A Study of Analytical Solution for the Special Dissolution Rate Model of Rock Salt

Yang, Xin; Liu, Xinrong; Zang, Wanjun; Lin, Zhiyong; Wang, Qiyun

doi:https://doi.org/10.1155/2017/4967913

Advances in Materials Science and Engineering

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Research Article | Open Access

Volume 2017 | Article ID 4967913 | https://doi.org/10.1155/2017/4967913

A Study of Analytical Solution for the Special Dissolution Rate Model of Rock Salt

Xin Yang,^1,2Xinrong Liu,^2,3Wanjun Zang,¹Zhiyong Lin,¹and Qiyun Wang¹

Academic Editor: Ying Li

Received24 Nov 2016

Revised03 Mar 2017

Accepted16 May 2017

Published08 Jun 2017

Abstract

By calculating the concentration distributions of rock salt solutions at the boundary layer, an ordinary differential equation for describing a special dissolution rate model of rock salt under the assumption of an instantaneous diffusion process was established to investigate the dissolution mechanism of rock salt under transient but stable conditions. The ordinary differential equation was then solved mathematically to give an analytical solution and related expressions for the dissolved radius and solution concentration. Thereafter, the analytical solution was fitted with transient dissolution test data of rock salt to provide the dissolution parameters at different flow rates, and the physical meaning of the analytical formula was also discussed. Finally, the influential factors of the analytical formula were investigated. There was approximately a linear relationship between the dissolution parameters and the flow rate. The effects of the dissolution area and initial volume of the solution on the dissolution rate equation of rock salt were computationally investigated. The results showed that the present analytical solution gives a good description of the dissolution mechanism of rock salt under some special conditions, which may provide a primary theoretical basis and an analytical way to investigate the dissolution characteristics of rock salt.

1. Introduction

Rock salt is produced under special natural conditions and contains minerals such as sodium, potassium, and magnesium sulfate, carbonate, and chloride. Its formation is completed under certain physicogeographical conditions depending on the persistent work of saturated multisource salt brine in a long geological time: the development process from salt crystallization to salt precipitation and then to salt accumulation, the evolution process of thousands or millions of years, and subsequent coverage protection. Exploitation of underground rock salt has a history of over 1000 years. Rock salt is often used as a medium for underground storage of petroleum, gas, and nuclear waste. This is because compared with other rocks it has the following five major advantages: () very low permeability ( m²); () a strong creeping property, which enables fissure self-repair and storage pressure self-adaption of rock salt; () a good trapping property (because underground rock salt caverns are located at the water-resisting layer) without contaminating the groundwater; () low costs for cavern leaching by dissolution compared with other rocks, including granite, mudstone, and basalt; and () absorbability of harmful nuclear radiation (a suitable medium for nuclear waste storage) and water solubility that supports convenient and economical construction of rock salt caverns (an ideal choice for deep burying of nuclear waste and the storage of petroleum and gas).

Rock salt caverns are leached by dissolution; specifically fresh water infuses on the ground and salt brine is discharged from the shaft mouth so that caverns of certain shapes and sizes are formed. Liang et al. [1] developed a coupling theory for the effects of dissolution and seepage and the mass transfer and mechanical properties of rock salt. Fairchild et al. [2] completed a field-test study of the dissolution characteristics of NaCl, and they found a positive correlation between the dissolution rate and the surface area of exposure. Weisbrod et al. [3] studied the effects of the composition, flow rate, and gravity of rock salt on its dissolution process. With a numerical method and based on a field survey, Zidane et al. [4] analyzed and calculated land subsidence resulting from rock salt dissolution. Yang et al. [5] investigated the risks of gas storage in bedded rock salt caverns. In an experiment, Alkattan et al. [6] investigated the dissolution characteristics of rock salt. Using a numerical method, Chen and Liu [7] discussed the evolution of the porosity and concentration of porous media during transmission. Rezaei et al. [8] attempted to establish a fluid transport model for calcite in a mixed area of tap water and salt water. Finneran and Morse [9] focused on the dissolution kinetics of calcite in salt water.

It should be pointed out that, in recent years, extensive studies have been conducted to investigate the chemical dissolution of porous rocks [10–15], for which the dissolution of rock salt only belongs to a special case. The theoretical and numerical results of these studies have clearly demonstrated that the dissolution of porous rocks is controlled by the coupling of the following processes: solute diffusion, pore-fluid flow, and specific chemical dissolution type. Although the chemical dissolution of porous rock is almost transient (i.e., time-dependent), the dissolution surface, which is commonly represented by the so-called dissolution front, can be in either a stable state or an unstable state, depending on both the Zhao number and the critical Zhao number of the dissolution system [10–21]. The existing studies have also demonstrated that many factors, such as mineral dissolution ratios [16], mechanical dispersion [17], medium/fluid compressibility [12, 18], reactive surfaces of particles [19], temperature effect [20, 21], and the permeability-porosity relationships [22], can affect the morphological evolution of reaction fronts within porous rocks in the field of the emerging computational geosciences [23]. As a result, the first monograph on this topic was published in the world [24], indicating that a complete theoretical framework has been successfully established on the chemical dissolution-front instability in porous media. In addition, such studies have been extended to investigate physical dissolution of porous media associated with groundwater pollution and contaminant site clean-up problems [25–28].

Despite the above-mentioned achievements in this particular research field, there is a lack of systematic investigation on an analytical solution for the dissolution rate equation of rock salt, so that more in-depth study is required. In the present study, we established an ordinary differential equation for the special dissolution rate model of rock salt under the assumption of instantaneous diffusion process. We then solved the equation mathematically to give an analytical solution and fitted the analytical solution using the results of a transient dissolution test to produce the dissolution parameters of the rock salt dissolution model when the dissolution surface is in a stable state. Finally, we calculated and analyzed the influential factors of the equation. This study provides a theoretical basis for understanding the dissolution mechanism of rock salt dissolution model under some special conditions.

2. Dissolution Mechanism of Rock Salt

2.1. Diffusion in Solution

The dissolution of rock salt is a process where rock salt molecules diffuse in water. Diffusion is caused by a gradient of the chemical potential, where molecules diffuse from the high-concentration zone to the low-concentration zone by their own thermal motion and eventually reach an approximate balance. Diffusion occurs even in the absence of macroscopic flow of the fluid. Diffusion can be described by Fick’s first law: the diffusion rate is directly proportional to the concentration gradient. The diffusion flux is defined as the flux passing through the unit-area reference surface per unit time, and it can be expressed as a first-order partial differential equation:where is the diffusion flux (g·cm⁻²·min⁻¹), namely, the mass flow of a substance passing through the unit area vertical to the normal direction, is the substance concentration (g·cm⁻³), namely, the amount of substance per unit volume, and is the diffusion coefficient (cm²·min⁻¹). The diffusion coefficient of solutes is generally related to the solvent and temperature, but, in certain cases, it can also be the function of rock porosity [22–25].

2.2. Dissolution Boundary Layer

During fluid transport for cavern leaching by dissolution, there is a dissolution boundary layer on the internal wall surface of the cavern, where the mass exchange between the solvent and rock salt (the dissolution process of rock salt) is completed.

Based on the study of Durie and Jessen [29], the concentration distribution profile at the salt dissolution boundary layer has a parabolic shape. The concentration distribution at the dissolution boundary layer can be expressed by the following equation:where is the concentration of the solution at the wall surface of rock salt, is the average concentration of the solution outside the boundary layer, is the concentration of the solution in the boundary layer, δ is the thickness of the boundary layer, and is the distance from the wall surface of rock salt (cm).

The concentration distribution profile of the rock salt solution in the dissolution boundary layer is shown in Figure 1.

2.3. Dissolution Rate Equation of Rock Salt

Some researchers [30, 31] proposed that the dissolution rate equation of rock salt can be established according to the mass balance principle:where R is the dissolved radius (cm), t is the dissolution time (min), D is the diffusion coefficient (cm²·min⁻¹), is the density of rock salt (g·cm⁻³), is the concentration of the rock salt solution (g·cm⁻³), is the concentration gradient of the rock salt surface, and is the dissolution surface.

If it is assumed that the diffusion rate is very quick, the following equation can be derived from (2):

It is generally believed that the concentration of the wall is the saturation concentration in the solution mining process of rock salt, so that we can get

is a time-dependent parameter, so that, under the assumption of , the following equation can be obtained:

Consequently, we can getwhere is the saturated concentration of the rock salt solution (g·cm⁻³) and is the concentration of the rock salt solution at time (g·cm⁻³).

Substituting (9) into (3) gives the following equation:

This equation is an ordinary differential equation for describing the dissolution rate of rock salt.

The dissolution characteristics of rock salt in Changshan area [32, 33] have been systematically investigated with the instrument shown in Figure 2. The experimental procedure was as follows. () During dissolution, the salt dissolution surface was parallel to the water level in the dissolution vessel, and the other dissolution surfaces were sealed with paraffin. () The flow rate was controlled by the knob of the resin column (the flow rate was adjusted in place before the experiment). () Slow stirring was frequently performed during the experiment to eliminate the effect of the concentration gradient, a small water sample was removed from the dissolution vessel at a certain interval, and the concentration of the salt solution in the test sample was then measured.

Because there was a very low flow rate in the brine overflow tube, in the above experiment can be calculated using the following equation:where is the dissolved mass at time (g), is the dissolution area (cm²), is the initial volume of the solution (cm³), is the flow rate (cm³·min⁻¹), and the other symbols are as defined above.

Substituting (11) into (10) and under the initial condition at gives the ordinary differential equation:

The equation for a first-order linear ordinary differential equation () is where constant is the constant term in the analytic solution, which depends on the conditions.

With (13) and the known conditions, (12) can be solved to give an analytical solution, and then and at time can be calculated by

3. Validation of the Analytical Formula for Transient Dissolution

The concentration of a rock salt solution experiment [32, 33] has been measured at different time points with the different flow rates (Q), as shown in Figure 3 and Table 1. Figure 3 shows that the larger value, the higher concentration of the rock salt solution at the same time point, suggesting that dissolution of rock salt accelerates as increases.

The following test parameters have been reported [32, 33]: g·dm⁻³, g·cm⁻³, cm², and cm³. The fitting of the test data was performed using (9). At , 5, and 9 cm³ min⁻¹, , 0.03570, and 0.05382 cm min⁻¹, respectively, which shows that the dissolution parameters of rock salt are augmented and dissolution of rock salt accelerates as the flow rate () increases. The corresponding correlation coefficients are 0.9947, 0.9997, and 0.9923. The fitting results are shown in Figures 4–6.

4. Physical Meaning and Influential Factors of the Analytical Formula

4.1. Physical Meaning of the Analytical Formula

The equation for can be derived from (14):where represents the concentration of the rock salt solution at dissolution time and represents the dissolution rate of rock salt (the slope of the concentration curve). When the initial volume of solution is constant, the higher the flow rate, the higher the dissolution rate of rock salt.

After the test parameters were substituted into (14), the concentrations of the rock salt solutions at dissolution time were (190.667 g·dm⁻³), (155.944 g·dm⁻³), and (142.885 g·dm⁻³) at , 5, and 9 cm³ min⁻¹, respectively. The corresponding dissolution rates of rock salt are 0.004, 0.01, and 0.018 min⁻¹.

4.2. Influential Factors of the Analytical Formula

4.2.1. Effect of Parameter

In Section 3, values at , 5, and 9 cm min⁻¹ were 0.02327, 0.03570, and 0.05382 cm min⁻¹, respectively, as shown in Figure 7. It was found that gradually increases with increasing of Q, and an approximately linear correlation is obtained by fitting with the equation (correlation coefficient 0.9996).

4.2.2. Effect of the Dissolution Area

Assuming that the other parameters were constant, the effect of changing the dissolution area (A) was investigated with the following parameters: cm³ min⁻¹, cm min⁻¹, g·dm⁻³, g·cm⁻³, cm³, and , 30, 50, 100, 200, and 300 cm². The calculated results are shown in Figures 8 and 9. These figures suggest that when A increases, R gradually decreases while increases. This is because when the initial volume of solution (V) is constant, R and are directly proportional to the difference between and (). As A increases, increases, resulting in a progressive decrease of R (5). When A is very low, R and show an approximately linear correlation with the dissolution time. If A is gradually increased to 100 cm², there is a nonlinear relationship between R or and the dissolution time.

4.2.3. Effect of the Initial Volume

Assuming that the other parameters were constant, the effect of the initial volume (V) of the rock salt solution was investigated with the following parameters: cm³ min⁻¹, cm min⁻¹, g·dm⁻³, g·cm⁻³, cm², and , 300, 500, 1000, and 2000 cm³. The calculated results are shown in Figures 10 and 11. These figures reveal that with augmentation of V, R gradually increases while decreases. This is because when the dissolution area is constant, the larger the volume, the lower the concentration of rock salt. According to (5), is directly proportional to . decreases with increasing , leading to a gradual increase of . Furthermore, when is very low, and have a nonlinear correlation with the dissolution time. When is progressively increased to 1000 cm³, there is an approximately linear relationship between or and the dissolution time.

5. Conclusion

An analytical solution for the dissolution rate equation of rock salt under the assumption of an instantaneous diffusion process and its influential factors were investigated. The following conclusions can be drawn from this study.

() An ordinary differential equation for the dissolution rate of rock salt was established by calculating the concentration distribution of rock salt at the boundary layer. It was then solved mathematically to give an analytical solution. Thereafter, the analytic solution was used to fit transient dissolution test data of rock salt. The fitting results show that, at flow rates of 2, 5, and 9 cm³ min⁻¹, the transient dissolution parameters are 0.02327, 0.03570, and 0.05382 cm min⁻¹, respectively.

() The physical meaning of the analytical formula for the dissolution rate of rock salt was discussed. The analytical formula partly represents the concentration of that rock salt solution at the dissolution time and partly represents the dissolution rate of rock salt. This study demonstrates that when the initial volume of the rock salt solution is constant, the larger the flow rate, the higher the dissolution rate of rock salt.

() The influential factors of the analytical formula were investigated. The results showed that there is an approximately linear relationship between the dissolution parameter () and the flow rate (Q): (correlation coefficient 0.9996). The changes of the dissolved radius and concentration of rock salt with dissolution time for different dissolution areas and different initial volumes were also calculated using the analytical formula.

() The results of this study can provide the basic parameters for the leaching of rock salt caverns and also a theoretical reference for the transient dissolution of rock salt.

It must be pointed out that chemical dissolution of porous rocks including rock salt is a very complicated scientific problem [10–24]. The dissolution surface that is commonly represented by the dissolution front can exist in either a stable state or an unstable state, depending on the coupled behavior among the solute diffusion, pore-fluid flow, and chemical reaction type in the whole dissolution system [34]. This indicates that this study still has certain limitations, which should be improved in the future research.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study is supported by the Scientific Research Foundation of Fujian University of Technology (Grant no. GY-Z160051), the Fujian Provincial Natural Science Foundation Projects (Grants nos. 2013J01170, 2015J05090, 2016J01205, and 2017J05070), and the National Natural Science Foundation of China (Grant no. 51508097); the authors gratefully acknowledge these supports.

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Copyright © 2017 Xin Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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