Study on Sound Velocity and Attenuation of Underwater Cobalt-Rich Crust Based on Biot and BISQ Theories

Abstract

A prediction model of the sound velocity and sound attenuation of underwater cobalt-rich crusts (CRCs) was established to solve the problem that it is difficult to predict the sound velocity in thickness measurements of cobalt-rich crusts. Based on Biot theory and BISQ theory, a simplified Biot and BISQ model was proposed for the prediction of the sound velocity and sound attenuation of CRCs by using the Kozeny–Carman (KC) equation. The models could calculate the sound velocity and attenuation by the porosity and detection frequency. Based on the physical and mechanical properties of CRCs, a similarity model of the sound velocity and sound attenuation of CRCs was made by using the similarity theory to solve the problem that it is difficult to measure the acoustic propagation characteristics of CRCs. The sound velocity and sound attenuation of CRC similarity models with different porosities were measured by an underwater transmission experiment and the results of the simplified model calculation and experimental measurements were compared. The results showed that the simplified Biot model was suitable for the CRC sound velocity prediction and the simplified BISQ model was suitable for the CRC sound attenuation prediction, which had a high prediction accuracy.

1. Introduction

Cobalt-rich crusts (CRCs) are seabed minerals located on the surface of seamounts between 800 and 3000 m and are rich in a variety of metal resources with a high metal grade content [1]. Cobalt plays an important role in industry and it is of great economic value to exploit deep-sea CRCs. Mining changes the seabed environment to a certain extent and affects the growth of seabed organisms. Accurate mining can effectively reduce the mining area and reduce the impact of mining on the environment. The exploration of cobalt-rich crusts is the key to promote the accurate mining of CRCs and the premise to realize CRC commercial exploitation. A manganese crust is usually detected and studied by an acoustic method [2]. Based on the acoustic echo signal, the CRC recognition and substrate classification are realized by machine learning [3,4] and deep learning [5]. Furthermore, the thickness measurement is one of the key techniques for accurate mining. A prototype probe [6], a high-resolution dual-channel acoustic probe [7] and a seabed profiler [8] have been used to measure the thickness of CRCs. In order to improve the thickness measurement accuracy, a combination method of echo signal and image processing technology has been proposed [9]. The sound velocity also determines the accuracy of the thickness measurement. The CRC sound velocity is usually obtained by measuring CRC samples. However, the CRC sound velocity varies within a certain range during the actual thickness measurement. Therefore, it is necessary to study the sound propagation characteristics of CRCs in order to realize the sound velocity prediction of CRCs, which is of great significance for measuring the thickness of CRCs.

A CRC is a type of biphasic pore mineral generated by hydrated minerals, sediments and other substances after long-term chemical reactions [10]. Therefore, the acoustic propagation characteristics of CRCs can be studied by using the theory of a biphasic pore medium. The Biot theory is commonly used to study the acoustic propagation characteristics of biphasic pore media such as sandy sediments [11,12], sandstone reservoirs [13], composite nanomaterials [14] and sandy soil [15]. The Biot–Stoll model has a good applicability to study the high-frequency sound attenuation of sediments [16]. The Biot theory has been used to estimate the sound velocity and sound attenuation of sediments [17]. In fluid–solid pore media, sound propagation not only has a diffusion flow mechanism but also has a jet flow mechanism. Based on the Biot theory and the jet flow mechanism, the BISQ theory was established [18]. As it is difficult to measure the jet flow length in the BISQ theory, the parameters that are difficult to calculate and measure are removed by improving the BISQ theory [19]. Permeability and porosity are independently measured and calculated in the Biot and BISQ theories. Permeability is difficult to measure. The Kozeny–Carman (KC) equation of porous media permeability [20,21] is widely used in the field of permeability. Therefore, the KC equation can be used to simplify the Biot and BISQ models.

However, a CRC is brittle and loose, so it is difficult to obtain standard samples. It is also difficult to measure the sound propagation characteristics of CRCs. A similarity model of CRCs can made by using the similarity theory to study the method of crushing CRCs [22]. Thus, the propagation characteristics of CRCs can be tested by the CRC similarity model. Unfortunately, the crushing similarity model only creates a similar model from the highest mechanical strength; this model is not suitable for acoustic studies. At present, there are few studies on the acoustic propagation characteristics of CRCs, but there are many studies on the acoustic propagation characteristics of sediments and porous rocks. The regression relationship between the sound velocity and porosity was established through sediment measurements and analyses [23]. The acoustic characteristics of sediments have been measured and analyzed and the influence of porosity on the sound velocity has been explored [24]. There is a strong correlation between the physical characteristics of carbonate rocks and the P-wave velocity, which follows a linear function [25]. The influence of rock strength characteristics was predicted by saturation and the P-wave velocity; it was shown that the P-wave velocity decreased with an increase in saturation [26]. Therefore, the sound propagation characteristics of CRCs are mainly related to the physical and mechanical properties.

In order to solve the problem that the sound velocity and attenuation of CRCs are difficult to predict and that it is difficult to obtain standard samples from CRCs, the aim of this paper was to establish a prediction model for the sound velocity and sound attenuation of CRCs. Based on Biot and BISQ theories, a sound propagation model of CRCs was studied. The KC equation was used to establish the relationship between the porosity and permeability of CRCs to simplify the theoretical models of Biot and BISQ. Based on the physical and mechanical properties of CRCs, a similarity model was made. The accuracy of the sound propagation model of CRCs was verified and discussed through a similarity model experiment.

2. Sound Propagation Model in a Biphasic Pore Medium

2.1. Sound Velocity and Sound Attenuation Calculation Formula

The analytical expression of sound velocity and sound attenuation in Biot and BISQ theories [18] is:

$$\{\begin{array}{c}{V}_p=\frac{1}{Re\left(\sqrt{Y}\right)}\hfill \\ \alpha =-\omega Im\left(\sqrt{Y}\right)\hfill \end{array} , Y=Y_{Bi},Y_{BS}$$

(1)

where $V_p$ is the velocity of the P-wave, $\alpha$ is the acoustic attenuation coefficient, ω is the angular frequency of the acoustic wave, $Y_{Bi}$ is the analytical expression of the Biot theory and $Y_{BS}$ is the analytical expression of the BISQ theory.

The analytical expression of $Y_{Bi}$ in the Biot theory is [18,27]:

$$Y_{Bi}=-\frac{B}{2A}-\sqrt{{\left(\frac{B}{2A}\right)}^2-\frac{C}{A}}$$

(2)

where:

$$A=\frac{\varnothing F{M}_{Bi}}{{\rho }_2^2}$$

(3)

$$B=\frac{F\left(2a-\varnothing -\varnothing \frac{{\rho }_1}{{\rho }_2}\right)-\left(M_{Bi}+F\frac{a^2}{\varnothing }\right)\left(1+\frac{{\rho }_a}{{\rho }_2}+i\frac{{\omega }_c}{\omega }\right)}{{\rho }_2}$$

(4)

$$C=\frac{{\rho }_1}{{\rho }_2}+\left(1+\frac{{\rho }_1}{{\rho }_2}\right)\left(\frac{{\rho }_2}{{\rho }_a}+i\frac{{\omega }_c}{\omega }\right)$$

(5)

$$\{\begin{array}{c}{\rho }_1={\rho }_{11}+{\rho }_{12}=\left(1-\varnothing \right){\rho }_s\hfill \\ {\rho }_2={\rho }_{22}+{\rho }_{12}=\varnothing {\rho }_f\hfill \\ {\rho }_a=-{\rho }_{12}\hfill \end{array}$$

(6)

where ${\rho }_s$ is the density of the CRC matrix, ${\rho }_f$ is the density of the fluid phase, $\varnothing$ is the porosity of the CRC, $i$ is an imaginary number unit and ${\rho }_a$ is the coupling density. $a$ is the pore elasticity coefficient, $a=1-K/K_s$. $K_s$ is the bulk modulus of the solid phase, ${\omega }_c$ is the characteristic angular frequency, $\mu$ is the viscosity coefficient of the fluid phase and $\kappa$ is the permeability. $M_{Bi}$ is the uniaxial stress modulus of a dry CRC and $M_{Bi}=K+4G/3$. $K$ and $G$ are the volumetric modulus and shear modulus of a dry CRC, respectively. $F={\left(1/{\rho }_f{c}_f^2+1/\varphi Q\right)}^{-1}$. $c_f$ is the velocity of sound. $Q^{-1}=K_s^{-1}\left(1-\varphi -K/K_s\right)$:

$${\omega }_c=\frac{\mu \varnothing }{\kappa {\rho }_f}$$

(7)

In the BISQ theory, Dvorkin studied the two-dimensional axisymmetric flow of fluid caused by a solid deformation, obtained the fluid pressure based on the solution of the mass balance equation of the fluid and connected the diffusion flow mechanism with the jet flow mechanism [18]:

$$\{\begin{array}{c}{F}_{sq}=F\left[1-\frac{J_1\left(\lambda R\right)}{\lambda R{J}_0\left(\lambda R\right)}\right]\hfill \\ {\lambda }^2=\frac{{\rho }_f{\omega }^2}{F}\left(\frac{\varphi +{\rho }_a/{\rho }_f}{\varphi }+i\frac{{\omega }_c}{\omega }\right)\hfill \end{array}$$

(8)

However, due to the complexity of the measurement and calculation of the jet flow length R, it is difficult to apply in practice. Diallo and Appel proposed an improved BISQ theory that eliminated the jet flow length R and finally obtained an improved BISQ theoretical model [19]:

$$\{\begin{array}{c}{Y}_{BS}=\frac{{\rho }_a\left(1-\varphi \right)+{\rho }_f\varphi +{\left(\omega {\rho }_f\right)}^2\theta }{M_{BS}+F\frac{a_0^2}{\varphi }}\hfill \\ \theta =-\frac{\varphi }{{\omega }^2{\rho }_f}{\left[\frac{{\rho }_a/{\rho }_f+\varphi }{\varphi }+j\frac{\eta \varphi }{\kappa {\rho }_f\omega }\right]}^{-1}\hfill \end{array}$$

(9)

where $a_0=a+2\varphi /3$, $M_{BS}={\left[\varphi /K_f+\left(a-\varphi \right)/K_s\right]}^{-1}$ and $K_f$ is the volume modulus of the fluid.

2.2. Simplification of the Forecasting Model

The permeability of rock depends on the microstructure of the porous medium and is a function of porosity and a specific plane. It is commonly characterized by the classical KC model:

$$\kappa =c\frac{{\varphi }^3}{a_v^2}$$

(10)

where $a_v$ is the specific surface of the solid phase and $c$ is the KC constant. Carman [28] pointed out that the KC constant of uniform spherical particles is 4.8 ± 0.3.

The KC model was further characterized as [29]:

$$\kappa =\frac{{\varphi }^3{\overline{d}}^2}{36c{\left(1-\varphi \right)}^2}$$

(11)

where $\overline{d}$ is the average particle size. By substituting (11) into the Biot theory (8) and the BISQ theory (9), the Biot and BISQ models of simplified permeability could be obtained.

3. Experimental Verification

3.1. Experimental Device and Measurement Method

The sound velocity and attenuation of CRCs were measured by an ultrasonic transmission system. The ultrasonic transmission system was composed of an industrial computer, an arbitrary waveform generator, a power amplifier, a probe and a signal receiving circuit, as shown in Figure 1. The center frequency of the signal transmitting and receiving probe was 90 kHz and the sampling frequency $f_c$ of the system was 6 MHz. Transmission measurements of similar samples of CRCs, clean water and an aluminum block were carried out in a water tank. As shown in Figure 2, the transmitting probe and receiving probe were 0.1 m apart. When measuring the similar samples and the aluminum block, the two probes were close to the sample.

When measuring the sound velocity, $c_g=L_t/t_g$ is often used to calculate the sound velocity of CRCs. However, due to the delay phenomenon of the acoustic wave emitted by the probe, the transit time of acoustic waves in CRCs cannot be accurately obtained. The difference of the first transmitted wave time between the clean water and the CRC was used to calculate the sound velocity of the CRC, which was the difference between the transit time of the sound waves $\Delta t$ in the clean water and the CRC. The problem that the delay of the detection system affected the transit time was solved. Therefore, the formula for calculating the sound velocity of the CRC was:

$$c_g=\frac{c_w{L}_t}{L_t-c_w\Delta t}$$

(12)

where $L_t$ is the thickness of the CRC, $c_w$ is the sound velocity of the water, $\Delta t=\left(t_w-t_g\right)/f_c$, $t_w$ is the time point of the first transmitted wave of the clean water and $t_g$ is the time point of the first transmitted wave of the CRC.

The acoustic attenuation coefficient is usually calculated by the received acoustic wave amplitude. The reference sample method is usually used to solve the acoustic attenuation coefficient in a laboratory. The reference sample is required to have known acoustic attenuation characteristics and the same size as the test samples. The acoustic attenuation coefficient is calculated by comparing the acoustic amplitude of the two samples. An aluminum block is usually considered to be the standard medium without attenuation [30], which was taken as the reference sample. The formula [31] for calculating the acoustic attenuation coefficient $\alpha$ was:

$$\alpha =-\frac{20}{L_t}ln\frac{U_0}{U_1}$$

(13)

where $U_0$ and $U_1$ are the signal amplitudes of the transmitted aluminum block and the CRC, respectively.

3.2. CRC Similarity Model

It can be seen from the Introduction that the sound propagation characteristics of CRCs are determined by their physical and mechanical properties, among which the porosity and permeability are the main factors affecting the sound propagation characteristics. Over the years, scholars have measured the physical and mechanical properties of CRCs [32,33,34,35,36], which are integrated into

Table 1. Results of CRC physical and mechanical properties measured by scholars.

Density (g/cm3)	Porosity (%)	Saturated Water Absorption (%)	Compressive Strength (MPa)	Elasticity Modulus (GPa)	Poisson Ratio	P-Wave Velocity (m/s)
1.6~2.2	25~60	15~30	0.5~15	0.5~8	0.2~0.5	2045~3390

.

The similarity first principle can be applied: for similar phenomena, the similarity index is 1 and the value of the similarity criterion is the same.

The similarity model for the study of sound propagation characteristics mainly requires the similarity of physical properties. Three similarity models for CRCs were made by using the similarity first principle. The similarity model was a cube with dimensions of 0.1 m × 0.1 m × 0.1 m. The measurement results of the physical and mechanical properties of the similar models are shown in

Table 2. Measurement results of physical and mechanical properties of CRC similarity model.

Test Group	Density (g/cm3)	Porosity (%)	Saturated Water Absorption (%)	Compressive Strength (MPa)	Elasticity Modulus (GPa)	Poisson Ratio
1	1.97	25.21	16.35	7.08	7.16	0.25
2	1.83	30.95	18.36	3.74	2.68	0.31
3	1.71	42.51	21.13	2.36	2.15	0.41

. The porosity was calculated from the apparent density ${\rho }_{ad}$ and the true density ${\rho }_{td}$. $\varnothing =\left(1-{\rho }_{ad}/{\rho }_{td}\right)\times 100\%$. ${\rho }_{ad}$ was the dry CRC similarity model density. ${\rho }_{td}$ was measured by a 3H-2000 TD1 automatic true density analyzer.

It could be seen that the physical and mechanical properties of the similarity model were in the range of the CRC and the similarity model was consistent with the physical and mechanical properties of the CRC.

3.3. Data Processing

One set of sample data collected from the similarity models, clean water and aluminum block is shown in Figure 3, Figure 4 and Figure 5, where (a) is the original data graph and (b) is the absolute value graph of the original data. In this article, we took the absolute value through the original data $f\left(n\right)$ to obtain $\left|f\left(n\right)\right|$. The amplitude and position of the first transmitted wave were obtained by maximizing $\left|f\left(n\right)\right|$. The transmitted wave amplitudes and positions of the similarity model data were solved, as shown in Figure 3b. The transmission wave position of the clean water data was solved, as shown in Figure 4b. The transmitted wave amplitude of the aluminum block data was solved, as shown in Figure 5b.

3.4. Experimental Results and Discussion

Three CRC similarity models were placed in clear water for 24 h to force them to become saturated water-bearing similarity models. Two transmission measurements were carried out for each similarity model. The measurement results of the sound velocity and sound attenuation of the CRC similarity models are shown in

Table 3. Measurement results of sound velocity and attenuation of CRC similarity model.

Serial Number	Porosity (%)	Sound Velocity (m/s)	Sound Attenuation (dB)
1.1	25.21	2678.57	−0.867
1.2	25.21	2678.57	−0.925
2.1	30.95	2575.11	−1.787
2.2	30.95	2575.11	−1.859
3.1	42.51	2448.98	−5.568
3.2	42.51	2448.98	−5.588

.

As can be seen from Table 3, the larger the porosity of the CRC, the smaller the sound velocity; the larger the porosity of the CRC, the larger the sound attenuation. The measurement results of the sound velocity were relatively stable; the measurement results of the sound attenuation had a certain fluctuation. The reason was that the transmitted wave amplitude was much larger than the environmental interference, so the maximum position of the transmitted wave amplitude did not shift. The transit time measurement value was stable and the sound velocity measurement value was stable. However, the superposition of the environmental interference and transmitted wave slightly affected the amplitude of the transmitted wave resulting in inconsistent acoustic attenuation measurement results.

According to Equation (1), the prediction results of the sound velocity and sound attenuation of CRCs were obtained. The measurement results of the CRC similarity model are plotted in Figure 6 and Figure 7.

Figure 6 and Figure 7 show that the sound velocity measured by the CRC similarity model and predicted by the simplified Biot model conformed to the range of the sound velocity measured by the CRC in Table 1. The simplified Biot model was more accurate in predicting the sound velocity of the CRC, but the error in predicting the sound attenuation of the cobalt-rich crust was large. The conclusion of the Biot theory in predicting the characteristics of sediment acoustic propagation was consistent [37]. The simplified BISQ model could better evaluate the sound attenuation of the CRC and the predicted sound velocity was much higher than the measured value of the similarity model. Therefore, the simplified Biot model was suitable for predicting the CRC sound velocity and the simplified BISQ model was suitable for predicting the CRC sound attenuation.

The porosity and detection frequency were inputted into the prediction models; the sound velocity prediction results of the simplified Biot model and the sound attenuation prediction results of the simplified BISQ model are shown in

Table 4. The sound velocity prediction results of the simplified Biot model and the sound attenuation prediction results of the simplified BISQ model.

Porosity (%)	Sound Velocity (m/s)	Absolute Error (m/s)	Relative Error (%)	Sound Attenuation (dB)	Absolute Error (dB)	Relative Error (%)
25.21	2673.94	4.63	0.173	−0.888	0.008	0.893
30.95	2574.99	0.12	0.005	−1.830	0.007	0.384
42.51	2447.14	1.84	0.075	−5.721	0.143	2.564

. In Table 4, the absolute and relative errors were obtained by measuring the mean values and the model-predicted values. The SSE, MSE and R-square of the predicted and measured acoustic propagation characteristics could be calculated from Table 3 and Table 4. The SSE, MSE and R-square of the sound speed were 13.1800, 2.1967 and 0.9998, respectively. The SSE, MSE and R-square of the sound attenuation were 0.4160, 0.0693 and 0.9831, respectively. Therefore, the simplified Biot model could accurately predict the sound velocity of the CRC and the simplified BISQ model could accurately predict the sound attenuation of the CRC.

4. Conclusions

In this paper, a method for predicting the sound velocity and attenuation of underwater CRCs was presented. Based on the Biot theory and an improved BISQ theory, a sound velocity and sound attenuation prediction model of a CRC was established. The relationship between the permeability and porosity of the CRC was established by KC model and simplified CRC sound velocity and sound attenuation prediction models were established. It is difficult to obtain standard samples of CRCs. The factors affecting the sound propagation characteristics of CRCs were analyzed in this paper. The measurement results of the physical and mechanical properties of CRCs were summarized. A similarity model of CRCs was made by using the similarity first theory. Transmission experiments of CRC sound propagation characteristics were carried out and the measured results were compared with the predicted results. The results showed that the simplified Biot model had a high prediction accuracy for the sound velocity of CRCs and the R-square of the sound velocity prediction model was 0.9998. The simplified BISQ model was more accurate in predicting the sound attenuation of CRCs and the R-square of the model was 0.9831. This sound velocity and sound attenuation prediction method could promote acoustic detection research such as thickness measurements of CRCs.

The experimental verification was only carried out at a detection frequency of 90 kHz; the prediction effect of the model at other detection frequencies was not discussed. In future work, we will increase the detection frequency types of the underwater transmission system and measure the sound velocity and attenuation of cobalt-rich crusts at various frequencies to expand the application range of the prediction model.