is unique value, but the parameters a and f0 are unique values in previous studies. For clayey soil, it will obviously show the characteristics of soil swelling and drying in the process of hydrological cycle such as rainfall infiltration, soil moisture evaporation, irrigation and drainage, so its porosity f0 will change with the SVWC changing. However, Brutsaert’s model is based on Biot theoretical model, whose porous medium is skeleton at the initial stage of establishment. Therefore, in Brutsaert’s model, porosity f0 is a fixed value. Tables 4 and 5 show that the saturated water content of soil samples measured in experiments is similar to the porosity of soil samples at the initial drying stage. According to the conversion formula of three-phase index, the value of soil saturation S is the ratio of SVWC (θv ) to porosity ( f0 ) (through Eq. (3)). Only the porosity value of soil samples in the initial drying stage is substituted into Eq. (3), the saturation value S is closed to 1. Therefore, we use the porosity value to verify the prediction accuracy of theoretical models in the initial drying stage.

Fig. 3. Relationship between soil volumetric water content and acoustic velocity for four types of soil sample. S-w in the figure represents the acoustic velocity curve measured by sand sample at the wetting stage. S-d in the figure represents the acoustic velocity curve measured by sand sample at the drying stage. L-w in the figure represents the acoustic velocity curve measured by loam sample at the wetting stage. L-d in the figure represents the acoustic velocity curve measured by loam sample at the drying stage. SL-w in the figure represents the acoustic velocity curve measured by sandy loam sample at the wetting stage. SL-d in the figure represents the acoustic velocity curve measured by sandy loam sample at the drying stage. CL-w in the figure represents the acoustic velocity curve measured by clay loam sample at the wetting stage. CL-d in the figure represents the acoustic velocity curve measured by clay loam sample at the drying stage.

4.3.1. Brutsaert’s model The value of parameter a is related to soil type. In order to study the effect of parameter a on the results of Brutsaert’s model under different soil types, the value of a were adjusted according to the acoustic velocity values by experiments. Five theoretical Brutsaert’s model curves which are closed to the measured acoustic velocity values were selected to explore the value of a . The relationship curves are shown in Fig. 5. The theoretically computed and experimentally measured values ofVp have been plotted against θv in Fig. 5. When the volumetric water content of soil sample is less than its residual water content, Brutsaert’s model can’t calculate the theoretical acoustic velocity. Thus, Brutsaert’s model can only calculate the theoretical acoustic velocity between the residual water content and the saturated water content. From Fig. 5, it

The main reason is that the drying rate of surface soil is faster than that of inner soil, which affects the uniformity of water phase in soil samples. As the water content levels decrease and the water phase becomes discontinuous the compressibility of soils increases, it leads to the Poisson’s ratio of drying shrinkage becomes smaller under the same SVWC. In addition to the influence of soil water content, soil plasticity is also affected by soil structure. The initial compaction conditions and confining pressures of soil will affect soil structure. Soil water content is not the only factor to affect soil Poisson’s ratio. Therefore, the relationship between Poisson’s ratio and SVWC obtained in this paper is under the condition of relatively stable soil structure. 8

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Fig. 5. Comparison of Brutsaert’s model and measured acoustic velocity in four types of soil sample. (a) Sand sample (b) Loam sample (c) Sandy loam sample (d) Clay loam sample.

RMSE and ea∗ in four types of soil samples were mainly between 16–30 and 3–8, respectively. The minimum values of RMSE and ea∗ calculated for each soil type correspond to the optimum parameter a of the soil sample, but the advantage is not obvious. For Brutsaert’s model, it is difficult to fill in the gap between theoretical and measured acoustic velocity values by adjusting the value of parameter a . Therefore, it is difficult to extend the traditional Brutsaert’s model to the actual agricultural production.

can be seen that both measured and theoretical acoustic velocities decrease withθv increasing. In the expected range of θv , the measured acoustic velocity values have a larger span than the theoretical values. At the lower stage of θv , the measured acoustic velocity is closed to the theoretical curve when parametera adopts a higher value. With θv increasing, the measured acoustic velocity is closed to the theoretical curve when parameter a holds a lower value. From residual water content to saturated water content, the measured values of acoustic velocity across over five theoretical curves, which correspond to the values of five parameters a . When θv of the four soil types is between the residual water content and saturated water content, the theoretical acoustic velocity calculated by Brutsaert’s model does not vary intensively. This result is consistent with the relationship curve between the theoretical acoustic velocity and soil saturation calculated by Adamo et al. (2004). The root mean square error (RMSE) and the average relative error (ea∗) are adopted to calculate each curve which corresponds to the relationship between the theoretical value and the measured value of Vp in the range of the residual water content and the saturated water content of θv . The five calculated curves verify the variation degree of both the theoretical value and the measured value of Vp under different values of parameter a , and the validation results were shown in Table 6. From the verification results in Table 6, we found that the values of

4.3.2. Improved Brutsaert’s model In order to study the feasibility of the improved Brutsaert’s model (through Eq. (10)) which considers the change of soil dynamic Poisson’s ratio to predict soil water content, we need to introduce both P-wave velocity and S-wave velocity. We choose effective acoustic velocity (Ve ) instead of the irrational expression of P-wave velocity and S-wave velocity in Eq. (10), so Ve can be expressed as: 1

Ve = [3·(3vp2 − 4vs2)] 2

(12)

The relationship between the volumetric water content of four soil types and the measured and theoretical values of effective acoustic velocity is shown in Fig. 6. As the same with Brutsaert’s model, improved Brutsaert’s model can only calculate the theoretical value of Ve 9

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Table 6 Model verification results of Brutsaert’s model.ea∗ represents the average relative error between the theoretical value and the measured value of Vp . RMSE represents the root mean square error between the theoretical value and the measured value of Vp . Soils

Curve 1

Sand

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

Loam

Sandy loam

Clay loam

Curve 2 0.08 22.43 6.97 0.10 29.30 7.82 0.10 20.49 4.95 0.17 29.51 6.29

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

Curve 3 0.09 13.15 4.25 0.11 19.41 4.96 0.11 14.82 4.22 0.18 21.56 4.97

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

Curve 4 0.10 16.26 4.46 0.12 16.48 4.72 0.12 18.66 6.10 0.19 17.10 4.13

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

Curve 5 0.11 26.74 4.59 0.13 22.29 6.29 0.13 27.66 9.06 0.20 17.85 3.68

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

0.12 38.46 13.94 0.14 31.85 9.08 0.14 37.92 12.52 0.21 23.35 4.30

in the predictable range, compared with the traditional Brutsaert’s model, the theoretical value calculated by improved Brutsaert’s model coincides with the measured value. The four theoretical curves of each soil type in Fig. 6 correspond to four a values. The root mean square error (RMSE) and the average relative error (ea∗) are adopted to calculate each curve which corresponds to the relationship between the

between the residual water content and saturated water content of θv , so this area belongs to predictable range of θv . Fig. 6d shows that there is a large deviation between the theoretical value and measured value of Ve when the SVWC of clay loam sample is less than 0.174(m3·m−3), so the predictable range of Ve is between 0.174 and 0.440(m3·m−3). In general, the measured value Ve decreases withθv increasing. When θv is

Fig. 6. Comparison of Improved Brutsaert’s model and measured effective acoustic velocity in four types of soil sample. 10

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Table 7 Model verification results of Improved Brutsaert’s model.ea∗ represents the average relative error between the theoretical value and the measured value of Ve . RMSE represents the root mean square error between the theoretical value and the measured value of Ve . Soils

Curve 1

Sand

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

Loam

Sandy loam

Clay loam

Curve 2 0.19 14.55 2.66

Curve 3

Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/% Value a RMSE ea∗/%

0.31 11.11 1.37 0.32 15.24 2.12 0.40 99.89 12.53

0.20 3.71 0.06

Curve 4

Value a RMSE ea∗ /% Value a RMSE ea∗ /% Value a RMSE ea∗ /% Value a RMSE ea∗ /%

0.32 6.02 0.23 0.33 6.71 0.56 0.45 51.52 6.09

0.21 13.74 2.35 0.33 14.09 1.75 0.34 8.77 0.93 0.50 9.43 0.65

Value a RMSE ea∗ /% Value a RMSE ea∗ /% Value a RMSE ea∗ /% Value a RMSE ea∗ /%

0.22 26.65 4.59 0.34 24.47 3.20 0.35 17.84 2.36 0.55 38.73 4.04

The selected soil samples in this paper are typical farming soils, so it is representative for our improved Brutsaert’s model. When we apply the improved Brutsaert’s model to the three soil types (loam, sandy loam and clay loam), we can observe that the theoretical value of the effective acoustic velocity calculated by the improved Brutsaert’s model coincides with the measured value. However, we need to further explore can the improved Brutsaert’s model be applied to the actual measurement of SWC. For this purpose, the sampling regions without planting of loam, sandy loam and clay loam samples were selected for each plot. Each plot was divided into six small areas averagely, and each small area was randomly humidified. After water permeation, two square grooves with a depth of 100 mm and span 200 mm were cut out from each area. Acoustic wave velocities of P-wave and S-wave were measured by TH204 acoustic parameter device in each small area, then measured the SVWC of each small area with cutting ring method. Measured P-wave velocity and S-wave velocity in each small soil area were substituted into Eq. (12) to calculate the effective acoustic velocity. According to the improved Brutsaert’s model, the predicted values of SVWC in each small soil area were calculated, and compared it with the measured values. Table 8 shows the results. When the SVWC is in the predictable range, the improved Brutsaert’s model has a good predictive effect on the three types of agricultural soil. The absolute error between the predicted value and the measured value is controlled at about 5%. The results show that our proposed improved Brutsaert’s model can be used to calculate the SVWC by predicting the change between the acoustic P-wave velocity and the acoustic S-wave velocity.

theoretical value and the measured value of Ve in the predictable range of θv .The four calculated curves verify the variation degree of both the theoretical value and the measured value of Ve under different values of parameter a . The optimum parameter a of four soil types were obtained, and the validation results were shown in Table 7. Table 7 shows that compared with the Table 6, the values of RMSE and ea∗ calculated for each soil type have obvious minimum values. When the parameter a of sand, loam, sandy loam and clay loam samples are 0.20, 0.32, 0.33 and 0.50 respectively, the RMSE andea∗ are the smallest. It indicates that the theoretical value of Ve calculated by improved Brutsaert’s model holds the smallest difference from the measured value under the current parameter a value. The results show that the improved Brutsaert’s model of sand sample has the highest accuracy, and the clay loam sample has the lowest accuracy. This result coincides with the initial theoretical model established by using rigid skeleton as porous medium. The results of Figs. 5 and 6 show that, compared with the traditional Brutsaert’s model, the improved Brutsaert’s model is more reliable in studying the relationship between SVWC and acoustic velocity (effective acoustic velocity). However, we need to determine the values of parameter a in advance. Therefore, the water content of soil can be estimated by improved Brutsaert’s model based on the measured values of P-wave and S-wave velocities of soil pulses. 4.4. Verification of the theoretical models Soil is the basis of agriculture and is one of the basic elements of crop growth. The three soil types (loam, sandy loam and clay loam) selected in our study are located in Guangzhou City, Guangdong Province, China. This area belongs to the south subtropical zone, and it holds south climate characteristics such as warm and rainy weather, sufficient sunlight and small temperature difference. The loam sample mainly planted citrus and litchi, sandy loam sample mainly planted tomatoes and eggplant, and clay loam sample mainly planted paddy.

5. Conclusions In this study, the traditional Brutsaert’s model is extended by introducing the dynamic solution model of Poisson’s ratio, and obtains the improved Brutsaert’s model to study the corresponding relationship between the soil volumetric water content (SVWC) and the effective

Table 8 Estimation absolute errors of volumetric water content for three types of soil under improved Brutsaert’s model.e1∗ represents the absolute error of the SVWC prediction model based on improved Brutsaert’s model, θvm represents the measured value of SVWC, θvp represents the predicted value of SVWC, * represents invalid values. Test groups

1 2 3 4 5 6

Loam

Sandy loam

Clay loam

θvm /m3·m−3

θvp /m3·m−3

e1∗ /%

θvm /m3·m−3

θvp /m3·m−3

e1∗ /%

θvm /m3·m−3

θvp /m3·m−3

e1∗ /%

0.145 0.161 0.224 0.248 0.297 0.353

0.172 0.178 0.273 0.269 0.265 0.387

2.7 1.7 4.9 2.1 3.2 3.4

0.137 0.188 0.216 0.258 0.302 0.331

0.159 0.170 0.175 0.232 0.321 0.287

2.2 1.8 4.1 2.6 1.9 4.4

0.142 0.196 0.244 0.273 0.312 0.367

* 0.180 0.227 0.239 0.288 0.311

* 1.6 1.7 3.4 2.4 5.6

11

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acoustic velocity of natural fine silica sand (sand) and three representative farming soils (loam, sandy loam and clay loam). The accuracy of improved Brutsaert’s model was also evaluated. The main findings of the work are shown as follows.

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Frequency domain versus travel time analyses of TDR waveforms for soil

• In our experiments, the soil porosity increases when injecting water

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into this soil, and decreases when drying this soil. What the state the soil sample lies in, the acoustic velocities of P-wave and S-wave decrease monotonously with the increase of SVWC. There is a key point in the curve between S-wave velocity and SVWC, which is related to soil matrix potential. In general, the acoustic propagates faster in the state of drying shrinkage than in the state of wet swelling, which indicates that the soil structure changes differently in the state of water absorption and drying. The highest velocity was obtained in clay loam soil, followed by the loam and sandy loam soil, and sand is the lowest. It indicates that soil texture will affect the acoustic velocity in soil. The higher the clay content, the faster is the acoustic velocity. With the SVWC increasing, the dynamic Poisson’s ratios of four types soil showed a monotonic increasing and the trend is slowsteep-slow. In general, the Poisson’s ratio of soil measured at the stage of wet swelling is larger than that measured at the stage of drying shrinkage. The main reason is related to the homogeneity of water phase within soil. The relationship curve between SVWC and theoretical P-wave velocity calculated by Brutsaert’s model is consistent with that calculated by Adamo et al. (2004). The experimental results show that the measured P-wave velocity values have a larger span than the calculated value. From residual water content to saturated water content, the measured values of P-wave acoustic velocity span the theoretical curves of five parameters a . Therefore, the Brutsaert’s model is not suitable for describing the relationship between SVWC and P-wave velocity. Compared with Brutsaert’s model, the Brutsaert’s model with Poisson’s ratio dynamic solution model is more suitable for describing the relationship between SVWC and effective acoustic velocity in the predictable range of SVWC. When the parameters a of the improved Brutsaert’s model describing sand, loam, sandy loam and clay loam samples are 0.20, 0.32, 0.33 and 0.50 respectively, the difference between the theoretical value of effective acoustic velocity and the measured value is the smallest. The results show that the improved Brutsaert’s model of sand sample has the highest accuracy and the clay loam sample has the lowest accuracy. This coincides with Brutsaert’s initial theoretical model for porous media with rigid skeleton. The results of field experiments show that the theoretical value of effective acoustic velocity calculated by improved Brutsaert’s model is in a good performance, which is relevance to the measured value within the predictable range of SVWC, and its detection accuracy is about 5%. It proves that the improved Brutsaert’s model proposed in this paper is suitable for the measurement of SWC.

It can be seen that the improved Brutsaert's model can be used in several applications; however, the main challenge is that it does not consider the soil temperature. In addition, the artificial operation would affect the measure accuracy, which makes the performance of the model limited. The change of SVWC is a complex process for soil microstructure, so it is necessary to explore the influence of soil microstructure change on the extraction of soil acoustic parameters in future research. Acknowledgments This research was supported by National Key R&D Program of China (Accession No. 2018YFD0201100), Guangdong Key R&D Program (Accession No. 2019B020223002) and Guangdong Province Modern 12

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