**1. Introduction**

Coral sands are a kind of biogenic soil in marine environments, originating from the fracturing and sedimentation of coral skeletons by wind and waves [1]. The primary features of coral sands include a high content of calcium carbonate—often more than 90%—and a relatively short sediment transport distance, which results in particles frequently retaining the inner pore structure of coral bones, and are characterized by irregular shapes, high angularity, and fragility [2–4]. The sediments formed by coral sands have a high void ratio and display hydrogeological characteristics different from those of terrigenous sediments. Islands in the South China Sea are all composed of coral sediments, with these sediments being the only carriers in the formation of fresh groundwater aquifers in the South China Sea islands and reefs. Given that the dispersion coefficient of coral sands is a key factor affecting the conservation of fresh groundwater, uncovering the dispersion pattern of groundwater solute in coral sands will provide the basic parameters and theoretical basis for the numerical simulation of the formation and evolution of fresh groundwater aquifers in the South China Sea islands and reefs, as well as the conservation and utilization of these aquifers.

Solute dispersion in hydrated porous media has been extensively studied. The earliest work was performed by Taylor, who used a capillary tube model to investigate this topic and proposed a method for calculating the longitudinal dispersion coefficient of porous media [5]. Taylor considered only convection, and therefore derived the molecular diffusion coefficient by measuring the longitudinal dispersion coefficient [5]. Klotzd et al. explored the relationship between the longitudinal dispersion coefficient and average pore flow velocity, the fluid viscosity coefficient, and the characteristic parameters of soil media by conducting a large number of field and laboratory experiments [6]. Gupta proposed a solute transport mechanism for unsaturated porous media, suggesting that two types of pores exist in unsaturated soils, namely, "backbones" and "dead ends," with solute mainly moving by convection in the former and by diffusion in the latter [7]. De Arcangelis et al. studied dispersion calculation methods for network models of porous media, deduced the precise law of tracer motion under the combined action of molecular diffusion and convection, and introduced an effective probability propagation algorithm, which permitted an exact calculation of the distribution of the first-passage-time of the tracer as it flowed through the medium [8]. Sahimi studied hydrodynamic dispersion in two types of heterogeneous porous media, one in which a fraction of the pores did not allow material transport to take place, and the other in which the permeabilities of various regions of the pore space were fractally distributed [9]. Lowe calculated the dispersion coefficient of tracer particles in the fluid of a porous medium randomly filled with spheres, finding that at high Peclet numbers, the tracer motion was mainly determined by convection, and the dispersion process was abnormal with a divergent dispersion coefficient [10]. Zhang proposed a calculation method for hydrodynamic dispersion parameters of adsorptive solutes, deriving formulas for calculating hydrodynamic dispersion coefficients of saturated and unsaturated soils [11]. LI improved the flexible-wall permeameter to make it suitable for finding the dispersion parameters of low-permeability soils, namely, determining the dispersion coefficients through numerical inversion of the breakthrough curve [12]. Shao conducted a one-dimensional dispersion test with silt loam, calculated the hydrodynamic dispersion coefficient of unsaturated silty loam using the soil water and salt dynamics measured in a vertical soil column solute transport experiment, and established the relationship between hydrodynamic dispersion coefficients and pore flow velocities for this type of soil [13]. Jensen adopted the nonlinear least-squares optimization code CXTFIT developed by Parker and van Genuchten to perform curve fitting, thereby obtaining parameters for different forms of the convection–dispersion equation (CDE) [14].

The underground freshwater of islands are the basis for the normal operation of their ecosystems. The generation of underground fresh water is closely related to island size and stratum characteristics. Restricted by the scale of research objects and experimental conditions, the research in this aspect is mostly conducted by means of numerical simulation. The permeability coefficient, dispersion coefficient, and specific yield are three important parameters that must be assigned to strata in numerical simulations. All three parameters can be obtained using field tests. However, the data obtained represent only the island studied, and are not universally representative. In addition, the artificial islands in the South China Sea are not open to the public, making it difficult to conduct field experiments there. In our study, with coral sands as the research subject, which have different granular morphologies and sediment characteristics than terrigenous sediments, laboratory model tests were employed to uncover the dispersion mechanisms of groundwater solute in coral sands, as well as the main factors influencing these mechanisms. According to the results of our study, the empirical value of dispersion coefficients can be provided for calcareous soils with common gradation and compactness. In this way, a more stratigraphic collocation design can be considered in the numerical simulation, so as to find the optimal stratigraphic design scheme that can promote the formation of underground fresh water.

In addition, the study of hydrodynamic dispersion presented in this paper can also be applied to many other aspects. In the simulation and prediction of groundwater pollution, the dispersion coefficient is an important parameter needed for simulation, which provides a quantitative basis for groundwater resource management and groundwater pollution reconstruction. Regarding the intrusion of seawater into coastal aquifers, the study of hydrodynamic dispersion is helpful in studying the migration of the transition zone of brackish water. In terms of water and salt transport in the vade-zone, the study of hydrodynamic dispersion is helpful in solving the problem of the effect of fertilizers and pesticides on underground water quality in islands. Finally, in terms of sewage treatment, hydrodynamic dispersion is helpful in solving the problem of the impact of sewage discharge from living and production on the underground freshwater quality in islands. discharge from living and production on the underground freshwater quality in islands. **2. Test Scheme**  The coral sands used for testing were taken from a natural reef in the South China Sea, whose

treatment, hydrodynamic dispersion is helpful in solving the problem of the impact of sewage

#### **2. Test Scheme** gradation characteristics are shown in Figure 1. The coefficient of curvature Cc was 2.12, which is

The coral sands used for testing were taken from a natural reef in the South China Sea, whose gradation characteristics are shown in Figure 1. The coefficient of curvature C<sup>c</sup> was 2.12, which is within the range of 1–3. The coefficient of uniformity C<sup>u</sup> was 45.45, which is much greater than 5, indicating that the reef consisted of a type of coral sand with good gradation. The coral sand was screened to obtain six kinds of coral sand with a single particle size, as shown in Figure 2. More specifically, the test scheme was composed of three steps: (1) using a self-designed one-dimensional dispersion test device, solute dispersion tests were conducted on a total of 14 groups of coral sands of a single particle size and dry density under various conditions to explore how the dispersion coefficient varies as a function of particle size and dry density; (2) using a custom-designed pore tortuosity test device, where pore tortuosity tests were conducted on a total of five groups of coral sands of a single particle size under various conditions to explore how the pore tortuosity varies as a function of particle size; and (3) using a custom-designed molecular diffusion and mechanical dispersion test device, where for a total of six test groups, molecular diffusion tests were conducted with different concentrations of injected solutions under the condition of the pore fluid having a flow velocity. In addition, molecular diffusion and mechanical dispersion tests were conducted with pore flow velocity increasing in a stepwise manner in order to explore how the weight of molecular diffusion versus mechanical diffusion varies with a stepwise increase in pore flow velocity. The test scheme is shown in Table 1. within the range of 1–3. The coefficient of uniformity Cu was 45.45, which is much greater than 5, indicating that the reef consisted of a type of coral sand with good gradation. The coral sand was screened to obtain six kinds of coral sand with a single particle size, as shown in Figure 2. More specifically, the test scheme was composed of three steps: (1) using a self-designed one-dimensional dispersion test device, solute dispersion tests were conducted on a total of 14 groups of coral sands of a single particle size and dry density under various conditions to explore how the dispersion coefficient varies as a function of particle size and dry density; (2) using a custom-designed pore tortuosity test device, where pore tortuosity tests were conducted on a total of five groups of coral sands of a single particle size under various conditions to explore how the pore tortuosity varies as a function of particle size; and (3) using a custom-designed molecular diffusion and mechanical dispersion test device, where for a total of six test groups, molecular diffusion tests were conducted with different concentrations of injected solutions under the condition of the pore fluid having a flow velocity. In addition, molecular diffusion and mechanical dispersion tests were conducted with pore flow velocity increasing in a stepwise manner in order to explore how the weight of molecular diffusion versus mechanical diffusion varies with a stepwise increase in pore flow velocity. The test scheme is shown in Table 1.

**Figure 1.** Gradation curve of coral sands used in the tests.

**Figure 1.** Gradation curve of coral sands used in the tests.

**Figure 2.** Calcareous sand of various grain sizes.

#### **Figure 2.** Calcareous sand of various grain sizes. **Table 1.** Test scheme.




3 0.25–0.5 mm (100%) 1.36 × 10−4 1.3 60 4 0.25–0.5 mm (100%) 6.61 × 10−4 1.3 60 5 0.25–0.5 mm (100%) 1.60 × 10−3 1.3 60 6 0.25–0.5 mm (100%) 6.16 × 10−3 1.3 60

#### **3. Test Method 3. Test Method**

#### *3.1. One-Dimensional Dispersion Test 3.1. One-Dimensional Dispersion Test*

The hydrodynamic dispersion coefficient is a tensor related to the average pore velocity and the characteristics of the porous medium. Studies show that the dispersion is directional even in an isotropic medium and is more complex when the medium is anisotropic [15]. Given this information, only the one-dimensional dispersion characteristics of a solute in coral sands were explored in this study. The test was conducted with a custom-designed one-dimensional dispersion test device, as shown in Figure 3. The complete test device consisted of four parts (denoted by numbers in the figure): dispersion columns (01), a freshwater supply tank (02), a tracer supply tank (03), and a data acquisition system (04). The dispersion columns were composed of several organic glass columns; each column was 8 cm in inner diameter and 50 cm in height. The preparation and tests of the porous medium samples were performed in the dispersion columns, with a 5-cm-thick buffer layer of glass beads placed separately at the top and bottom of the samples. The sensors used in the data acquisition system were CS655 multi-parameter sensors (Campbell Scientific, State of California, America), which can simultaneously measure volumetric moisture content, temperature, conductivity, dielectric constant, signal propagation time, and signal attenuation. The tracer was a 20 g/L NaCl solution and the samples were single-grained coral sands. The test was conducted by continuously injecting the tracer and collecting the data at a fixed location. The samples in the dispersion columns were first saturated with fresh water, then the valve of the freshwater supply tank was shut, followed by opening the valve of the tracer supply tank and starting data acquisition in a synchronous manner. The relative concentration of NaCl was calculated according to *C* = (*Ccj* − *C*0)/(*Cmax* − *C*0), with *Ccj* denoting the NaCl concentration at time *j*. *C*<sup>0</sup> was the NaCl concentration at the initial time, and *Cmax* was the maximum concentration (final steady concentration) in the test. The hydrodynamic dispersion coefficient is a tensor related to the average pore velocity and the characteristics of the porous medium. Studies show that the dispersion is directional even in an isotropic medium and is more complex when the medium is anisotropic [15]. Given this information, only the one-dimensional dispersion characteristics of a solute in coral sands were explored in this study. The test was conducted with a custom-designed one-dimensional dispersion test device, as shown in Figure 3. The complete test device consisted of four parts (denoted by numbers in the figure): dispersion columns (01), a freshwater supply tank (02), a tracer supply tank (03), and a data acquisition system (04). The dispersion columns were composed of several organic glass columns; each column was 8 cm in inner diameter and 50 cm in height. The preparation and tests of the porous medium samples were performed in the dispersion columns, with a 5-cm-thick buffer layer of glass beads placed separately at the top and bottom of the samples. The sensors used in the data acquisition system were CS655 multi-parameter sensors (Campbell Scientific, State of California, America), which can simultaneously measure volumetric moisture content, temperature, conductivity, dielectric constant, signal propagation time, and signal attenuation. The tracer was a 20 g/L NaCl solution and the samples were single-grained coral sands. The test was conducted by continuously injecting the tracer and collecting the data at a fixed location. The samples in the dispersion columns were first saturated with fresh water, then the valve of the freshwater supply tank was shut, followed by opening the valve of the tracer supply tank and starting data acquisition in a synchronous manner. The relative concentration of NaCl was calculated according to *C* = (*Ccj* − *C0*)/(*Cmax* − *C0*), with *Ccj* denoting the NaCl concentration at time *j*. *C0* was the NaCl concentration at the initial time, and *Cmax* was the maximum concentration (final steady concentration) in the test.

(a) Schematic Diagram (b) Real Product

**Figure 3.** One-dimensional dispersion device (01: one-dimensional soil column, 02: freshwater supply tank, 03: tracer supply tank, 04: data acquisition system). **Figure 3.** One-dimensional dispersion device (01: one-dimensional soil column, 02: freshwater supply tank, 03: tracer supply tank, 04: data acquisition system).

The concentration distribution was derived by solving the one-dimensional steady flow problem in a semi-infinite soil column with a constant concentration at one end. With *t* (s) as the time since the tracer has been turned on, *x* (cm) as the distance of the multi-parameter sensor from the tracer injection port, *E*(*x*, *t*) (dS/m) as the conductivity measured at time *t*, *E0* (dS/m) as the conductivity of the tracer, and *u* (cm/s) as the average flow velocity of the pore fluid in the soil column, the longitudinal dispersion coefficient *DL* (cm2/s) is calculated as given below, and the coordinate system is shown in Figure 4. The concentration distribution was derived by solving the one-dimensional steady flow problem in a semi-infinite soil column with a constant concentration at one end. With *t* (s) as the time since the tracer has been turned on, *x* (cm) as the distance of the multi-parameter sensor from the tracer injection port, *E*(*x*, *t*) (dS/m) as the conductivity measured at time *t*, *E*<sup>0</sup> (dS/m) as the conductivity of the tracer, and *u* (cm/s) as the average flow velocity of the pore fluid in the soil column, the longitudinal dispersion coefficient *D<sup>L</sup>* (cm<sup>2</sup> /s) is calculated as given below, and the coordinate system is shown in Figure 4.

**Figure 4.** Boundary problem with a constant concentration at one end of the one-dimensional soil column.

**Figure 4.** Boundary problem with a constant concentration at one end of the one-dimensional soil column*.*  The above problem is converted into a mathematical equation as follows:

$$\begin{cases} \frac{\partial E}{\partial t} = D\_L \frac{\partial^2 E}{\partial x^2} - u \frac{\partial E}{\partial x} \\ E(x,0) = 0, x \ge 0 \\ E(0,t) = E\_0, t > 0 \\ E(\infty, t) = 0, t > 0 \end{cases} \tag{1}$$

 *E(x,0) = 0 x 0* , ≥ (1) The Laplace transform of Equation (1) with respect to t gives

 

 

$$\begin{cases} D\_L \frac{d^2 \overline{E}}{dx^2} - u \frac{dp \overline{E}}{dx} = p \overline{E} \\ \overline{E}(0, p) = \frac{E\_0}{p}, t > 0 \\ \overline{E}(\infty, p) = 0, t > 0 \end{cases} \tag{2}$$

 <sup>−</sup> <sup>=</sup> 2 2 *pE dx dpE u dx <sup>d</sup> <sup>E</sup> DL* where *E*(*x*, *p*) is a function of *x*, and *p* is a parameter. The original problem is converted into the following definite solution problem as an ordinary differential equation:

$$D\_L \frac{d^2 \overline{E}}{d\mathbf{x}^2} - \mu \frac{d p \overline{E}}{d\mathbf{x}} - p \overline{E} = 0 \tag{3}$$

 (∞, ) = 0,t > 0 *E p* The solution of the second-order homogeneous linear ordinary differential equation is:

$$\frac{E(\mathbf{x},t)}{E\_0} = \frac{1}{2}\sigma f c \left[\frac{\mathbf{x} - ut}{2\sqrt{D\_L t}}\right] + \frac{1}{2}\sigma^{\frac{\mathrm{tr}}{D\_L}}\sigma f c \left[\frac{\mathbf{x} + ut}{2\sqrt{D\_L t}}\right] \tag{4}$$

following definite solution problem as an ordinary differential equation: 2 *dpE <sup>d</sup> <sup>E</sup> DL* (3) where *erfc* is the complementary error function, and ξ in *er f c*(*u*) = 1 − 2 √ π R <sup>u</sup> 0 e −ξ <sup>2</sup>*d*ξ is the mathematical expectation value.

*u*

*dx*

0 <sup>2</sup>

<sup>−</sup> <sup>−</sup> *pE* <sup>=</sup> *dx*

When *x* or *t* is large:

(*E* x,*p*)

where

$$\frac{E(\mathbf{x},t)}{E\_0} = \frac{1}{2}erfc\left[\frac{\mathbf{x}-\mathbf{u}t}{2\sqrt{D\_Lt}}\right] = \frac{1}{\sqrt{\pi}}\int\_{\frac{\mathbf{x}-\mathbf{u}t}{2\sqrt{D\_Lt}}}^{\infty}e^{-\epsilon2}d\xi\tag{5}$$

( ) + + <sup>−</sup> <sup>=</sup> *<sup>D</sup> <sup>t</sup> x ut <sup>e</sup> erfc <sup>D</sup> <sup>t</sup> <sup>x</sup> ut erfc <sup>E</sup> E D ux L L* 2 2 1 2 2 x, t 1 0 (4) Let ξ <sup>2</sup> = η <sup>2</sup>/2 and dξ = 1/ √ 2*d*η (where η is the substitute for the mathematical expectation value ξ and does not have practical meaning), then we have:

$$\frac{E(\mathbf{x},t)}{E\_0} = 1 - \frac{1}{\sqrt{2\pi}} \int\_{-\infty}^{\frac{\mathbf{x}-\mu t}{\sqrt{2D\_L}}} e^{-\frac{\eta^2}{2}} d\eta = 1 - N \left[ \frac{\mathbf{x} - \mu t}{\sqrt{2D\_L t}} \right] \tag{6}$$

*L*

mathematical expectation value. Then, according to Equation (6):

$$0.1587 = 1 - N \left[ \frac{\chi - \mu t\_{0.1587}}{\sqrt{2 D\_L t\_{0.1587}}} \right] \\ 0.8413 = N \left[ \frac{\chi - \mu t\_{0.1587}}{\sqrt{2 D\_L t\_{0.1587}}} \right]$$

According to the normal distribution table, *N*(−1) = 0.1587, *N*(1) = 0.8413, thereby leading to:

$$\frac{\chi - \iota t t\_{0.1587}}{\sqrt{2D\_L t\_{0.1587}}} = 1$$

$$\frac{\chi - \iota t t\_{0.8413}}{\sqrt{2D\_L t\_{0.8413}}} = -1\tag{7}$$

Solving this equation leads to the dispersion coefficient as follows:

$$\left[\frac{\mathbf{x} - \mathbf{u}t\_{0.1587}}{\sqrt{2D\_L t\_{0.1587}}} - \frac{\mathbf{x} - \mathbf{u}t\_{0.8413}}{\sqrt{2D\_L t\_{0.8413}}}\right]^2 = 4\tag{8}$$

Here, *t*0.1587 is the time when (*Ccj* − *C*0)/(*Cmax* − *C*0) = 0.1587 s and *t*0.8413 is the time when (*Ccj* − *C*0)/(*Cmax* − *C*0) = 0.8413 s.
