An Experimental Study on the Influence of the Fractal Characteristics of X80 Steel Surface Morphology on Water Ring Stability

Abstract

The surface morphology of X80 steel with hydrophilic underwater oleophobic characteristic is described greater comprehensively and quantitatively in this work by combining fractal dimension and multifractal. X80 steel with hydrophilic underwater oleophobic surface characteristics was constructed using a chemical etching method. Then, with the aid of three wettability parameters—contact angle, rolling angle, and adhesion work—this study investigated the relationship between the surface fractal dimension of X80 and the stability of the water ring in the core annular flow. The results showed that: (1) the fractal dimension of X80 steel specimens increased first and then decreased with the increase of reaction time. Besides that, the value of it was greater than 2, indicating that the surface had obvious fractal characteristics. The spectral difference, Δf(α), and the spectral width, Δα, supplemented the description of the X80 steel surface morphology, which was consistent with the scanning electron microscope results. (2) When the maximum fractal dimension was 2.0808, the minimum contact angle of distilled water on its surface was 50.2°, and the maximum contact angle of underwater oil droplets was 166.4°. The larger the fractal dimension of X80 steel with hydrophilic underwater oleophobic properties, the more hydrophilic and underwater oleophobic it is. This illustrated that there was a strong binding force between the water and the X80 steel pipe wall, and hence the quality and efficiency of the core annular flow was improved, which was more conducive to the promotion of this technology in the field of heavy oil transportation.

1. Introduction

Heavy oil poses significant transportation and exploitation difficulties due to its high density, high viscosity, high asphaltene and resin content, and poor flow performance at room temperature. Therefore, improving the transportation efficiency of heavy oil is a technical problem in the oil and gas storage and transportation industry all the time. Traditional transportation methods of heavy oil mainly include heating [1], emulsification [2], dilution [3], etc. However, these methods have many problems, such as larger loss, lower oil quality, higher cost, and bringing much trouble in subsequent treatments. Thus, researchers proposed a core annular flow (CAF) [4] to reduce the friction loss during the transportation of heavy oil and improve the transportation efficiency.

Core annular flow is a technology based on the formation of a flow pattern in which the oil is transported in the central region of the pipe, surrounded by a thin annular aqueous film formed near the wall, lubricating the flow and decreasing the friction damage. Du et al. [5] had carried out dedicated experiments to study the effects of the phase’s density, viscosity, velocity, and interfacial tension on the core annular flow. It turned out that water ring transportation can greatly reduce the transportation resistance of heavy oil on the premise that heavy oil is confined to the central area of the pipeline. Based on these experimental findings, this team conducted a second study in which they used numerical simulations to analyze the effects of these four variables on water ring stability and pressure drop [6]. The results showed that there was basically no velocity gradient in the core area of heavy oil during heavy core annular flow, which could be approximated as the elastic solid, and the water phase friction had a greater impact on the pipe pressure drop. Their research all emphasized the heavy oil flow in the center of pipe, where a water ring was in touch with the inner pipe wall; thus, it can be seen, the stability of the water ring is a significant factor in the application of this technology. However, once disturbance occurs in the pipe during transportation, the oil droplets in the center of the pipe are easily dispersed and the eccentric oil ring is formed subsequently. If so, the functions of the water ring will be greatly weakened. For this reason, how to maintain the stability of the water ring has become the most critical issue in the formation of core annular flow.

There are quite a lot of factors affecting the stability of the water ring, such as the water ring thickness, water content, oil–water density difference, and wettability of the pipe wall, etc. The wettability, here, includes both oil and water spreading over the inner pipe surface, usually using the contact angle (θ) to measure surface wettability. The degree of wetting (i.e., wettability) is determined by the cohesive force (Wc) between liquid molecules and the adhesion work (Wa) generated by the molecular interaction between liquid and solid. If the contact angle between the pipe wall and water is smaller, the adhesion work will be greater and the pipe wall will adhere more firmly to the water, making the water ring more stable. In addition to contact angle and adhesion work, the rolling angle (α) is also an important variable characterizing surface wettability. The rolling angle refers to the inclination angle of a surface at which an approximately spherical droplet can just roll off it. The greater the rolling angle between the pipe wall and the water, the more difficult it is for water to roll off from the wall of the pipe, and the more stable the water ring will be.

The following studies have shown that changing the wettability of the pipe wall could improve the efficiency of the core annular flow. Silva et al. [7] treated the inner surface of four different materials by KMnO₄ oxidation and increased the roughness of it. The contact angle of the treated materials was less than 20°, which indicated that a more stable water ring was formed between the pipe wall and the oil, avoiding the adhesion of the oil to the pipe wall. Santos et al. [8] found that after removing asphaltene and naphthenic acid from crude oil, the wetting state of the pipe wall changed from lipophilic to hydrophilic. When the water phase contained 1 wt% sodium metasilicate and sodium chloride, the contact angle was less than 60°, which could effectively maintain the stability of the water ring. Shi et al. [9] reported that when the contact angle between the oil phase and the pipe wall was reduced in the core annular flow, the oil phase was more likely to adhere to the pipe wall and the pipe wall was easier to be polluted by oil. As seen above, if the hydrophilic underwater oleophobic properties of the pipe wall are improved, the water phase can be firmly adhered to the pipe wall and the stability of the water ring can be promoted greatly. Meanwhile, the possibility of mixing at the oil–water interface will be reduced, and the transportation efficiency of heavy oil will be improved obviously as well.

Current studies have shown that the morphology of solid surface is an important factor affecting the wettability of the solid–liquid interface [10]. Buczko et al. [11] constructed a superhydrophobic aluminum mechanical surface and found that the greater the current density and the longer oxidation time, the greater the surface roughness of the aluminum substrate and contact angle, and the weaker the wetting performance. Similarly, Li et al. [12] prepared alumina film on a non-smooth convex aluminum and discussed the influence of morphology on wettability. The experimental results showed that the water contact angle of the substrate surface could distinctly increase from 80.6° to 133° due to the change of the surface morphology. Tiny columns and pores could more easily trap air into the pores and enhance surface hydrophobicity. Tong et al. [13] designed and characterized microscale square texture and nail shaped texture on silicon surface. It was found that the root mean square roughness of the silicon surface after the chemical modification increased from 0.14 nm to 1.24 nm, and the water contact angle of the surface increased to 104.2° synchronously. Wang et al. [14] prepared three-dimensional porous micro-nano hierarchical copper film and revealed the influence law of morphology on contact angle. They concluded that a longer deposition time would increase the pore size and reduce the surface roughness and contact angle. Sajid et al. [15] investigated the influence of corrosion on the water wettability of ASTM A36 steel. During the experiment, contact angles were observed to decrease from 88° to 15°—dramatically decreasing by 83%—as the average roughness of specimens increased from 12 nm to 59 μm. It was concluded that the surface roughness greatly affected the wettability of steel. In another study, Sajid et al. [16] treated the surface of ASTM A572 steel specimens with different room-temperature ionic liquids (RTIL). They found that the average initial water contact angle of steel specimens decreased by 24−87%, and the average surface free energy increased by 49−100% after surface treatment with ionic liquids, which rendered the RTIL-treated steel surfaces more wettable.

From the above, it can be found that the surface roughness, microstructure scale and other parameters were generally used to characterize the surface morphology. However, these parameters were actually constrained by the resolution and sampling length of the instrument and cannot fully describe the structure, distribution, and morphology of surface roughness [17]. Therefore, researchers introduced fractal theory, and use fractal dimension to characterize the surface morphology. Huang et al. [18] proposed a general machine learning framework for surface wetting, arguing that only contour or area roughness parameters are not sufficient to describe surface morphology characteristics. For this reason, they introduced fractal dimension, two-dimensional entropy and periodicity to describe the complexity and randomness of surface morphology besides the surface roughness. Karl et al. [19] certificated the relationship between the mean roughness values and the fractal dimension of rough and self-affine PTFE surfaces. The results demonstrated a strong association between the mean roughness value and the fractal dimension estimated using the cube counting method, with a correlation coefficient R² of 0.888, which was higher than the fractal dimension value obtained via the height difference correlation function. Xing et al. [20] characterized the morphology of calcium carbonate fouling crystals by a direct fractal method. They hold that it is feasible to quantitatively describe the overall morphological features of fouling crystals through fractal dimension. Das et al. [21] investigated the effect of Al doping on the wettability of the TiN surface. The results demonstrated that the wettability could be tuned by adjusting the fractal dimension, Hurst exponent, root mean square slope, and other parameters of the surface. More importantly, the variation of curvature pressure with Al doping concentration verified the dependence of wettability on fractal parameters. Mwema et al. [22] conducted fractal analysis on the morphology of aluminum films sputtered on glass. They found that the roughness increased, while the fractal dimension decreased slightly with the increase of the substrate temperature (55–95 °C). When Zhang et al. [23] used fractal dimension to describe rough surface, they proposed that fractal dimension still had its limitations in describing the local morphology of surface. Therefore, how to seek other parameters to characterize the morphology comprehensively and quantitatively on the basis of a single fractal is an urgent problem for researchers.

Ţălu et al. [24] recorded the changes of surface morphology of sputtered indium tin oxide thin films with atomic force microscope. They believed that the multifractal method related to the surface analysis of stereo measurement was an effective tool to quantify the changes of three-dimensional surface micro-texture. Modabberasl et al. [25] used a multifractal spectrum to characterize the surface morphology of diamond-like carbon (DLC) films deposited on silicon substrate. Their analysis illustrated that DLC films deposited at different substrate temperatures had multifractal properties. With the increase of substrate temperature, all the width of the multifractal spectrum, strength of multifractality, and surface roughness will increase. Ţălu et al. [26] conducted an experimental study on silver in diamond-like carbon nanocomposites. It was found that the results of generalized fractal dimension, multifractal singularity spectrum, and statistical function were consistent with the X-ray diffraction analysis of surface morphology and UV–Vis spectrum measurement results. Shakoury et al. [27] studied the morphology characteristics of 304 stainless steel coated with manganese films annealed at different temperatures. The results of the multifractal analysis indicated that the spatial complexity of surface decreased with increasing the annealing temperature, while the vertical roughness increased with increasing the height difference of multifractal spectrum. Although many scholars have confirmed that multifractal is an effective way to characterize the morphology characteristics, few studies combine the single fractal and multifractal parameters to explore the fractal characteristics of the morphology on the hydrophilic underwater oleophobic surface and its influence on the stability of water ring.

Rather than depending on variables, like roughness, that are constrained by instrument resolution and sampling duration, this paper combined multifractal and single fractal theories to conduct a thorough quantitative investigation of the surface morphology of X80 steel specimen with hydrophilic and hydrophobic properties under water. This was the greatest innovation of this paper. Second, based on the medium state of oil and water phases in the core annular flow, the relationship between the surface fractal dimension and wettability of both water and oil was investigated. Three parameters of contact angle, rolling angle, and adhesion work were used to indirectly evaluate the water ring stability, and the effect of different morphologies on the above parameters were also discussed. In contrast, the wetting of steel by oil droplets in this paper took place underwater, not in the air, and it occurred in a liquid–liquid–solid form. This creative contact angle measurement of underwater oil droplets was another significant innovation of this article. Finally, the X80 steel utilized in the experiment is a popular steel used for heavy oil transportation at oilfield sites, but there was limited research on its hydrophilic underwater oleophobic performance, thus the experimental results in this study have reference and theoretical guiding relevance.

2. Experiments and Methods

2.1. Construction of Morphology on X80 Steel

In this paper, hexamethylenetetramine and zinc nitrate were used as reaction solutions. To obtain X80 steel specimens with hydrophilic underwater oleophobic properties, zinc oxide compounds of different morphologies were prepared on the surface of X80 steel by changing the reaction time (t). The surface morphology of X80 steel specimens was collected by the JSM-6390A scanning electron microscope (SEM, Japan Electronics Co., Ltd., Tokyo, Japan). It is worth noting that the chemical modification method used in this article has good durability. The specific experimental steps are as follows:

(1)

Specimen cleaning: cut the X80 steel sheet into several test specimens with the size of 15 × 10 × 2 mm by the JMQ-60Z automatic precision cutting machine (Shanghai Metallurgical Equipment Company Ltd., Shanghai, China), place the test specimens in a mixed solution of acetone and anhydrous ethanol (volume ratio of 1:1) to ultrasonic cleaning for 10 min and dried at room temperature.

(2)

Etching solution preparation: prepare 50 mL of a 0.3 mol/L zinc acetate solution, 50 mL of a 0.1 mol/L zinc nitrate solution, and 50 mL of a 0.1 mol/L hexamethyltetramine solution, respectively. Then, uniformly mix the zinc nitrate solution and the hexamethyltetramine solution.

(3)

Surface etching: place the X80 steel specimens pretreated by step (1) in the zinc acetate solution and take them out after standing for 15 min, 20 min, 25 min, 30 min, 35 min, 40 min, 45 min, and 50 min, respectively. Then, place all specimens in a vacuum drying oven at 100 °C for 30 min. After that, calcine them in a resistance furnace at 400 °C for 2 h. Finally, take them out and cool them to room temperature.

(4)

Surface hydrophilic underwater oleophobic modification: pour the mixed solution of zinc nitrate and hexamethyltetramine and place the pretreated X80 steel specimens successively into the crucible of the resistance furnace. React together under a closed condition of 105 °C for 3 h. After the reaction, take out the test specimens, wash the floating matter on the surface with deionized water, and dry it with nitrogen.

(5)

Surface morphology collection: paste the specimen on the sample table of the sample room with conductive adhesive, observe from the low multiple, and gradually increase the multiple so that the surface morphology can be clearly and intuitively observed. Then, take photos of the surface morphology for preservation.

2.2. Fractal Characteristics of Morphology

In this paper, the fractal theory was introduced. The fractal dimension D and multifractal spectrum were used to characterize the surface morphology of materials comprehensively and quantitatively. The multifractal spectrum was expressed by two parameters, spectral width Δα and spectral difference Δf(α). The box counting dimension was selected to calculate the fractal dimension.

The target fractal image is covered by a small square lattice with the scale δ. The partition function χ(δ) is a power function of δ [25,27]:

$$\chi \left(\delta \right)\propto {\delta }^{-{\tau }_q}$$

(1)

where τ(q) is the mass index. If τ(q) is linear with q, the image is single fractal. If τ(q) is a convex function with q, the image is multifractal. The Legendre transformation can be obtained as [28]:

$$\left\{\begin{array}{c}\alpha =\frac{\mathrm{d}\tau \left(q\right)}{\mathrm{d}q}\\ f\left(\alpha \right)=q\alpha -\tau \left(q\right)\end{array}\right.$$

(2)

where f(α) is a multifractal spectrum, which is defined as the fractal dimension of a subset of the same singularity index α. A reflects the singularity degree of each cell in the image. q is a weight factor, q > 0 indicates that the higher surface area has a larger weight. On the contrary, q < 0 means that the lower surface area has a larger weight. It can be seen from Equation (2) that α was a function of q. So, α can be written as α(q).

The multifractal spectrum is usually characterized by two parameters, namely, the spectral width Δα and the spectral difference Δf(α) [25].

$$\Delta \alpha ={\alpha }_{\max }-{\alpha }_{\min }$$

(3)

$$\Delta f\left(\alpha \right)=f\left({\alpha }_{\min }\right)-f\left({\alpha }_{\max }\right)$$

(4)

where α_max and f(α_max) are the maximum probability subset dimension and the sum of the maximum probability subset dimension, respectively; α_min and f(α_min) are the minimum probability subset dimension and the sum of minimum probability subset dimension, respectively. The spectral width Δα is used to characterize the uniformity of the probability distribution. The smaller the Δα is, the more uniform the distribution is. The spectral difference Δf(α) is used to characterize the difference between the sum of the minimum probability and the maximum probability subset dimension. When Δf(α) > 0, the multifractal spectrum is left hook and the surface morphology height is relatively low. When Δf(α) < 0, the multifractal spectrum is right hook, and the surface morphology height is relatively high.

Firstly, import the SEM image into the current folder of MATLAB and eliminate the uneven brightness of images based on the background gray subtraction. Next, the threshold segmentation by the iterative method. The black and white bitmap can be obtained from the SEM image after the binarization processing. Finally, calculate the fractal dimension, singularity index α, and multifractal spectrum f(α) by box counting method and programming, respectively. The calculation flowchart of the fractal dimension and multifractal spectrum based on the MATLAB is shown in the Appendix A. The programming calculation of the last two parameters related to multifractal spectrum is according to the Equations (1)–(4).

2.3. Characterization of Water Ring Stability

The stability of the water ring is indirectly evaluated by the contact angle, rolling angle, and adhesion work.

2.3.1. Contact Angle

For the core annular flow, the smaller the contact angle between the pipe and water, the greater the contact angle of the underwater oil droplets, the more stable the water ring formed. Therefore, this paper designed a set of experimental devices specifically to measure the contact angle (θ) of distilled water and underwater oil droplets on the X80 steel surface at 28 °C, and indirectly evaluated the stability of the water ring. The schematic diagram of the contact angle measurement in multiple media environments is shown in Figure 1. Gasoline and diesel are selected for the oil droplets to be measured and the volume is 3 μL, which is precisely controlled by a liquid inlet needle. Each specimen surface was measured 6 times at different positions, and the results were averaged.

The device consists of four parts, including liquid injection, temperature monitoring, pressure control, and optical imaging. The measurement of the contact angle in this article is carried out under 28 °C and atmospheric pressure conditions. First of all, fix the X80 steel specimen on the upper surface of the sampling platform when the water contact angle is measured in the air, then regulate the height of the adjusting base so that the sampling platform and sapphire crystal eyepiece are at the same height. After that, fill the whole second cavity with silicone oil at a preset temperature of 28 °C; the multi-medium cavity is fixed on the JC2000D2 contact angle instrument (Shanghai Zhongchen Digital Equipment Co., Ltd., Shanghai, China) with the help of a clamping groove, and the sapphire crystal eyepiece and the contact angle instrument lens are required to be kept at the same level. Then, insert the liquid inlet needle with a micro injection pump externally into the preset hole in the center of the end cover. After setting the injection parameters, drop the water droplets to the specimen surface when the temperature of the stainless steel thermometer reaches 28 °C. Finally, the contact angle images are obtained by camera through the lens and fed back to the connected computer in real time. However, different from measuring the contact angle of water droplets in the air, it is proper to fix specimen on the lower surface of the sampling platform when the contact angle of the underwater oil droplets is measured. Moreover, distilled water is added into the first cavity from the second liquid inlet until the water line exceeds the lower surface of the sampling platform at least. Meanwhile, when the temperature of the stainless steel thermometer reaches 28 °C, drop the tested oil droplets to the specimen surface by using a liquid inlet bent needle. During the measurement, the gas outlet is always open to maintain the normal pressure. This set of equipment can also measure the liquid contact angle under pressure conditions. Then, open the gas inlet valve while closing the outlet valve and fill the first cavity with a certain gas medium. Stop pressurizing and close the inlet valve when the pressure P is reached.

2.3.2. Rolling Angle

The rolling angle refers to the inclination angle of a surface at which an approximately spherical droplet can just roll off it. The greater the rolling angle between the pipe wall and the water, the more difficult it is for water to roll off from the wall of the pipe, and the more stable the water ring will be. Therefore, the rolling angle (α) can be used to indirectly evaluate the stability of the water ring. It was also measured based on the JC2000D2 contact angle instrument with the device shown in Figure 1 at 28 °C during this experiment. The difference between measuring the contact angle and the rolling angle of distilled water in the air is whether the platform is rotated or not. After dropping distilled water on the specimen surface, the platform is rotated at a speed of 180°/min from the horizontal until the water droplets have a rolling tendency during the rolling angle measurement. The value of the rolling angle can be read off from the droplet picture collected at this instant. The volume of distilled water was precisely controlled to 20 μL by a liquid inlet needle. Each specimen surface was measured 6 times at different positions, and the results were averaged.

2.3.3. Adhesion Work

The adhesion work is defined as the reversible thermodynamic work that is needed to separate the interface from the equilibrium state of two phases to a separation distance of infinity. The greater the adhesion work between the pipe wall and the water, the more firmly the water adheres to the pipe wall, and the less likely the water and oil phases are to be mixed. Thus, the adhesion work (W_a) can also be used to indirectly evaluate the stability of the water ring.

When both the apparent contact angle of liquid on the solid surface and the surface tension of liquid are known, the adhesion work at the liquid–solid interface can be calculated from Equation (5) [29].

$$W_{\mathrm{a}}={\gamma }_{\mathrm{gl}}\left(\cos \theta +1\right)$$

(5)

where W_a is the adhesion work, γ_gl is the interfacial tension between gas and liquid, and θ is the apparent contact angle. The smaller the contact angle, the greater the adhesion work. This suggests that the larger the minimum value of the work required to pull the contact solid and liquid phases away from the junction, the stronger the liquid–solid adhesion and the more the liquid will wet the solid surface.

The cohesion work W_c is defined as that capable of creating an interface within a liquid and separating them into two independent surfaces against vapor [29].

$$W_{\mathrm{c}}=2{\gamma }_{\mathrm{gl}}$$

(6)

3. Results and Discussion

3.1. Surface Morphology at Different Reaction Times

Hydrophilic and underwater oleophobic surfaces with different morphologies were prepared by tuning the reaction time of X80 steel and zinc acetate solution. The morphologies of X80 steel surfaces were collected by a scanning electron microscope with a magnification of 400 times. Each specimen was scanned at 6 different positions. The fractal dimension of 1 specimen was the average value of the fractal dimension calculated from 6 positions. The most typical SEM image was selected as the specimen surface morphologies for different reaction times. The resulting images are shown in Figure 2.

Figure 2 shows that only pretreated X80 steel did not react with any chemical reagent, so the surface morphology did not change and it was the flattest among them. When the reaction time was 15~20 min, pitting pits with a size of about 10 μm began to appear on the surface of the X80 steel specimen. This may be due to the fact that the reaction between the X80 steel surface and the chemical reagents had just started at this time. In addition, the surface of the specimen was accompanied by granular protrusions with a side length of about 5 μm. This may be attributed to a short reaction time. When the reaction time was 25 min, the surface of the X80 steel specimen was distributed with blocky protrusions with a side length of about 20 μm. This is possibly for the agglomeration of the granular protrusion parts with the extension of reaction time. When the reaction time was 30 min, the surface of the X80 steel specimen consisted of irregular flakes. It was speculated that the reaction between the X80 steel and chemical reagent was more complete due to the extended reaction time. Then, the reaction products covered the flaky ferrite on the surface of the specimen [30], thus forming irregular, but flat flakes. Meanwhile, when the reaction time lengthened to 45 min, some flakes dissolved into needles, forming a snowflake structure. A too long reaction time may lead to the pitting corrosion of the X80 steel surface morphology. When the reaction time was up to 50 min, only a small number of fabric-like protrusions appeared on the surface of the X80 steel specimen, and the bulk of the surface tended to be flat. Since the reaction time continued to lengthen, the pitting corrosion of the surface morphology also continued, resulting in a depression of the protrusions and a reduction of the surface roughness.

3.2. Fractal Parameters under Different Reaction Time

The fractal dimension of the X80 steel specimens was calculated and plotted in Figure 3 as a function of reaction time. The multifractal spectrum of the X80 steel specimens with different reaction times was drawn, as shown in Figure 4.

The multifractal spectrum parameters Δα and Δf(α) of the X80 steel specimens were calculated according to Equations (3) and (4), and the results are shown in

Table 1. Multifractal spectrum parameters and fractal dimensions of X80 steel under different reaction time.

t/min	D	Δf(α)	Δα
0	2.0228	0.0117	0.1665
15	2.0397	0.0552	0.2294
20	2.0508	0.0391	0.1833
25	2.0624	0.0272	0.2149
30	2.0808	0.0227	0.2585
35	2.0765	0.0184	0.2633
40	2.0668	0.0221	0.2054
45	2.0697	0.0383	0.2289
50	2.0613	0.025	0.1818

.

It can be seen from Figure 3 that the fractal dimensions of the X80 steel specimens are all greater than 2, indicating that the surface has obvious fractal characteristics [31]. Furthermore, with the increase of reaction time, the fractal dimension of the X80 steel specimen increased first and then decreased, reaching the maximum value of 2.0808 at t = 30 min. Combined with Figure 2, it can be seen that the gradually increasing reaction time caused the change process of “not rough—rough—not rough” on the surface of the X80 steel specimens. When t = 30 min, the surface morphology of the X80 steel specimen is relatively rough, resulting in a large fractal dimension [32]. When the reaction time reached 45 min, the fractal dimension of the surface of the X80 steel specimen increased slightly, by 0.14% compared to the 40 min case. This is consistent with the roughness of the corresponding specimen in Figure 2. What is more, on the basis of Table 1, the ∆α of the surface of the X80 steel specimen at a reaction time of 45 min also increased by 11.44% compared to that at 40 min. In the light of the definition, Δα is the difference between the maximum and the minimum probability subset dimension probability. Thus, the spectral width Δα can be used to characterize the uniformity of the probability distribution. The larger the Δα, the more uneven the distribution. This is a supplementary description of the material surface from the perspective of multifractal spectrum parameters.

As shown in Table 1 and Figure 4, the spectral difference Δf(α) on the surface of the X80 steel specimens is greater than 0 and the multifractal spectrum shows a left hook shape. This indicates that the number of maximum probability subset dimensions is less than that of the minimum probability subset dimensions. As a result, the proportion of small-scale morphology on the surface of X80 steel specimen is large in the whole reaction process. On the other hand, the uniformity characterized by the spectral width Δα of the X80 steel specimen surface is consistent with the SEM observation results. According to Table 1, the Δα of reaction time 20 min decreased by 20.23% compared with that of the reaction time of 15 min, while the fractal dimension increased by 0.54% under the corresponding reaction time. The reason for this phenomenon may be that the clusters on the surface of the X80 steel specimens reduced a bit at this time, and the surface is slightly more uniform than the morphology at the reaction time of 15 min. Therefore, the analysis above proves that the single fractal dimension cannot fully and accurately characterize the morphology. Additionally, it is necessary to combine the multifractal spectrum parameters to comprehensively characterize the morphology.

3.3. Effect of Fractal Parameters on the Stability of Water Ring

3.3.1. Effect of Fractal Parameters on the Contact Angle

The contact angles of the X80 steel specimens with distilled water, underwater gasoline, and underwater diesel at different reaction times are shown in Figure 5. The fractal dimensions of the X80 steel surface was linearly fitted with the contact angles of distilled water, underwater gasoline, and underwater diesel, respectively. The results are shown in Figure 6.

From Figure 5 and Figure 6, it could be seen that the contact angle of distilled water on the surface of the X80 steel specimens decreased with the increase of the surface fractal dimensions, and there was a significant negative correlation between them. However, the change of the contact angle of underwater oil droplets with the fractal dimension was exactly the opposite, and there was an obviously positive correlation between them. The linear fitting factors R² of water droplets, gasoline droplets, and diesel droplets were 0.65995, 0.61948 and 0.82914, respectively. From the above analysis, it was consistent with the law that the rougher the surface of lyophilic (lyophobic) materials, the larger the fractal dimension and the more lyophilic (lyophobic) the surface [28]. Under the same fractal dimension, when the distilled water was on the surface of the X80 steel, the maximum relative deviation between the fitted and measured values of the contact angle appeared at D = 2.0808, reaching 23.54%, the minimum relative deviation occurred at D = 2.0228, which was 0.21%, the average relative deviation is 7.53%. When the underwater oil droplets were on the surface of the X80 steel, the average relative deviation of contact angles were 26.12% (gasoline) and 22.95% (diesel), respectively. Meanwhile, when the fractal dimension increased from 2.0613 to 2.0624, the corresponding contact angle of the distilled water decreased by 13.31% when the increase was only 0.053%. Combined with the multifractal spectrum parameters of the X80 steel specimens in Table 1, it was speculated that the reason for the large difference in the contact angle may have been the large increase in Δα, reaching 18.21%. Clearly, this complemented the variation of the fractal dimension from the perspective of the multifractal spectrum parameters.

Figure 5 showed the measured contact angle standard deviation for different liquids on the X80 steel specimens. The maximum standard deviation of distilled water contact angle was 4.77, and the minimum was 2.07. The maximum standard deviation of underwater gasoline contact angle was 4.76, and the minimum was 0.22. The maximum standard deviation of the underwater diesel measured contact angle was 4.89, and the minimum was 2.24. When t = 30 min, the fractal dimension D of both distilled water and oil droplets on the surface of the X80 steel specimens reached the maximum value of 2.0808. At this time, the contact angle of distilled water was the smallest, with a value of 50.2°, which could wet the surface well, while the contact angle of underwater oil droplets was the largest, with a value of 166.4°, which could hardly wet the surface. Therefore, for the core annular flow, the larger the fractal dimension of the pipe surface, the smaller the contact angle of distilled water, and the larger the contact angle of oil droplets. Consequently, the pipe wall would be more hydrophilic and oleophobic, and the water ring stability would be better. When t = 0 min, D was the minimum value of 2.0228, and the minimum contact angles of underwater gasoline and underwater diesel were 28.1° and 19.5°, respectively, which were significantly smaller than their corresponding fitted values of 67.2° and 43.3°. At this time, the relative deviation between the measured and fitted values of contact angles were the largest, which were 139.15% and 122.05%, respectively. This may be due to that the X80 steel specimen at this time was only a pretreated specimen without a rough structure, so the fractal dimension reached the minimum value. According to the Cassie–Baxter model, the “air cushion” effect cannot be formed to reduce the contact between the water and the surface due to the lack of groove structure on the specimen surface, so the measured value of underwater oil droplets is small when D = 2.0228.

3.3.2. Effect of Fractal Parameters on the Rolling Angle

The rolling angle of distilled water on the surface of the X80 steel specimens was measured by the tilt method when the reaction time was 20 min, 30 min, and 45 min. The results are shown in Figure 7.

As exhibited in Figure 7, distilled water had a rolling angle of 37°, 48°, and 42° on the surface of the X80 steel specimens with reaction times of 20 min, 30 min, and 45 min. The highest rolling angle of water droplets occurred when the reaction time was 30 min, demonstrating that it was the most difficult to remove distilled water from the surface of the X80 steel specimen. This was in accordance with the rule in Figure 5a, which stated that the X80 steel specimen with a reaction time of 30 min had the smallest surface contact angle and was most likely to have water attach to it. According to the analysis in Figure 3, the rolling angle of water droplets increased with an increasing fractal dimension, and the maximum fractal dimension D of the X80 steel specimen surface with a 30 min reaction time was 2.0808. Therefore, the larger the surface fractal dimension, the less likely the water droplets were to roll off the pipe wall, making the water ring more stable.

3.3.3. Effect of Fractal Parameters on the Adhesion Work

The surface tension of distilled water at room temperature was 71.68 mN/m, as measured by an automatic tensiometer. Substituting the surface tension and the contact angle into Equation (5), the adhesion work between distilled water and the X80 steel specimens could be calculated. Figure 8 shows the variation of the adhesion work with the reaction time for the distilled water and the X80 steel specimens, and the linear fitting of the fractal dimension and adhesion work for the X80 steel specimens is shown in Figure 9.

As can be seen from Figure 8, the adhesion work between the distilled water and the X80 steel specimens was largest at 117.53 mJ/m² for a reaction time of 30 min. The surface of X80 steel specimen was most firmly adhered to distilled water in the moment.

The linear fitting factor R² of the fractal dimension and the adhesion work for the X80 steel specimens shown in Figure 9 was 0.63826. The adhesion work increased with the increase of surface fractal dimension, and they showed a positive correlation. For the X80 steel specimens, the maximum relative deviation between the fitted value and the calculated value of fractal dimension and adhesion work was 11.38%, the minimum relative deviation was 1.20%, and the average relative deviation was 5.87%.

The cohesion work of distilled water calculated by Equation (6) was 143.36 mJ/m², which was greater than the adhesion work on the X80 steel specimen. The adhesion work between the distilled water and the X80 steel specimen surface was the maximum at D = 2.0808, and the difference between adhesion work and cohesion work was the minimum at this time. As a result, the contact angle of the distilled water on the surface of the X80 steel specimen was small and the binding force with the surface was strong, so the slip phenomenon was not easily generated. This allowed the water to adhere more firmly to the wall of the pipe, and the water ring was more stable. On the contrary, the adhesion work was the smallest and the difference with cohesion work was the largest at D = 2.0808. The mutual attraction of distilled water itself was greater than that on the surface of the specimen in the moment. Thus, distilled water was difficult to spread over the surface, and the large contact angle formed is apt to cause slippage [33].

3.4. Discussion

The stability of the water ring was crucial for the efficiency of heavy oil transportation by the core annual flow. After chemical etching, the surface of the X80 steel had achieved hydrophilic underwater oleophobic with obvious fractal characteristics. Multifractal spectral parameters, both Δf(α) and Δα, provided quantitative and more comprehensive characterization of surface morphology compared with a single fractal dimension. It was necessary to measure the surface contact angle, rolling angle, and adhesion work of the X80 steel specimen. The correlations between these three variables and the fractal dimension verifies that the fractal dimension and water ring stability were inseparable.

4. Conclusions

In this paper, X80 steel specimens with different morphologies were prepared firstly, which had the characteristics of hydrophilic underwater oleophobic. Additionally, the surface morphologies were collected by using the SEM. Then, the surface morphologies of the X80 steel specimens were characterized by introducing the fractal theory and using the MATLAB to calculate fractal parameters. Finally, the water ring stability was indirectly evaluated by exploring the effects of the fractal dimension on the three parameters of contact angle, rolling angle, and adhesion work. The main conclusions were as follows:

(1)

The fractal dimension of the X80 steel specimens first increased and then decreased with the increase of reaction time and was greater than 2, indicating that the surface had distinct fractal characteristics. In addition, the spectral difference Δf(α) was greater than 0, demonstrating that the proportion of the small-scale morphology on the surface of the X80 specimens was large. The homogeneity characterized by the spectral width Δα was in overall agreement with the SEM observations. Therefore, the combination of multifractal spectrum parameters provided a quantitative and more comprehensive characterization of surface morphology compared to a single fractal dimension.

(2)

The contact angle of distilled water on the surface of the X80 steel specimens decreased with the increase of fractal dimension, while the contact angle of underwater oil droplets increased with the increase of the fractal dimension. When the fractal dimension reached the maximum value of 2.0808, the measured contact angle of distilled water on the surface of the X80 steel specimens reached the minimum value of 50.2°, and the contact angle of underwater oil droplets was 166.4° at maximum. Furthermore, both the rolling angle and the adhesion work increased with the increase of the fractal dimension.

(3)

For the core annular flow, the larger the fractal dimension of the pipe surface, the stronger the hydrophilic underwater oleophobic properties of the pipe wall, the more stable the water ring would be. Consequently, increasing the fractal dimension of the pipe surface was beneficial to enhancing the stability of the water ring, and the efficiency of the core annular flow to transport heavy oil would be improved.