*5.1. Molecular Di*ff*usion in Coral Sands*

Figure 13 depicts a curve showing the diffusion concentrations of solute molecules in coral sands having different particle sizes. *ENaCl* is the conductivity measured by the left sensor and *Ewater* is measured by the right sensor, with *EMed* = (*ENaCl* + *Ewater*)/2. When there was a concentration gradient in the saturated porous medium and the pore flow velocity was 0, the NaCl solution concentrations gradually decreased and increased in the left and right samples, respectively, until the two concentrations reached a relative equilibrium. It is noteworthy that the concentrations of the NaCl solution on both sides were not necessarily the same when equilibrium was reached; this was due to the fact that when the concentration gradient was low (defined as the limiting concentration gradient at equilibrium, ranging from 0–1, with larger values representing larger concentration gradients at equilibrium), the "obstruction" effect of the porous medium occurred, and the concentration gradient would not continue to decrease. Porous media with different particle sizes (pore sizes) had different limiting concentration gradients at equilibrium. As shown by Figure 13, when the particle size was larger than 0.1 mm, the time spent prior to reaching the limiting concentration gradient at equilibrium gradually decreased with increasing particle size, concomitant with a gradual decrease in the limiting concentration gradient at equilibrium. When the particle size was larger than 1 mm, the limiting concentration gradient at equilibrium was infinitely close to 0. When the particle size was smaller than 0.1 mm, the molecular driving force generated by the concentration gradient was less than the "obstruction" effect of the porous medium; thus, the limiting concentration gradient at equilibrium was rapidly achieved and relatively large (approaching 1). Figure 14 illustrates how the amount of time taken to reach the limiting concentration gradient at equilibrium varied with particle size. As shown in Figure 14, 0.1–0.25 mm was still the characteristic particle size of coral sands for molecular dispersion, and the group of coral sands with this particle size range took the longest time to reach the limiting concentration gradient at equilibrium. This finding is in agreement with the conclusion drawn from the one-dimensional dispersion test conducted in this study. *J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 14 of 21 equilibrium), the "obstruction" effect of the porous medium occurred, and the concentration gradient would not continue to decrease. Porous media with different particle sizes (pore sizes) had different limiting concentration gradients at equilibrium. As shown by Figure 13, when the particle size was larger than 0.1 mm, the time spent prior to reaching the limiting concentration gradient at equilibrium gradually decreased with increasing particle size, concomitant with a gradual decrease in the limiting concentration gradient at equilibrium. When the particle size was larger than 1 mm, the limiting concentration gradient at equilibrium was infinitely close to 0. When the particle size was smaller than 0.1 mm, the molecular driving force generated by the concentration gradient was less than the "obstruction" effect of the porous medium; thus, the limiting concentration gradient at equilibrium was rapidly achieved and relatively large (approaching 1). Figure 14 illustrates how the amount of time taken to reach the limiting concentration gradient at equilibrium varied with particle size. As shown in Figure 14, 0.1–0.25 mm was still the characteristic particle size of coral sands for molecular dispersion, and the group of coral sands with this particle size range took the longest time to reach the limiting concentration gradient at equilibrium. This finding is in agreement with the conclusion drawn from the one-dimensional dispersion test conducted in this study.

**Figure 13.** Concentration variation curves during molecular diffusion in coral sands. **Figure 13.** Concentration variation curves during molecular diffusion in coral sands.

soils [22].

**Figure 14.** Varying amounts of time required to reach the limiting concentration.

**Figure 14.** Varying amounts of time required to reach the limiting concentration**.**  In order to quantitatively characterize the "obstruction" effect of porous media on molecular diffusion, earlier studies introduced the concept of pore tortuosity (*θ*), i.e., the ratio of the length of the porous medium sample to the actual path traveled by the fluid particles through a sample of that length [20]. Thus, *θ* = <sup>∗</sup> ⁄ , where is the dispersion coefficient in the open water body and <sup>∗</sup> is the dispersion coefficient in the porous medium. Given the assumption that the limiting concentration gradient at equilibrium infinitely approaches 0 for molecular diffusion in an open and still water body, the *θ* of a porous medium should range between 0 and 1. According to the definition of pore tortuosity, *θ* = *S*/(*v* × *t*), *S* (cm) is the length of the porous medium sample, *v* (cm/s, taken as scalar without considering direction) is the velocity of fluid particle movement, and *t* (s) is the time duration of fluid particle movement, namely, the time spent by the fluid particles prior to reaching the limiting concentration gradient at equilibrium. Based on the data in Figure 13, the time *t* in each particle size group of coral sands could be obtained, with *t* set to positive infinity in the case of particle sizes less than 0.1 mm. In addition, based on the time *t* and the known distance of 4.9 cm between the sensor and the central partition, it was possible to obtain the relationship curve of pore tortuosity In order to quantitatively characterize the "obstruction" effect of porous media on molecular diffusion, earlier studies introduced the concept of pore tortuosity (θ), i.e., the ratio of the length of the porous medium sample to the actual path traveled by the fluid particles through a sample of that length [20]. Thus, θ = *D*0/*D*<sup>∗</sup> , where *D*<sup>0</sup> is the dispersion coefficient in the open water body and *D*<sup>∗</sup> is the dispersion coefficient in the porous medium. Given the assumption that the limiting concentration gradient at equilibrium infinitely approaches 0 for molecular diffusion in an open and still water body, the θ of a porous medium should range between 0 and 1. According to the definition of pore tortuosity, θ = *S*/(*v* × *t*), *S* (cm) is the length of the porous medium sample, *v* (cm/s, taken as scalar without considering direction) is the velocity of fluid particle movement, and *t* (s) is the time duration of fluid particle movement, namely, the time spent by the fluid particles prior to reaching the limiting concentration gradient at equilibrium. Based on the data in Figure 13, the time *t* in each particle size group of coral sands could be obtained, with *t* set to positive infinity in the case of particle sizes less than 0.1 mm. In addition, based on the time *t* and the known distance of 4.9 cm between the sensor and the central partition, it was possible to obtain the relationship curve of pore tortuosity versus particle size, as shown in Figure 15, where *v* is the velocity of fluid particle movement at a certain temperature. *J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 16 of 21

**Figure 15.** Relationship curve between pore tortuosity and particle size of coral sands.

**Figure 15.** Relationship curve between pore tortuosity and particle size of coral sands.

Dispersion consists of molecular diffusion and mechanical dispersion. When the flow velocity is greater than 0, the two processes usually coexist, albeit with different weights on the overall

Figure 16 presents the variation curves of relative conductivity at a fixed pore flow velocity of 0 for two different concentrations of tracer NaCl solution (20 g/L versus 60 g/L). Here, *EC* is the sensormeasured conductivity, *E0* is the initial conductivity before the injection of the tracer, and *Emax* is the maximum conductivity measured by each sensor. The rate of increase (slope) of relative conductivity was greater for the 60 g/L NaCl solution than for its 20 g/L counterpart, suggesting that the higher

**Figure 16.** Molecular diffusion curves of different concentration gradients.

dispersion coefficient under different conditions (i.e., concentration and flow velocity).

the concentration gradient, the greater the molecular diffusion rate.

*5.2. Mechanical Dispersion in Coral Sands* 

A previous study on the dispersion characteristics of terrigenous sediments found that θ ranges from 0.01–0.5 [21]. In addition, this parameter has been proposed to be 0.1 for clays and 0.7 for sandy soils [22]. **Figure 15.** Relationship curve between pore tortuosity and particle size of coral sands. *5.2. Mechanical Dispersion in Coral Sands* 

*J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 16 of 21

#### *5.2. Mechanical Dispersion in Coral Sands* Dispersion consists of molecular diffusion and mechanical dispersion. When the flow velocity is

Dispersion consists of molecular diffusion and mechanical dispersion. When the flow velocity is greater than 0, the two processes usually coexist, albeit with different weights on the overall dispersion coefficient under different conditions (i.e., concentration and flow velocity). greater than 0, the two processes usually coexist, albeit with different weights on the overall dispersion coefficient under different conditions (i.e., concentration and flow velocity). Figure 16 presents the variation curves of relative conductivity at a fixed pore flow velocity of 0 for two different concentrations of tracer NaCl solution (20 g/L versus 60 g/L). Here, *EC* is the sensor-

Figure 16 presents the variation curves of relative conductivity at a fixed pore flow velocity of 0 for two different concentrations of tracer NaCl solution (20 g/L versus 60 g/L). Here, *E<sup>C</sup>* is the sensor-measured conductivity, *E*<sup>0</sup> is the initial conductivity before the injection of the tracer, and *Emax* is the maximum conductivity measured by each sensor. The rate of increase (slope) of relative conductivity was greater for the 60 g/L NaCl solution than for its 20 g/L counterpart, suggesting that the higher the concentration gradient, the greater the molecular diffusion rate. measured conductivity, *E0* is the initial conductivity before the injection of the tracer, and *Emax* is the maximum conductivity measured by each sensor. The rate of increase (slope) of relative conductivity was greater for the 60 g/L NaCl solution than for its 20 g/L counterpart, suggesting that the higher the concentration gradient, the greater the molecular diffusion rate.

**Figure 16.** Molecular diffusion curves of different concentration gradients.

**Figure 16.** Molecular diffusion curves of different concentration gradients. Figure 17 shows the variation curves of relative conductivity at different flow velocities, with Figure 17A–D depicting the variation curves of measured conductivities from the sensors EC\_1 to EC\_3. Figure 17a–d depict the variation curves of measured conductivities from the sensors EC\_4 to EC\_6. In relation to the tracer injection port, the sensors EC\_1 to EC\_3 were downstream, while the sensors EC\_4 to EC\_6 were upstream. The dispersion process downstream from the port would involve both mechanical dispersion and molecular diffusion, with both taking place in the same direction. In contrast, in the dispersion process occurring against the flow direction upstream from the port, mechanical dispersion would take place in a different direction than molecular diffusion. Therefore, the conductivity change detected by the sensor would be a net conductivity change of the solutes with their molecular diffusion overcoming their mechanical dispersion. Generally, when mechanical dispersion dominates, the shape of concentration–time curve will be as shown in Figure 18a. In this situation, concentration attenuation occurs after the maximum concentration has been reached, with *t<sup>p</sup>* denoting the time taken to reach the peak point of the curve. When molecular diffusion dominates, the shape of the concentration–time curve is as shown in Figure 18b. In this case, the concentration remains constant after reaching the maximum value, with *t<sup>s</sup>* denoting the time taken to reach the stable point of the curve.

Figure 17 shows the variation curves of relative conductivity at different flow velocities, with Figure 17A–D depicting the variation curves of measured conductivities from the sensors EC\_1 to EC\_3. Figure 17a–d depict the variation curves of measured conductivities from the sensors EC\_4 to EC\_6. In relation to the tracer injection port, the sensors EC\_1 to EC\_3 were downstream, while the sensors EC\_4 to EC\_6 were upstream. The dispersion process downstream from the port would involve both mechanical dispersion and molecular diffusion, with both taking place in the same direction. In contrast, in the dispersion process occurring against the flow direction upstream from the port, mechanical dispersion would take place in a different direction than molecular diffusion. Therefore, the conductivity change detected by the sensor would be a net conductivity change of the solutes with their molecular diffusion overcoming their mechanical dispersion. Generally, when mechanical dispersion dominates, the shape of concentration–time curve will be as shown in Figure 18-a. In this situation, concentration attenuation occurs after the maximum concentration has been reached, with *tp* denoting the time taken to reach the peak point of the curve. When molecular diffusion dominates, the shape of the concentration–time curve is as shown in Figure 18-b. In this case, the concentration remains constant after reaching the maximum value, with *ts* denoting the time

**Figure 17.** Dispersion mechanisms in coral sands having different pore flow velocities. (Figure 17A– **Figure 17.** Dispersion mechanisms in coral sands having different pore flow velocities. (Figure 17A–D depicting the variation curves of measured conductivities from the sensors EC\_1 to EC\_3. Figure 17a–d depict the variation curves of measured conductivities from the sensors EC\_4 to EC\_6.) *J. Mar. Sci. Eng.* **2019**, *7*, x FOR PEER REVIEW 18 of 21

D depicting the variation curves of measured conductivities from the sensors EC\_1 to EC\_3. Figure

**Figure 18.** Schematic of the typical curves corresponding to different dispersion mechanisms.

**Figure 18.** Schematic of the typical curves corresponding to different dispersion mechanisms. It can be seen that when the flow velocity was 1.36 × 10−4 cm/s (Figures 17A and 17a), the curve characteristics of the left figure fell between those of Figure 18-a and Figure 18-b, indicating the simultaneous presence of the two mechanisms of molecular diffusion and mechanical dispersion downstream from the port. Data comparison between the left and right figures indicates that the time to reach the maximum conductivity was similar in the two figures, suggesting that the two It can be seen that when the flow velocity was 1.36 <sup>×</sup> <sup>10</sup>−<sup>4</sup> cm/s (Figures 17A and 17a), the curve characteristics of the left figure fell between those of Figures 18a and 18b, indicating the simultaneous presence of the two mechanisms of molecular diffusion and mechanical dispersion downstream from the port. Data comparison between the left and right figures indicates that the time to reach the maximum conductivity was similar in the two figures, suggesting that the two mechanisms were comparable in terms of the dispersion process under this condition. The relative conductivity decreased

gradually similar to that in Figure 18-a, indicating that mechanical dispersion was gradually enhanced. This was manifested by the gradual shortening of the displacement stage and the continuous increase in the diffusion rate. In contrast, the curve shape in the right figure (Fig.17-a,b,c and d) became increasingly atypical, and the time taken to reach the maximum value gradually increased and was much longer than the corresponding time in the left figure, indicating that molecular diffusion was increasingly subject to the flow velocity. By the time the flow velocity reached 6.16 × 10−3 cm/s, the molecular diffusion upstream from the tracer injection port was basically

Based on a comprehensive analysis of the above test results, the velocity boundaries of the pore

fluids that control the dispersion mechanisms were derived, as shown in Table 3.

displacement effect of the pore fluids.

negligible.

mechanisms were comparable in terms of the dispersion process under this condition. The relative conductivity decreased after the peak value in the right figure, which was likely the result of the after the peak value in the right figure, which was likely the result of the displacement effect of the pore fluids.

With the increase in flow velocity, the curve shape in the left figure (Figure 17A–D) becomes gradually similar to that in Figure 18a, indicating that mechanical dispersion was gradually enhanced. This was manifested by the gradual shortening of the displacement stage and the continuous increase in the diffusion rate. In contrast, the curve shape in the right figure (Figure 17a–d) became increasingly atypical, and the time taken to reach the maximum value gradually increased and was much longer than the corresponding time in the left figure, indicating that molecular diffusion was increasingly subject to the flow velocity. By the time the flow velocity reached 6.16 <sup>×</sup> <sup>10</sup>−<sup>3</sup> cm/s, the molecular diffusion upstream from the tracer injection port was basically negligible.

Based on a comprehensive analysis of the above test results, the velocity boundaries of the pore fluids that control the dispersion mechanisms were derived, as shown in Table 3.


**Table 3.** Velocity threshold value of pore fluids that control the dispersion mechanisms.
