Transient Calculation Studies of Liquid–Solid Collision in Jet Descaling

Abstract

Sichuan is gradually being transformed and is utilizing groundwater and thermal resources. However, this investigation found that the high mineralization rate of geothermal resources in the Sichuan Basin is common and efficient, and environmentally friendly descaling technology is the key to promoting the utilization of thermal resources in low-yield oil and gas wells. Due to the high efficiency, low cost, and lack of pollution of high-pressure jet descaling, it has attracted more and more attention recently, but the mechanism of jet descaling is still unclear. The key to jet descaling is the stress concentration in the scale caused by the impact of droplets from the jet. In this paper, the process of jet descaling is simplified as a 2D droplet–scale collision with a detailed theoretical analysis of the stress on the scale. A circular droplet was simulated to impact the surface of the scale. By using numerical methods for transient calculations, we couple the pressure of the droplets and the scale strain. We acquired transient equivalent stress fields inside scales and pressure distributions inside the water droplet. As a result of the impact, areas of high stress in the scale appeared. Due to the stress superposition, the highest stress is concentrated in two areas: the contact edge and the shaft. These results can identify the mechanism for high-pressure jet descaling and help improve the efficiency of high-pressure water-jet descaling.

1. Introduction

There are scaling problems in the industrial processes of geothermal energy, oil, and gas, especially the scaling of geothermal wells, which seriously restricts the development of China’s geothermal industry [1]. In a geothermal power plant, during the steam production process, due to the pressure drop and the release of steam and CO₂, the pH value and ion concentration gradually increase, resulting in more serious scaling and affecting the efficiency of the power plant. If fouling is severe, downtime may be required to clean the pipes and equipment before the plant is fully operational, increasing maintenance costs and reducing plant capacity [2]. Therefore, the descaling of geothermal power plants deserves global attention [3]. The Sichuan Basin has rich geothermal resources, but because most of the geothermal fluids in the oilfield area of the Sichuan Basin are characterized by high salinity, the transformation of the gas field faces the bottleneck problem of descaling. This issue seriously restricts the development and utilization of geothermal resources. Through the investigation of 18 abandoned oil and gas wells in the Moxi gas field, it was found that the average temperature of these 18 wells is 96.7 °C, and the average salinity of the formation level is about 58,300 mg/L. High salinity geothermal water will increase the risk of corrosion and fouling of good geothermal pipelines, well pumps, transmission pipelines, and other equipment, which greatly affects the utilization of thermal energy. Therefore, in the development process of the conversion of waste oil and gas wells into geothermal wells in Sichuan Province, it is necessary to develop descaling technology.

High-pressure jet descaling technology has the advantages of not requiring the addition of any chemical reagents, being very friendly to the environment, not causing chemical damage to the geological environment and equipment, and having the potential to play a role in geothermal wells. The researchers attempted to investigate whether jetting could be used for the descaling of geothermal wells [4]. A large number of experimental results and practical applications show that high-pressure water-jet technology is very significant in both cost and benefit in industrial cleaning [5]. It can provide an effective guarantee for the service life and cleaning quality of industrial equipment. The velocity and pressure distribution of the jet have an important influence on descaling. Surveys such as that conducted by Zhou (2022) have shown that using the downhole rotary jet tool with controllable rotational speed generates high-pressure water jets to flush the wellbore directly [4]. In addition, high-frequency oscillation hydraulic waves and cavitation noise can achieve descaling physically, thus relieving or unblocking the blockage status of the wellbore.

The experimental research related to high-pressure jet descaling and rust removal is mainly divided into two research directions. On the one hand, it is an experimental verification of descaling and rust removal. On the other hand, the flow characteristics of the jet and the nozzle structure design are studied through observation. The experimental verification of descaling and rust removal mainly includes the influence on the width and uniformity of descaling and rust removal in terms of jet distance, pressure, angle, and mixed abrasives in the jet [6]. For linear jets, studies have shown that in the case of fixed pressure, the speed of movement and the number of times the jet moves to affect the width of descaling/decorating is fundamentally due to the contact time of the jet on the solid surface; while the rotating jet can increase the descaling/decorating width by the movement of the nozzle [7]. Further research found that erosion is the primary mechanism for water jet technology to remove scale [8]. Wang et al. investigated the use of high-pressure jets for the descaling of coal mine drainage pipes through experiments and optimized the parameter structures of water jets and nozzles [9]. Zhang et al. used Fluent software to simulate the fluid field inside the convergence nozzle to obtain axial pressure, turbulent kinetic energy, and speed, and recommended the best nozzle convergence angle and the length-to-diameter ratio [10]. Tamura et al. experiments studied the change of descaling jet structure and the attenuation of droplet velocity along the jet distance, observed the jet structure with a high-speed camera, and measured the water-drop velocity and diameter with a phase-Doppler analyzer [11]. Wang et al. published their research on the descaling of coal drainage pipes involving the measurement of the tensile strength of scale via a tensile test, which is one of the very few studies that have focused on the mechanical properties of the scale [12].

The high-speed droplets hit the scale surface, and the stress concentration leads to breakage of the scale. Therefore, the research on the stress on the scale is the core issue in developing efficient geothermal-well jet-descaling technology. The main weakness of the previous study was the lack of research on the stress on the scale. Meantime, it hasn’t clarified the mechanism of jet descaling from a kinetic point of view.

To date, several studies have investigated liquid–solid collision. However, there are few studies on the dynamics of collision between liquid and solid, most of which focus on the liquid jet and lack the analysis of stress in the solid. We call the typical liquid–solid impact problem “water hammer”. “Water hammer” or “water-hammer pressure” refers to the impact pressure at the liquid–solid interface. Cook (1928) first proposed the water-hammer force [13], followed by Engel [14]. Heymann (1969) confirmed that shock waves form in water droplets and are separated from water droplets at specific moments [15]; after that, the lateral jet is formed, the impact pressure is released, and the energy becomes momentum. At present, a large number of experimental studies on liquid–solid collisions have been carried out (based on high-speed photography) [16,17,18,19], and most of the theories [20,21,22,23,24] and numerical values have been obtained [25,26,27,28,29,30,31].

In summary, to illustrate the mechanism of high-pressure jet-descaling technology in geothermal wells, we will study the dynamic behavior of the impact of water droplets and scale surfaces through numerical simulation, and discuss the stress concentration of scale. The calculations in this paper are transient.

2. Droplet–Scale Impact Model

2.1. Physical Analysis

The schematic diagram of the droplet impacting the surface of the scaling process is shown in Figure 1. Due to the compressibility of the water droplet, a shock wave is generated when the spherical droplet collides with the surface of the scale. Droplets create a pressure distribution in the scaling layer that changes with time and space coordinates, causing stress waves to pass through the scale. The high-pressure region may exist near the contact edge point and along the axis of the water droplet, which may be the cause of microcracks and crack propagation in brittle materials such as glass. For numerical analysis, it has been made into a physical model suitable for droplet–scale impacts [32,33]. Some scientists have detailed that the most extreme strains occur before the shock wave leaves the droplet [34,35,36]. When the shock wave leaves the water droplet, the influence of the pressure on the scale surface on the scale damage is reduced. Therefore, in this article, we focus on the critical value of the shock wave before it leaves the water droplet.

This paper focuses on the basic problems of jet descaling and the characteristics of the transient stress field. The emphasis of the study on jet descaling is to obtain the transient stress field, peak stress, impact duration, and the location of the most dangerous substance points inside the scale.

By using the mathematical models and numerical calculations in our research [37,38], the entire process of the impact can be described when coupling the pressure inside the water droplet and the stress inside the scale. The properties of the droplet and the elastic deformation of the scale can be also taken into account. Based on previous research, this paper discusses the problem of jet descaling through numerical simulation.

2.2. Coupling of 2D Droplet–Scale Impact Model

Here we ignored the viscosity of the droplet and considered the compressibility of the droplet in the sound wave velocity. This hypothesis has been proven correct by previous studies [37]. The two-dimensional wave model of the physical process in the droplet area during the impinging jet descaling process can be written as follows:

$$\{\begin{array}{c}{\nabla }^2\Psi =\frac{1}{c^2}\frac{{\partial }^2\Psi }{\partial t^2} \\ \mathrm{p}=\mathrm{A}{\mathsf{\rho }}^{\mathsf{\kappa }}-\mathrm{B}\end{array}\hspace{1em}\hspace{1em}\hspace{1em}\{\begin{array}{c}\mathrm{{\rm P}}=-\mathsf{\rho }\frac{\partial \Psi }{\partial t}\\ c=\sqrt{\mathsf{\kappa }\frac{\mathrm{p}+\mathrm{B}}{\mathsf{\rho }}}\end{array}$$

(1)

In the formula, $\Psi$ represents the velocity potential function, $c$ is the sound velocity in the droplet, A = 1.0147663$\times {10}^{-19}$, B = 2.858987 $\times {10}^8$ Pa, $\mathsf{\kappa } =7$.15, and $\mathsf{\rho }$ is the density. P (on the right up) represents the pressure, and A and B represent the constants in Tait’s water state equation. The physical procedure in the droplet region of the impact was cleared in the model.

The physical problem of droplets hitting the scaling layer can be viewed as an axisymmetric problem. The displacement of the scale could be obtained from the velocity of the droplets, coupling the water and the scale. When it comes to the inside of the scale, the elastic equation can be referenced directly. The Lame equation was used to solve the dynamic behavior of the scale. The two-phase coupling equation is as follows:

$$\{\begin{array}{c}\frac{{\partial }^2\Psi }{\partial {\mathrm{x}}^2}+\frac{1}{\mathrm{r}}\frac{\partial }{\partial \mathrm{r}}\left(\mathrm{r}\frac{\partial \Psi }{\partial \mathrm{r}}\right)=\frac{1}{{\mathrm{c}}^2}\frac{\partial \Psi }{\partial t^2}\\ {\Psi |}_{t=0}=0\\ {\frac{d\Psi }{dx}|}_{x=0}={\frac{d{\mathcal{U}}_s}{dx}|}_{x=0}\end{array}$$

(2)

$$\{\begin{array}{c}\left(\lambda +2\mu \right)\frac{{\partial }^2{\mathcal{U}}_s}{\partial x^2}+\mu \frac{{\partial }^2{\mathcal{U}}_s}{\partial r^2}+\left(\lambda +\mu \right)\frac{{\partial }^2{\mathcal{V}}_s}{\partial x\partial r}+\lambda \frac{\partial }{\partial x}\left(\frac{{\mathcal{V}}_s}{r}\right)+\frac{\mu }{r}\left(\frac{\partial {\mathcal{V}}_s}{\partial x}+\frac{\partial {\mathcal{U}}_s}{\partial r}\right)={\rho }_s\frac{d^2{\mathcal{U}}_s}{d{t}^2}\\ \mu \frac{{\partial }^2{\mathcal{V}}_s}{\partial x^2}+\left(\lambda +2\mu \right)\frac{{\partial }^2{\mathcal{V}}_s}{\partial r^2}+\left(\lambda +\mu \right)\frac{{\partial }^2{\mathcal{U}}_s}{\partial x\partial r}+\lambda \frac{\partial }{\partial r}(\frac{{\mathcal{V}}_s}{r})+\frac{2\mu }{r}(\frac{\partial {\mathcal{V}}_s}{\partial r}-\frac{{\mathcal{V}}_s}{r})={\rho }_s\frac{d^2{\mathcal{V}}_s}{d{t}^2}\\ {{\mathcal{U}}_s|}_{t=0}=0\end{array}$$

(3)

$$\{\begin{array}{c}{{\mathcal{V}}_s|}_{t=0}=0\\ {\frac{d{\mathcal{U}}_s}{dx}|}_{x=0}=\frac{1}{{\mathrm{E}}_1}{\frac{d\Psi }{dt}|}_{x=0}\\ {\frac{d{\mathcal{V}}_s}{dx}|}_{x=0}=0\end{array}$$

(4)

In the above formula, x represents the transmission distance of the transverse stress wave in the scale, r represents the radius of water droplets, and u_s represents the longitudinal displacement velocity in the scale. In Equation (3), λ (Young’s modulus of the scale) and μ (Poisson’s ratio of the scale) are constants of the Lame equation, and v_s represents the lateral displacement velocity in the scale. The strain can be calculated as follows:

$$\{\begin{array}{c}{\epsilon }_x=\frac{\partial {\mathcal{U}}_s}{\partial x}\\ {\epsilon }_r=\frac{\partial {\mathcal{V}}_s}{\partial r}\\ {\epsilon }_{\varphi }=\frac{{\mathcal{V}}_s}{r}\end{array}\hspace{1em}\hspace{1em}\{\begin{array}{c}{r}_{xr}=\frac{\partial {\mathcal{V}}_s}{\partial x}+\frac{\partial {\mathcal{U}}_s}{\partial r}\\ \theta ={\epsilon }_x+{\epsilon }_r+{\epsilon }_{\varphi }\end{array}$$

(5)

In the above formula, ε_x represents the longitudinal strain in the scale, ε_r represents the transverse strain in the scale, and ε_Φ represents the angular deformation rate in the scale. The stress can be described as follows:

$$\{\begin{array}{c}{\sigma }_x=\lambda \theta +2\mu {\epsilon }_x\\ {\sigma }_r=\lambda \theta +2\mu {\epsilon }_r\end{array}\hspace{1em}\hspace{1em}\{\begin{array}{c}{\sigma }_{\varphi }=\lambda \theta +2\mu {\epsilon }_{\varphi }\\ {\tau }_{xr}=\mu {\gamma }_{xr}\end{array}$$

(6)

In the formula, σ_x represents the longitudinal stress of the scale, σ_r represents the transverse stress of the scale, and σ_Φ represents the tangential stress of the scale. After the stress distribution in the solid region is obtained, since the solid internal stress is a tensor with many components, it is not convenient for further discussion. Therefore, the fourth strength theory is used to analyze the stress of solid materials, and all stress components are integrated into an equivalent stress for discussion, which is expressed by σ_e. It is defined that the dimensionless equivalent pressure (P) is equal to the pressure in the liquid divided by the water-hammer force in the undisturbed liquid, and the dimensionless equivalent stress (σ_eq) is equal to the solid equivalent stress divided by the water-hammer force in the undisturbed liquid. The term “pressure” or “stress” mentioned later refers to dimensionless pressure or dimensionless equivalent stress. The coordinates x and r and the time t were compared with the droplet’s initial dimensions r₀ and the drop’s initial sound speed c₀ to obtain dimensionless variables.

Through the discretization of the above governing equations, the pressure distribution of the droplets and the stress distribution of the scale layer in each time layer were calculated. Then the maximum stress in each time layer in the scale was obtained. Finally, the change law of the maximum stress with time in the transient process of the droplet collision was obtained.

2.3. Solving Droplet–Scale Impact Model

The original droplet density ρ₀ was 1000 kg/m³ (high-pressure jet after jet pressure is atmospheric pressure; water is generally normal temperature, so the density range is very close to 1000 kg/m³ for the convenience of calculation; thus, it is a unified 1000 kg/m³ calculation). The water sound speed c₀ was 1430 m/s. When t = 0, the initial static water droplets produced a certain impact velocity after being impacted by the rigid scale plane. The particle velocity at the contact interface rose from 0 to v₀ instantaneously, while the pressure on the contact surface rose rapidly from 0 to the peak, then fluctuated, and eventually reached a stable number of 1.43107 Pa.

Solution of droplet–scale coupling: firstly, we set the undisturbed droplet region and the displacement to 0 scale area. The motion velocity of the droplet boundary node was then instantaneously changed to the impact velocity. The wave equation in the droplet phase was solved to obtain the pressure distribution at the droplet and droplet–scale interface. Then, using the pressure distribution on the droplet–scale interface, the velocity of the scale particles on the interface was calculated as the displacement boundary condition to solve the displacement in the scale. The above solution process required iterative iterations between the droplet and scale regions until all the physical quantities converged. Then we entered the next layer to continue the calculation. Through the above calculation process, we could obtain the water-droplet pressure field and the stress field in the scale.

3. Simulation Results, Analysis, and Conclusions

Table 1. The physical properties of scale and droplet.

	Scale	Droplet
Density (m3/kg)	4090	1000
Sonic speed (m/s)	4268	1430
Acoustic impedance (kg/m2s)	1.746 × 107	1.430 × 106
Young’s modulus (Pa)	74.5 × 109
Poisson’s ratio	0.28
Flexure (tensile) strength (MPa)	75.4
Compressive strength (MPa)	1300

shows the main physical properties of the scale parent material and water studied in this paper. According to the current technology of jet descaling, we chose 2 mm water droplets for calculation [39]. The relevant parameters of the droplet refer to the literature on high-pressure jets [40,41]. The physical parameters of the scale come from the investigation of the hydrothermal resources of the low-yield oil and gas wells in the Sichuan Basin.

Impact of the Droplet on the Scale

Figure 2 shows the simulation results of jet descaling, mainly showing the pressure distribution in droplets at different time layers (10–60 ns) and the stress distribution in the scale. The position of X = −1 is the droplet–scale interface, and the white area is the outer contour of the spherical droplet. The main messages shown in the figure are as follows:

• The pressure value in the droplet keeps increasing. On the contact surface of the water droplet and the scale, the point of highest pressure always occurs at the edge of the interface (the shock front). Figure 3a–f clearly shows that the internal shock wavefront of the droplet forms a flat crown in the disturbance area. Because the density of water increases and the elastic modulus increases in the disturbed area, the speed of sound in the disturbed area is greater than that in the undisturbed area. Therefore, the pressure wave in the disturbed area can catch up with the initial pressure wavefront to form a superposition of the wavefront, thereby forming a shock wave.
• The pressure wave is transmitted faster in the radial direction than in the axial direction. Assuming a contact angle of θ, for a first-order approximation, the speed of motion at the contact edge point of the droplet with the scale is:

  $${\mathsf{\nu }}_{\mathrm{e}}={\mathsf{\nu }}_0\cot \mathsf{\theta }$$

  (7)
  For the collision process at the nanosecond level, if the impact velocity is considered constant, the equation is as follows:

  $${\mathsf{\nu }}_{\mathrm{e}}={\mathsf{\nu }}_0\frac{{\mathrm{r}}_0-{\mathsf{\nu }}_0\mathrm{t}}{\sqrt{{\mathrm{r}}_0^2-{\left({\mathrm{r}}_0-{\mathsf{\nu }}_0\mathrm{t}\right)}^2}}$$

  (8)
  For r₀ = 1 mm, v₀ = 210 m/s, and t = 10 ns, the estimated moving speed of the edge point is v_e = 3235 m/s, but the speed of sound in water is about 1430 m/s. At this time, the moving speed of the edge point is not only much greater than the impact speed but also greater than the speed of sound. Therefore, in the radial direction, the disturbance source moves faster than the speed of sound, but the pressure wave can only travel at the speed of sound. At this time, the maximum pressure point in the droplet appears inside the droplet. However, due to the lack of contact with the scale, it is not our main concern.
• Since the speed of sound in the scale (4268 m/s) is higher than the moving speed of the edge point, the propagation of the stress wave in the scale is less affected by the disturbance source, so the stress-wave influence area in the solid is spherical. There are two areas with high-stress values. The first is located near the edge of the water-droplet contact with the scale, which is where the peak pressure in the water droplet occurs. The second is located at a certain depth on the axis and is superimposed by the axially symmetrically distributed stress waves. This area is the source of stress at which scale erosion occurs at a certain depth.

Figure 3. Cloud diagrams of the pressure distribution inside the droplet and the stress distribution inside the scale under diverse impact velocities (r₀ = 1 mm; the unit of x and R in the figure is mm).

Figure 3. Cloud diagrams of the pressure distribution inside the droplet and the stress distribution inside the scale under diverse impact velocities (r₀ = 1 mm; the unit of x and R in the figure is mm).

Figure 3 shows the pressure distribution in the droplet and the stress distribution in the scale under different impact velocities, the impact velocities are 50 m/s, 100 m/s, 150 m/s, and 200 m/s, respectively. The pattern of the two-time layers t = 10 ns and t = 20 ns is also shown.

Under all the impact velocity conditions, the pattern characteristics of the t = 10 ns and t = 20 ns time layers are similar, except that the disturbance area of the t = 20 ns time layer is larger.

Different impact velocities have a great influence on the characteristics of the stress field in the scale. First of all, the magnitude of the stress is different. When the impact speed is high, the stress in the scale is greater. Second, the frequency of the stress waves is different. When the impact speed is low, the ripple of the stress wave is thinner. When the impact speed is higher, the pattern of the stress wave is denser, in addition to the fact that the sound velocity in the scale does not change much, so the denser ripple indicates that the frequency is higher. The above effect is due to the large variation in the density of the liquid with pressure. During impact, if the impact velocity is high, the density of the liquid is higher, the speed of sound is increased, and the frequency of the pressure wave increases, increasing the fluctuation frequency of the pressure source transmitted to the solid. Consequently, the stress wave frequency will also increase. This is the result of typical liquid–solid interactions. In summary, these results show that the faster the impact of the water droplets, the stronger the pressure source in the scale.

Figure 4 shows the variation of the maximum equivalent stress with time. The maximum internal stress in the scale increases with the increase of impact time, and the overall trend is that the oscillation increases linearly. When the stress concentration exceeds its tensile strength, the stress concentration area breaks and expands, eventually causing it to fall off.

4. Conclusions

In this paper, the theoretical problems of jet descaling were studied by numerical simulation. The axisymmetric single-wave control equation and two-dimensional droplet–scale impact coupling were established to solve the problem of jet impact removal and explore the concentration of stress in the scale.

In the process of jet descaling, due to the superposition of shock waves inside the water droplets, the momentum lost by the water droplets was finally converted into the strain of the water droplets. Then there was a stress field distribution that could not be ignored in the scale. The generation of internal strain and stress could be the root cause of scale fracture.

The data showed that under continuous impact, when the stress concentration in the scale exceeded its tensile strength, three areas of stress concentration may crack and expand, eventually leading to shedding. The main factors affecting the maximum stress are Young’s modulus of scale, Poisson’s ratio of scale, the impact velocity of the droplet, and the mass and density of scale. Through the numerical simulation of the dynamic behavior in droplet–scale impact, we obtained the theoretical process of scale shedding caused by jet descaling.