Introduction

Energy in the form of oil or natural gas is an essential factor in a country’s economy and affects to both national welfare and people’s livelihood. There are three main methods of energy storage: land storage tanks, maritime storage tanks, and underground storage. Of the three methods, underground storage is the safest way of energy storage, known as “the reserve warehouse of highly strategic security.” Compared with other rocks, rock salt has many advantages as a medium for energy underground storage in the form of rock salt caverns. At present, research on rock salt is focused mainly on aspects of stability analysis and the mechanical properties of rock salt. Tsang et al. (2005) used laboratory tests and theoretical analysis to investigate cavern excavation damage zone of rock salt. Wawersik (1981) and Cosenza et al. (1999) measured the permeability of rock salt in their study of long-term stability of gas storage. Stormont (1997) measured gas permeability of rock salt in laboratory tests and described the damage that occurred in the rock salt. Hampel and Schulze (2007) analyzed the basic mechanical parameters of a constitutive model of rock salt. Yang et al. (2013) examined the risks of gas storage in a laminar rock salt cave, and Liang et al. (2008) established the coupling theory of the dissolution and seepage effects on mass transmission and the mechanical properties of rock salt.

Many scholars have studied the dissolution characteristics of rock salt in different conditions (Liu et al. 2002; Jiang et al. 2009; Ma et al. 2010; Tang et al. 2012; Zhou et al. 2006; Jessen 1972; Liu et al. 2015), mainly through experimental investigations. In this paper, based on dissolution test data of rock salt, a dynamic dissolution model of rock salt subjected to gravity is established which is solved using the finite-difference method, which can provide basic parameters constructing rock salt cavities.

Experimental research on the dynamic dissolution properties of rock salt under gravitational effect

Experimental process

The specimens for the experiments were obtained from natural rock salt from the Himalayan Mountains in Pakistan. The bulk rock salt was processed into test specimens 50 × 100 mm. The specimens were comprised of 99.8 % soluble substance with an average density of 2059 kg/m3 (Table 1).

Table 1 Components of rock salt specimens (%)
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A water-passage pinhole of 0.6 cm diameter was drilled at the center of the sample’s axial position and processing conducted strictly in line with the experimental regulations. Figure 1 depicts the rock salt samples prepared for hole-boring.

Fig. 1
figure 1

Rock salt samples

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The apparatus for testing the dynamic dissolution was designed specifically for this experiment (Fig. 2). The testing equipment included the following: iron pedestal, beaker chain clip, 703 glue, two large beakers, liquid flowmeter (range 6–60 L/h), tissue, air blower, high precision electronic scales (precision 0.01 g), plastic pipe (outer diameter of 0.6 cm), and a vernier caliper.

Fig. 2
figure 2

Apparatus for testing dynamic dissolution of rock salt under the effect of gravity

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The experimental procedure is as follows:

  1. (1)

    Each drilled specimen is weight using electronic scales, and its length and diameter are measured using the vernier caliper.

  2. (2)

    Each specimen is coated with a waterproof sealant material (703 silicone glue) and then weighed by electronic scales.

  3. (3)

    The tap water flow from the faucet is firstly adjusted to 10 L/h.

  4. (4)

    The sealed specimen is firmly attached to the beaker chain clip, and the water pipe is inserted into the specimen for the rock salt dynamic dissolution test.

  5. (5)

    After 3 min, the water pipe is pulled out of the specimen and the pipe directed into another large beaker to ensure flow stability. The specimen is removed from the testing device, wiped dry, blow-dried with an air blower, and weighed with the electronic scales.

  6. (6)

    Step (4) is repeated until dissolved to the boundary of specimen.

  7. (7)

    Steps (4), (5), and (6) are repeated for water flows of 20 L/h, 30 L/h, and 40 L/h.

Test results

Table 2 Rock salt test specimens—pre-test parameters

Sample no.

Diameter (mm)

Length (mm)

Mass of specimen (g)

Mass of sealed and waterproofed specimen (g)

Density (g · cm−3)

B1

52.86

99.7

458.52

465.79

2.1241

B2

51.38

103.42

457.15

463.35

2.1625

B3

52.66

102.56

433.64

439.59

1.9679

B4

52.1

98.82

437.69

444.72

2.1435

Table 3 Test results of dynamic dissolution of rock salt subjected to gravity—specimen weight in (g)

Flow rate

10 L/h

20 L/h

30 L/h

40 L/h

Time (min)

B1

B2

B3

B4

0

465.79

463.35

439.59

444.72

3

461.18

456.25

428.93

427.73

6

455.42

444.67

413.06

408.95

9

448.13

429.95

393.98

383.35

12

446.57

411.97

376.52

360.15

15

436.22

392.48

353.38

330.96

18

426.5

371.08

326.76

305.35

21

414.58

353.1

306.63

277.67

24

402.57

338.18

288.78

 

27

389.32

319.47

  

30

374.94

293.45

  

33

359.49

266.23

  

36

 

238.08

  

Fig. 3 Specimen mass after dissolution under gravitational effect at different flow rates

figure a

Fig. 4 Specimen dissolved mass measured every 3 min under gravitational effect for different flow rates.

figure b

Fig. 5 Test specimens after dissolution test

figure c

Analysis of test results

The curves of the specimen mass, the dissolved mass with time, and test specimens after dissolution are shown in Figs. 3, 4, and 5, respectively. From Fig. 4, we can see the curve of the dynamic dissolution weight of the rock salt specimen is not smooth, but fluctuates with time. Moreover, as the flow rate Q increased, the dissolution rate of the specimen also increased and the dissolution weight change became more obvious, indicating that the movement of the solution accelerated its convection and diffusion.

Our results were consistent with those of Liu et al. (2015) for the three flow rates of 20, 30, and 40 L/h, which conducted in a non-gravity condition. The dissolution weight changes under the effect of gravity of this paper were larger than those of Liu et al. (2015) without considering the gravitational effect. Without gravity, the average dissolution rates at water flows of 20, 30, and 40 L/h were 2.8388, 3.7295, and 5.4354 g/min, while under gravitational effect, the average dissolution rates were 5.9733, 6.2838, and 7.9548 g/min, respectively. The average dissolution rates under gravity effect were 2.1041, 1.6848, and 1.4635 times of those without gravity for flow rates of 20, 30, and 40 L/h, respectively, indicating that the gravity effect increased the water flow and accelerated the dissolution of the rock salt. Because the dissolution mechanism of rock salt under gravity effect is the sum of two parts: dissolution without gravitational effect and dissolution generated by the increasing flow velocity under the effect of gravity, as shown in Fig. 6.

Fig. 6
figure 3

Illustration of dissolution mechanism under gravitational effect

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Dynamic dissolution model of rock salt subjected to gravity

As shown in the previous section, the dissolution rates under the effect of gravity were greater than without considering the gravitational effect. Increased dissolution rates would raise the concentration level of the water-salt solution. Hence, the solution concentration increased, leading to a faster dissolution.

To establish the dynamic dissolution model of rock salt under gravity, we considered two hypotheses.

  1. (1)

    The first hypothesis suggests that the effect of gravity increased the flow rate Q.

Liu et al. (2015) present the partial differential equations (Eq. (1)) of a dynamic dissolution model which conducted in a non-gravity condition describing the changes in the dissolved radius R and solution concentration C:

$$ \left\{\begin{array}{l}\frac{\partial R}{\partial t}=\frac{2}{\delta }D\frac{M}{\rho_s}\left({C}_s-{C}_{xt}\right)\\ {}\frac{\partial C}{\partial x}=\frac{2D}{\delta Q}\pi \left(d+2{R}_{xt}\right)\left({C}_s-{C}_{xt}\right)\\ {}{\left.{R}_{xt}\right|}_{t=0}=0\\ {}{\left.{C}_{xt}\right|}_{x=0}={C}_0\end{array}\right. $$
(1)

where, C 0 refers to the initial concentration, C s represents saturation concentration of the rock salt solution (mol · L−1), C xt refers to the concentration of solution at the section position of x with the time period of t, (mol · L−1), D represents the diffusive coefficient (cm−2 · s−1), δ refers to the boundary layer’s thickness (cm), ρ s refers to the density of rock salt (cm3 · s−1), M represents the molar mass of rock salt (g · mol−1), R is the dissolved radius of the rock salt specimen (cm), and d represents the diameter of the pinhole (cm).

Equation (1) indicates that if Q increases under gravitational effect, C would decrease. This is contradictory to our test results. According to Eq. (1), which showed a gradual rise in C along the direction of the flow movement, the dissolved radius R decreases gradually along the direction of the flow movement; this too does not reflect our test results.

Therefore, based on this hypothesis, the dissolution equation is inconsistent with the dynamic dissolution test results of rock salt under gravity, leading us to explore the second hypothesis.

  1. (2)

    The second hypothesis proposes that the effect of gravity raises the value of parameter D/δ.

It assumes that the parameter D/δ in Eq. (1) was not an invariable constant that increases as the flow rate Q becomes higher. As Q increases under the effect of gravity, C increases as D/δ increases, the dissolved radius R also increases along the direction of the flow movement due to the rise in the value of D/δ. Therefore, this hypothesis could well explain the test results of dynamic dissolution under gravity.

In Liu et al. (2015), the parameter D/δ was fitted for the various flow rates to obtain the values shown in Table 4. The relationship between the flow quantity Q and the parameter D/δ was acquired by linear fitting, where D/δ = 0.00005039Q + 0.0005421, with a correlation coefficient 0.9999. The research result showed that the flow quantity Q was linear with the parameter D/δ.

Table 4 Parameter D/δ for various flow rates (Liu et al. 2015)
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The linear relationship between Q and D/δ indicates that the parameter D/δ and the flow velocity follow a similar trend. In the dynamic dissolution process of rock salt, we introduce a transformation coefficient α which conformed to the following relation:

$$ \frac{\mathrm{D}}{\delta }=\upalpha {\mathrm{V}}_{\mathrm{g}} $$
(2)

where, V g is the flow velocity (cm · s−1) under the effect of gravity.

The flow velocity under gravity V g can be simplified as an acceleration process with an initial velocity Q/A and an acceleration of 980 cm · s−2 over a distance H (H is the position head). Thus, we obtain the following:

$$ {\mathrm{V}}_{\mathrm{g}}=\sqrt{{\left(\frac{\mathrm{Q}}{\mathrm{A}}\right)}^2+1960\mathrm{H}} $$
(3)

Substituting Eqs. (2) and (3) into Eq. (1), the dissolution model under gravitational effect can be acquired as follows:

$$ \left\{\begin{array}{l}\frac{\partial \mathrm{R}}{\partial \mathrm{t}}=\frac{\mathrm{M}}{\uprho_{\mathrm{s}}}\left[2\alpha \left(\sqrt{{\left(\frac{\mathrm{Q}}{\mathrm{A}}\right)}^2+1960\mathrm{H}}\right)\left({\mathrm{C}}_{\mathrm{s}}-{\mathrm{C}}_{\mathrm{x}\mathrm{t}}\right)\right]\\ {}\frac{\partial \mathrm{C}}{\partial \mathrm{x}}=\frac{\pi }{\mathrm{Q}}\left(\mathrm{d}+2{\mathrm{R}}_{\mathrm{x}\mathrm{t}}\right)\left[2\alpha \left(\sqrt{{\left(\frac{\mathrm{Q}}{\mathrm{A}}\right)}^2+1960\mathrm{H}}\right)\left({\mathrm{C}}_{\mathrm{s}}-{\mathrm{C}}_{\mathrm{x}\mathrm{t}}\right)\right]\\ {}{\left.{\mathrm{R}}_{\mathrm{x}\mathrm{t}}\right|}_{\mathrm{t}=0}=0\\ {}{\left.{\mathrm{C}}_{\mathrm{x}\mathrm{t}}\right|}_{\mathrm{x}=0}={\mathrm{C}}_0\end{array}\right. $$
(4)

Numerical solutions and analysis of the rock salt dynamic dissolution model under gravity

Numerical solution

Because of the interaction between the velocity field and concentration field, Eq. (4) is nonlinear and an analytical solution is difficult to obtain; therefore, to acquire a suitable difference scheme, the partial differential equations (Eq. (4)) were discretized and solved by a finite-difference method.

The finite-difference model grid is shown in Fig. 7, where the interval in x direction is h and the interval in t direction is τ. And

Fig. 7
figure 4

Finite-difference grid for the dissolution model

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$$ \left\{\begin{array}{l}R\left(x,t\right)=R\left(nh,i\tau \right)={R}_i^n\\ {}C\left(x,t\right)=C\left(nh,i\tau \right)={C}_i^n\\ {}\frac{\partial R\left(x,t\right)}{\partial t}=\frac{\partial R}{\partial t}\\ {}\frac{\partial C\left(x,t\right)}{\partial x}=\frac{\partial C}{\partial x}\end{array}\right. $$
(5)

We took the center difference to the area parameter A, and the difference scheme was established based on Eq. (4) as follows:

$$ \left\{\begin{array}{l}\frac{\mathrm{R}\left(\mathrm{n}\mathrm{h},\left(\mathrm{i}+1\right)\uptau \right)-\mathrm{R}\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)}{\uptau}=\frac{\mathrm{M}}{\uprho_{\mathrm{s}}}\left[2\alpha \Big({\mathrm{C}}_{\mathrm{s}}-\right.\\ {}\left.\left.\mathrm{C}\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)\right)\left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\mathrm{R}}^2\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)+\pi {\mathrm{R}}^2\left(0,\mathrm{i}\uptau \right)}\right)}^2+1960\mathrm{H}}\right)\right]\\ {}\frac{\mathrm{C}\left(\left(\mathrm{n}+1\right)\mathrm{h},\mathrm{i}\uptau \right)-\mathrm{C}\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)}{\mathrm{h}}=\frac{\pi }{\mathrm{Q}}\left(\mathrm{d}+2\mathrm{R}\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)\right)\left[2\alpha \right.\left({\mathrm{C}}_{\mathrm{s}}-\right.\\ {}\left.\left.C\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)\right)\left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\mathrm{R}}^2\left(\mathrm{n}\mathrm{h},\mathrm{i}\uptau \right)+\pi {\mathrm{R}}^2\left(0,\mathrm{i}\uptau \right)}\right)}^2+1960\mathrm{H}}\right)\right]\end{array}\right. $$
(6)

Setting \( \frac{2M}{\rho_s}{C}_s\tau =a,\frac{2M}{\rho_s}\tau =b,\frac{2}{Q}{C}_s\pi h=A,\frac{2}{Q}\pi h=B, \) we obtain the following:

$$ \left\{\begin{array}{l}{\mathrm{R}}_{\mathrm{i}+1}^{\mathrm{n}}=\alpha \left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\left({\mathrm{R}}_{\mathrm{i}}^{\mathrm{n}}\right)}^2+\pi {\left({\mathrm{R}}_{\mathrm{i}}^0\right)}^2}\right)}^2+1960\mathrm{H}}\right)\left(\mathrm{a}-\mathrm{b}{\mathrm{C}}_{\mathrm{i}}^{\mathrm{n}}\right)+{\mathrm{R}}_{\mathrm{i}}^{\mathrm{n}}\\ {}{\mathrm{C}}_{\mathrm{i}}^{\mathrm{n}+1}=\alpha \left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\left({\mathrm{R}}_{\mathrm{i}}^{\mathrm{n}}\right)}^2+\pi {\left({\mathrm{R}}_{\mathrm{i}}^0\right)}^2}\right)}^2+1960\mathrm{H}}\right)\left(\mathrm{A}-\mathrm{B}{\mathrm{C}}_{\mathrm{i}}^{\mathrm{n}}\right)\left(\mathrm{d}+2{\mathrm{R}}_{\mathrm{i}}^{\mathrm{n}}\right)+{\mathrm{C}}_{\mathrm{i}}^{\mathrm{n}}\end{array}\right. $$
(7)

Which leads to the following:

$$ \left\{\begin{array}{l}{\mathrm{R}}_{\mathrm{i}}^0=\alpha \left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\left({\mathrm{R}}_{\mathrm{i}-1}^0\right)}^2+\pi {\left({\mathrm{R}}_{\mathrm{i}-1}^0\right)}^2}\right)}^2+1960\mathrm{H}}\right)\left(\mathrm{a}-\mathrm{b}{\mathrm{C}}_{\mathrm{i}-1}^0\right)+{\mathrm{R}}_{\mathrm{i}-1}^0\\ {}{\mathrm{C}}_0^{\mathrm{n}}=\alpha \left(\sqrt{{\left(\frac{2\mathrm{Q}}{\pi {\left({\mathrm{R}}_0^{\mathrm{n}-1}\right)}^2+\pi {\left({\mathrm{R}}_0^0\right)}^2}\right)}^2+1960\mathrm{H}}\right)\left(\mathrm{A}-\mathrm{B}{\mathrm{C}}_0^{\mathrm{n}-1}\right)\left(\mathrm{d}+2{\mathrm{R}}_0^{\mathrm{n}-1}\right)+{\mathrm{C}}_0^{\mathrm{n}-1}\end{array}\right. $$
(8)

Using the initial conditions R(x,0) = 0 and C(0,t) = C 0, and the recursion formula (Eq. (8)), R n0 , C 0 i can be obtained; thus, all the initial conditions are obtained.

With the known initial condition R n0 and R n i obtained from explicit difference formula, \( \left\{\begin{array}{l}{C}_i^n\\ {}{R}_i^n\end{array}\right. \) can be derived by the chasing method and \( \left\{\begin{array}{l}{C}_i^{n-1}\\ {}{R}_i^{n-1}\end{array}\right. \), and the numerical solution to Eq. (4) is archived.

The test parameters were as follows: M = 58.5 g/mol, C 0 = 0 mol/cm3, C s  = 0.0054 mol/cm3, d = 0.6 cm, ρ s , Q, and D/δ were empirically derived and are shown in Tables 2, 3, and 4. Thus, the parameters a, b, A, and B could be calculated to acquire.

Parameters inversion

The formula of the dissolved radius and weight of the rock salt weight was established in Ren et al. (2011):

$$ \mathrm{R}\left(\mathrm{k}\right)=\sqrt{\frac{\mathrm{m}\left(\mathrm{k}\right)}{\uprho_{\mathrm{s}}\mathrm{l}\pi }+\frac{{\mathrm{d}}^2}{4}}-\frac{\mathrm{d}}{2} $$
(9)

where R (k) is the dissolved radius of rock salt for the kth test record, l is the height of the specimen, and m (k) is the dissolved mass for test record k.

The fitness function can be established as follows:

$$ \mathrm{f}=\frac{1}{\mathrm{N}}{\displaystyle \sum_{\mathrm{k}=1}^{\mathrm{N}}\left(\sqrt{\frac{\mathrm{m}\left(\mathrm{k}\right)}{\uprho_{\mathrm{s}}\mathrm{l}\pi }+\frac{{\mathrm{d}}^2}{4}}-\frac{\mathrm{d}}{2}-\mathrm{R}\right)} $$
(10)

where N is the number of test records, for tests recorded every 3 min, the total test time in seconds is t = N × 180.

Under gravitational effect, the dynamic dissolution model parameters of rock salt were obtained by parameters inversion (Table 5). A comparison of the experimental and simulation results for the dynamic dissolved radius of rock salt for flow rates of 10, 20, 30, and 40 L/h is shown in Figs. 8, 9, 10, and 11, respectively. The parameters inversion results show that the model and test results were in good agreement, indicating that under gravitational effect, the dynamic dissolution model of rock salt established in this paper provides a good estimate of the dynamic dissolution mechanism of rock salt. The results presented in this paper provide a theoretical basis and analytical means that can be used as a guide for studies on the dissolution mechanism of rock salt.

Table 5 Parameters for rock salt dissolution model from inversion results
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Fig. 8
figure 5

Dynamic dissolved radius of rock salt with gravitational effect over time for flow rate of 10 L/h. Experimental results (black squares) and simulation results (red line)

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Fig. 9
figure 6

Dynamic dissolved radius of rock salt with gravitational effect over time for flow rate of 20 L/h. Experimental results (black squares) and simulation results (red line)

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Fig. 10
figure 7

Dynamic dissolved radius of rock salt with gravitational effect over time for flow rate of 30 L/h. Experimental results (black squares) and simulation results (red line)

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Fig. 11
figure 8

Dynamic dissolved radius of rock salt with gravitational effect over time for flow rate of 40 L/h. Experimental results (black squares) and simulation results (red line)

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Conclusions

Based on the dissolution characteristics of rock salt obtained in a laboratory experiment, the dynamic dissolution model under gravitational effect was established for different rates of water flow, dynamic dissolution characteristics were thoroughly and systematically examined, and the following conclusions were drawn:

  1. (1)

    Using a custom-designed dynamic dissolution testing apparatus, the dynamic dissolution process of rock salt under the effect of gravity at different water flow rates was investigated. Under gravitational effect at flow rates of 20, 30, and 40 L/h, the average dissolution rates were 5.9733, 6.2838, and 7.9548 g/min, respectively, these dissolution rates were 2.1041, 1.6848, and 1.4635 times those calculated without considering the gravitational effect, indicating that gravity increased the flow velocity and accelerated the dissolution of rock salt.

  2. (2)

    Based on the dissolution characteristics of rock salt obtained in the experiment, the effect of different parameters of the dissolution model was analyzed. The analysis indicated that the parameter D/δ was linearly related to the flow velocity and established the dynamic dissolution model under the effect of gravity.

  3. (3)

    Using the finite difference method, a numerical solution of the dynamic dissolution model of rock salt was obtained. The simulation results based on parameter inversion were consistent with the experimental result, indicating that the dynamic dissolution model of rock salt established in this paper provides a reliable description of the dissolution mechanism of rock salt subjected to gravity.