Study on Meso-Structural Evolution of Bedrock Beneath Offshore Wind Turbine Foundation in Pressurized Seawater

Abstract

In recent years, offshore wind turbine technology has been widely developed, making a significant contribution to the advancement of renewable energy. Due to the predominant subsurface geological composition characterized by rocky formations in some marine areas, rock-socketed piles are commonly applied as offshore wind turbine foundations. Generally, rock-socketed piles need to be driven into rock layers that have not undergone significant weathering or erosion for optimal load-bearing capacity. This design is essential to ensure structural support for offshore wind turbines. However, during the long-term operation period of offshore wind turbines, the contact surface between the rock-socketed pile and the rock is prone to be detached under multiple dynamic loads. The generated channel makes seawater seep into the unweathered rock layer, resulting in the erosion of rock meso-structure and deterioration of mechanical properties. The reduced load-bearing capacity will adversely affect the operation of the offshore wind turbine. In this study, the meso-structural evolution of bedrock in pressurized seawater is investigated by X-ray CT imaging using tuff samples from the marine areas of an offshore wind farm in China. A cellular automata model is proposed to predict the long-term evolutionary process of tuff meso-structure. Results indicate that the porosity of the tuff sample in the pressurized seawater shows an upward trend over time. Based on the erosion rate of pores obtained from the CT scanning test, the proposed cellular automata model can predict the evolutionary process of tuff meso-structure and corresponding failure strength of the bedrock in the long term.

1. Introduction

The development of offshore wind turbine technology has facilitated offshore wind energy as a crucial component of renewable energy. In marine areas with thick soil layers (e.g., dense sands, silts, or clays), foundations like monopiles, gravity-based structures, and jacket foundations are often utilized for offshore wind turbines [1,2,3]. In contrast, in some marine areas, due to the presence of relatively thin and weak Quaternary sandy layers in the subsurface geology, the utilization of rock-socketed piles has become imperative in offshore wind turbine construction. This is commonly observed along the southeast coast of China, where the bedrock mainly consists of granite, sandstone, tuff, etc. [4]. The rock-socketed piles are required to be directly embedded into unweathered rock strata with superior load-bearing capacity, ensuring the stable operation of the offshore wind turbines. However, the piles are usually subjected to various dynamic loads in the long-term operation period of offshore wind turbines, including the wind turbine loads, wave loads, and wind loads [5,6], as Figure 1 shows. In this case, the pile–rock contact surface is damaged and separated continuously. The immersion of seawater will erode the pile–rock contact surface and the surrounding rock, posing a threat to the safety and long-term stability of the offshore wind turbine foundation. Specifically, bedrock is eroded by seawater through a combination of physical and chemical processes [7]. The constant motion of seawater wears away rock surfaces and open pores. Meanwhile, the salts and other dissolved minerals in seawater react with the mineral components of the bedrock, leading to dissolution or other forms of chemical weathering. Furthermore, many offshore wind turbines are located in deep seawater areas, where the high hydraulic pressure makes the seawater seep into the rock at a faster rate [8,9]. Therefore, it is necessary to evaluate the impact of pressurized seawater on the bedrock in the long-term operation period of offshore wind turbines.

Field and experimental studies have shown that rocks are in contact with water in a variety of conditions [10]. The resulting deterioration of rock mechanical properties has been widely reported [11,12,13,14]. For example, Zhang et al. [4] carried out a series of tests to analyze the effect of seawater on the rock, finding that the compressive strength and slake durability index of rock experienced a decline in the seawater. Liu et al. [15] studied the effects of water pressure on the mechanical properties of the rock and reported the decreased strength of the rock with the increasing water pressure. However, the deterioration mechanism of rock mechanical properties in the pressurized seawater, that is the meso-structural evolution of the rock, is usually ignored in previous studies.

Limited studies have focused on the meso-structure of the rock in the water or seawater by different analytical methods [16]. Among these analytical methods, the X-ray computed tomography (CT) technique [17,18,19] is commonly applied to analyze the distribution of pores and fractures in the rock mass. The utilization of the CT technique can realize the non-destructive and three-dimensional imaging of rock samples. It is helpful for understanding how seawater and pressure conditions affect bedrock at a meso-level. In previous studies, researchers mainly focused on the microcrack growth in the rock samples under high water pressure by CT, and the samples were immersed in the water for a short time. However, the meso-structural evolution of the rock, especially the change in pores in the pressurized seawater, is a long-term process. Most of the existing studies pay little attention to the evolution in the morphological characteristics of pores from a longer time dimension because this is rarely a concern in traditional rock engineering projects. With the rapid development of offshore wind power projects in recent years, engineers are increasingly giving importance to the enduring erosive impact of seawater on bedrock. Therefore, the long-term evolutionary process of the pores in the pressurized seawater continues to be an open problem.

Relying solely on the experimental studies to study the long-term meso-structural evolution results of rock in the pressurized seawater is cost-prohibitive. Combining experimental results spanning a period, typically ranging from several weeks up to a couple of months, with numerical methods provides a practical approach to forecasting long-term changes. The simulation results of the meso-structure in the long term can further serve as a foundation to predict the mechanical properties of rock, which can be used for the design of rock-socketed piles. The conventional methods used in the previous studies for the simulation of meso-structural evolution included mesh-based methods and mesh-free methods. As two common mesh-based methods, Finite Difference Method (FDM) and Finite Element Method (FEM) were often used in the simulation of meso-structural evolution [20,21], but the disadvantage of mesh distortion was difficult to solve when large material deformation occurred. Different from mesh-based methods, rock was regarded as a discontinuous medium in the discrete particle methods, such as the Discrete Element Method (DEM) [22,23,24,25], but it was computationally expensive and possibly failed for large-scale models. Compared with these numerical methods, cellular automata have been recognized as an effective method to simulate the meso-structural evolution and failure process of rock mass [26,27], which could reflect the heterogeneity, discontinuity, and anisotropy of rock materials with lower computational cost [28,29,30,31].

In summary, previous research has primarily focused on the short-term effects of seawater on rock mechanical properties, but the impact of long-term exposure to pressurized seawater has not been thoroughly investigated. While these studies utilized methods such as X-ray computed tomography to examine microcrack growth under high water pressure, they have not adequately considered the long-term meso-structural evolution of pores in pressurized seawater. Therefore, this study aims to analyze the meso-structural evolution of the bedrock beneath offshore wind turbine foundations in pressurized seawater. A typical lithology from the marine regions of an offshore wind turbine farm in China is collected to investigate the meso-structural evolution. The resulting meso-structural evolution will be used to develop a numerical model to predict rock mechanical properties in the long term. The remainder of this study is organized as follows. In Section 2, methodology and materials are discussed, including the preparation of rock samples and the CT scanning test. In Section 3, the evolutionary process of meso-structure in the rock will be analyzed in detail from the perspective of 2D and 3D, respectively. A cellular automata model is developed to predict the long-term meso-structural evolution and the deterioration of rock mechanical properties in Section 4.

2. Methodology and Materials

2.1. Preparation of Rock Samples

Eastern Fujian Province, coastal SE China is an important area for the development of offshore wind turbines owing to the abundant wind resources [32]. As Figure 2 shows, the Xinghua Bay Offshore Wind Farm is a prominent offshore wind energy farm in this marine area, comprising 59 turbines with a combined installed capacity of 357.4 megawatts [33]. Tuff from the second member of the Jurassic Nanyuan formation (J₃n^b) is widely found in this marine area [34], so it is selected and processed as rock samples for this study.

As Figure 3 shows, cube samples with each side length of 30 mm are used for the CT scanning test. Part of the tuff is sliced into thin sections for optical microscopy analysis. Thin section identification results show that the tuff sample is named crystalline lithic tuff, which shows typical features of volcanic tuff with variegated structure. The tuff sample contains many components, including crystal pyroclast, cuttings, and vitroclastic texture.

Table 1. Classification of mineral composition of tuff.

Mineral Classification		Content
Crystal pyroclast	Plagioclase	15%
	Quartz	10%
	Alkali feldspar	8%
	White mica	4%
	Chalcedony	3%
Cuttings		35%
Vitroclastic texture		15%
Interstitial material		10%

lists the specific classification and proportion of mineral composition in the tuff. The seawater used in the immersion tests is also taken from the marine area surrounding this offshore wind turbine farm. The density of the seawater is 1.045 kg/m³, and

Table 2. Ion composition in seawater.

Test Items	Unit	Value	Limit of Detection
K+	mg/L	539	0.05
Ca2+		441	0.02
Na+		8.70 × 103	0.03
Mg2+		1.05 × 103	0.003
Cl−		1.76 × 104	0.007
SO42−		4.63 × 103	0.018
CO32−		5.2	0.3
HCO32−		104	0.6

lists the main ion components in the seawater.

Three environmental conditions, including pure water at ordinary pressure, seawater at ordinary pressure, and pressurized seawater, are set, respectively, to study the effect of environmental medium and pressure. The ordinary pressure is denoted as OP in the following discussion. The pressurized immersion test is conducted in a seawater simulation system, as Figure 3 shows. This system consists of a stainless-steel cylinder with a diameter of 500 mm and a depth of 1200 mm. The electronic control system in the seawater simulation system can control pressure and time. The air compressor is used to provide the system with a pressure between 0 and 0.7 MPa. Considering that the seawater depth in the current offshore wind power project area is generally less than 50 m [35], the seawater pressure of 0.5 MPa is set as the pressure value for the simulation of pressurized seawater.

2.2. CT Scanning Test

The equipment used in the CT scanning test is a Yxlon X-ray tomography scanner, including an advanced X-ray source, a stage, and a flat-panel detector. The X-ray source can be configured with either a single or a dual tube, allowing for flexibility in the energy and intensity of the X-rays used during scanning. The stage is precisely controllable for accurate sample positioning, ensuring that images are taken at consistent angles and positions during the testing period. The flat-panel detector is designed for high resolution, which can detect fine details within the tuff samples. The scanner’s technical specifications include a range of adjustable parameters to optimize the imaging process.

Table 3. Technical parameters of Yxlon X-ray tomography scanner.

Model	Y. CT Precision S
Operation modes	CT	Volume scan (Cone beam geometry)
	DR	Digital radiography
Scan time	CT	Typical 10~30 min
	DR	Max 7.5 frames/sec
Max. tube power	320 W
High voltage range	10~225 kV
Tube current	0.01~3.0 mA
Pixel number	10242 Pixel

lists the main technical parameters of the equipment. The voltages and currents for the X-ray source can be fine-tuned to enhance contrast and resolution based on the density and composition of the samples. The detector has 1024 pixels along each dimension. This high number of pixels allows for detailed images with high spatial resolution, enabling the detection and analysis of small features in the scanned samples.

To investigate the meso-structural evolution of tuff in the three environments, three tuff samples are scanned in the CT scanning test before being immersed. Subsequently, these three tuff samples are immersed in the three environments, respectively. CT scanning tests are carried out again when three tuff samples are immersed for 30 days and 60 days, respectively.

Original CT images need to be preprocessed to make the subsequent analysis more accurate. Firstly, the position of the tuff samples is not completely consistent in each CT scanning test, resulting in slight changes of the same slice in different tests, so CT slices of the tuff samples in the second and third tests need to be adjusted based on the 3D outer contour of the tuff samples obtained in the first test. Specifically, the coordinates of the cubic sample’s edges are recorded from the initial CT scanning 3D model using an automated edge detection technique. In subsequent tests, rotational and translational transformations are applied to match the edge coordinates of 3D models in the later scans with that from the first test. Secondly, some tiny white noise points that exist in the CT images may interfere with the analysis of the images, so they need to be processed based on denoising methods, such as the Sigma filter, non-local mean filter, and median filter [36,37]. The median filter is applied herein to eliminate noise points from the CT images. It is realized by moving through the CT image pixel by pixel, replacing each value with the median value of neighboring pixels within a defined window. Thirdly, the defect of the CT scanner detector and the inadequate warm-up of the CT scanner before scanning may lead to black ring artifacts in some of the CT images, which also need to be removed before analysis. A correction algorithm is utilized to remove the black ring artifacts. This method can identify the concentric rings by their characteristic patterns and then mitigate their impact by interpolating the intensity values based on the surrounding image data. Details of artifact removal can be referred to in the literature [38,39].

3. Analysis of Meso-Structural Evolution of Tuff

3.1. Two-Dimensional Analysis Based on CT Slices

Figure 4 shows representative CT slices of three tuff samples before they are immersed in the three environments. These CT slices have been processed according to the procedures described in Section 2.2. In the grayscale images, the area with colors closer to white and black represents higher and lower density, respectively. The complex color distribution in the grayscale images indicates that many kinds of minerals exist in the tuff samples. This is consistent with the detection result reported in thin section identification. Although some of the pores can be observed in Figure 4, the shape and size of these pores are not clear in the grayscale images due to their small size. In this case, the evolution of pores is harder to notice due to the limitation of vision recognition.

The gray histogram provides a way to analyze the changes in grayscale images, which represents the number of pixels with a certain gray level in the CT slice images. Since the pores of the tuff samples at some positions are changed in the three environments, the overall change in gray value could be observed in the gray histogram, as Figure 5 shows. The abscissa represents the gray values, and the ordinate represents the frequency of the corresponding gray value. The red and blue areas in the gray histogram represent the increment and decrement of corresponding gray values, respectively. It should be noted that the total increment is equal to the total decrement because the total number of pixels remains unchanged. Since the ordinate of the gray histogram is expressed in logarithmic form, the area of the red columns and that of the blue columns in the gray histogram are not consistent. Figure 5 shows that the number of pixels with small gray values experiences an increasing trend on the 30th and 60th day in the three environments, indicating that the area corresponding to these pixels in the three tuff samples becomes larger. As mentioned before, a small gray value means the lower density in the specific region. This suggests that the pores in these three tuff samples are enlarged after immersion. Specifically, after 30 and 60 days of immersion in pure water, 0.673% and 0.704% of the pixels changed, respectively. This increase suggests a gradual progression of water-related effects on the sample’s structure over time. Similarly, when the tuff sample was immersed in seawater (OP), the pixel changes were pronounced, with 1.256% for the 30-day period and 1.312% after 60 days. For the sample immersed in pressurized seawater, the pixels changed by 0.876% and 0.982% for the 30-day and 60-day periods, respectively. It is worthwhile to mention that the variation in the percentage of pixel changes observed in the three environments may be attributed to the selection of CT slices. Some samples may have fewer pores susceptible to erosion at the specific slice locations chosen for analysis. This heterogeneity in pore distribution means that certain sections of the sample may exhibit different degrees of change when exposed to water or seawater.

3.2. Three-Dimensional analysis Based on Pore Model

Although the evolution of pores has been analyzed by the gray histogram in Section 3.1, it is limited to two dimensions and cannot reveal the distinct state of pores at different spatial positions in the rock samples. Therefore, 3D structural analysis is necessary to investigate the evolution of pores quantitatively. The pore model can be obtained by 3D reconstruction technique [40]. The changes in the total pore volume in the tuff samples are analyzed first based on this 3D pore model.

Table 4. Total pore volume and porosity of tuff at different times under three environments.

Environment	Time	Pore Volume/mm3	Porosity/%	Amplification/%
Pure water (OP)	Before immersion	413.52	1.53	---
	30th day	419.83	1.55	1.53
	60th day	424.17	1.57	2.58
Seawater (OP)	Before immersion	347.61	1.29	---
	30th day	353.09	1.31	1.58
	60th day	356.18	1.32	2.47
Pressurized seawater (0.5 MPa)	Before immersion	362.23	1.34	---
	30th day	381.76	1.41	5.39
	60th day	393.51	1.46	8.64

lists the total pore volume and porosity of tuff samples in the three environments at different times. The porosity of the tuff samples in the pure water (OP) and seawater (OP) has a similar growth rate after being immersed for 30 days and 60 days, while that in the pressurized seawater increases faster at the same time. This suggests that the effect of pure water and seawater on the tuff meso-structure is similar, and the erosion degree of the meso-structure increases with the increasing pressure. Specifically, the tuff sample in the pressurized seawater shows an amplification of pore volume of 8.64% after being immersed for 60 days. This tuff sample will be analyzed in detail in the following discussion.

The tuff sample immersed in pressurized seawater contains large numbers of pores. To simplify the analysis, only the pores with a volume larger than 0.1 mm³ are extracted, as Figure 6a shows. It is obvious that no large-scale fractures exist in this tuff sample, and water can only seep into the pores that are connected to the six surfaces. These pores are defined as effective pores in this study, which are marked in red in Figure 6. There are nine effective pores in this 3D pore model. These nine effective pores are numbered from one to nine from the largest to the smallest volume. Meanwhile, this tuff sample contains large numbers of non-effective pores that are not affected by water or seawater, which are marked in gray in Figure 6. Since the proportion of non-effective pores is much larger than that of effective pores, the limited amplification of the total pore volume of the tuff sample in the pressurized seawater can be explained.

After analyzing the change in total pore volume, the evolutionary process of each effective pore is investigated. Each effective pore is composed of many small pores that are collected with each other. To visually analyze these small pores in each effective pore, pore network models (PNMs) are applied in this study. By identifying the topological skeleton, i.e., the pore bodies and throats, the complex pore models can be simplified as an equivalent network where relatively large pore bodies at the nodes of the network are connected by smaller pore throats [41]. Figure 6b shows the PNMs of these nine effective pores in the tuff sample. Spherical models in the PNM represent the small pores that each effective pore contains, and stick models represent the throats that connect these small pores. The spatial arrangement and size distribution of these spheres and sticks within the network are important as they represent the permeability and flow characteristics of each pore. It can be found that the No. 1 effective pore contains the largest number of small pores among these nine effective pores. This complexity suggests a more complex pathway for seawater flow and potentially a higher level of interaction with the seawater. By contrast, the No. 6 effective pore can be simplified as a pore network model that is only collected by two small pores. The simplicity of this network may indicate a distinct erosion pattern relative to the more intricate pore networks.

Figure 7 shows the volume of these nine effective pores and the corresponding growth rate at different times. Among these nine effective pores, the volume of the largest (No. 1) and the smallest (No. 9) effective pores reaches 16.7 mm³ and 1.015 mm³, respectively, before the tuff sample is immersed in the pressurized seawater. The growth rate of each pore volume is not equal at the same time. The volume growth rate of the No. 2 effective pore is the highest on the 30th day and 60th day, reaching 87.41% and 165.97%, respectively. Taking the No. 1 effective pore as an example, the cause of unequal pore volume growth rate of these effective pores for the same immersion time is analyzed. As Figure 8 shows, No. 1 effective pore is composed of 55 small pores before the tuff is immersed, among which 15 pores with volume between 0.1 mm³ and 0.2 mm³ account for the largest proportion. Six more pores are included in this effective pore after the tuff sample is immersed for 30 days, indicating that this effective pore is connected to the surrounding independent pores when eroded in the seawater. Therefore, the volume growth of the No. 1 effective pore can be divided into two parts, including the growth of small pore volume caused by seawater erosion, and the increase in the number of small pores contained in the original effective pore. In the following discussion, the volume growth of an original effective pore caused by seawater erosion is defined as the real increment of pore volume. The volume growth of this original effective pore caused by the connection with surrounding small pores is defined as the fake increment of pore volume. The sum of the real increment of pore volume and the fake increment of pore volume is defined as a nominal increment.

When the reason for the volume growth of effective pores is known, the erosion rate of effective pores can then be calculated. It is one of the important indicators to predict the long-term meso-structural evolution of the tuff. One of the representative effective pores, the No. 8 effective pore, is extracted from a 3D pore model to show the calculation of the erosion rate at different times. Figure 9 shows the 3D models of the No. 8 effective pore at different times. When the tuff sample is immersed in the pressurized seawater for 30 days and 60 days, the volume in some positions, such as position A marked in Figure 9b,c, becomes larger. The increment in this region is the real increment of pore volume for the original No. 8 effective pore. Meanwhile, some positions also begin to connect with the surrounding independent pores, such as position B marked in Figure 9c. This increment is the fake increment of the pore volume.

To quantify the real erosion rate of the No. 8 effective pore and exclude the interference of surrounding pores, the real increment of pore volume at each time is calculated based on a series of Boolean operations. Denote the No. 8 effective pore before erosion as P, and after erosion as P′ in the following discussion. Figure 10 shows the specific process for the extraction of the real increment of pore volume. Key steps are summarized as follows.

Step 1: Initiate the procedure by generating a nominal increment model of P. This is accomplished by subtracting the pre-erosion 3D model of pore P from the post-erosion 3D model of pore P′. The result of this subtraction is a model that represents not only the actual enlargement of pore P but also includes any adjacent pores that have become connected due to the erosion process.

Step 2: Perform the intersection operation between the nominal increment model of P after seawater erosion and the total pore model before seawater erosion. The result of this operation is a model that represents only those independent pores that have merged with pore P due to erosion. This step is critical as it identifies and separates the new connections formed from the original volume increase.

Step 3: Subtract the identified independent pores that have been connected after seawater erosion in Step 2 from the obtained nominal increment model of P after seawater erosion in Step 1. The resulting model represents the real increment of P after seawater erosion.

This sequence of Boolean operations ensures that the calculated volume increment is a true measure of erosion on the specific pore. It removes effects from other adjacent pores and provides a foundation for the precise measurement of erosion rate. After extracting the real increment model of P, its real erosion rate in the pressurized seawater is calculated based on the real volume increment. The 3D model of the real increment of P consists of many local pore increments that have different shapes, so the real normal rate v of P eroded by seawater can be expressed by Equation (1):

$$v=\frac{{\displaystyle \sum }{V}_i}{S\times T}$$

(1)

where v (μm/d) is the real normal rate of P eroded by seawater; T (d) is the seawater erosion time; V_i (μm³) is the volume of local pore increment i after being eroded by seawater for T days; and S (μm²) is the total surface area of P before seawater erosion.

Results show that the total surface area of P before the seawater erosion is 2.382 × 10⁷ μm², and the total real incremental volume of P after the seawater erosion of 30 and 60 days is 7.190 × 10⁸ μm³ and 1.186 × 10⁹ μm³, respectively. Therefore, the real normal erosion rate v of P is 1.006 μm/d and 0.830 μm/d on the 30th and 60th day, respectively.

4. Prediction of Long-Term Deterioration by Cellular Automata

4.1. Model Concept and Governing Equations in Cellular Automata

The cell is the basic component of the cellular automata, which is distributed on a grid of discrete space. The state of each central cell depends on its state and that of its neighbor cells, and the common neighbor types include Von Neumann type, Moore type, and extended Moore type [28]. The evolution rule of cellular automata can be regarded as a dynamic function, which can capture the state of the cell at the next moment based on the current state. The change in a specific cell will affect the state of its neighbor cells, resulting in changes in the whole model, so the evolution rule of the model could be explored from the changing patterns.

The state of cells in the rock could be divided into “living” and “dead” in the traditional cellular automata, where the “living” state with a value equal to 1 represents the rock, and the “dead” state with a value equal to 0 represents pores or fractures. Only the pores that are connected to six surfaces of the tuff sample could be eroded by the seawater, so “dead” cells are also divided into “dead-I” and “dead-II” in the proposed cellular automata model. “Dead-I” represents those effective pores or fractures that could be eroded by the seawater. “Dead-II” represents the other non-effective pores or fractures. Figure 11 shows the process of simulating pore erosion by the cellular automata. The key steps required to develop a pore erosion code are summarized as follows.

Step 1: Connectivity analysis. A function is created to find the connected regions in binary images. These connected regions represent pores, including non-effective pores and effective pores, in the rock samples. They are labeled with different numbers larger than 1.

Step 2: Pore type discrimination. Based on the coordinates of each pore, the effective pores are sifted and the value of each element in these pore regions is set to 0, which represents the value of each dead-I cell.

Step 3: Edge detection. The edge of different pores or fractures is detected by the methodology of Canny edge detection.

Step 4: Pore erosion. Based on the results of edge detection, the edges of the effective pores are extended at a specific erosion rate v, so that the living cells with a value equal to 1 are transferred to the dead-I cells with a value equal to 0.

Step 5: Pore type transition. Once the non-effective pores are detected to be connected to the effective pores at some specific time step, the dead-II cells in these pores with values larger than 1 are transferred to the dead-I cells with values equal to 0.

Based on the results of the pore erosion in the cellular automata model, the numerical simulation of the uniaxial compressive test is carried out. “Dead-I” cells and “dead-II” cells are combined to dead cells with value 0 in the binary matrix, and the value of living cells remains unchanged. Failure thresholds E_max(x_i, y_j) of each cell are set based on the function of Weibull distribution [27], which can be expressed by Equation (2):

$$\phi \left(E_{max}\right)=\frac{m}{{\overline{E}}_{max}}{(\frac{E_{max}}{{\overline{E}}_{max}})}^{m-1}{e}^{-{(\frac{E_{max}}{{\overline{E}}_{max}})}^m}$$

(2)

where $\phi (E_{max})$ is the distribution probability density function; E_max is the failure thresholds of cells; ${\overline{E}}_{max}$ is the average failure thresholds of cells; and m is the homogeneity index of rock.

The effect of seawater is also included in the model by adding the reduction coefficient f(T) of the failure thresholds, which can be expressed by Equation (3):

$$E_{max}\left(T\right)=E_{max}\left(T_0\right)·f\left(T\right)$$

(3)

where E_max(T) is the failure threshold of cells after being immersed in seawater for T days; E_max(T₀) is the failure threshold of cells in the initial state; and f(T) is the reduction coefficient of the failure thresholds after being immersed in seawater for T days.

Assuming that the initial energy of each cell is zero and the energy is input at every time step, the living cells are transferred to the dead cells once the total energy E(x_i, y_j) is up to E_max(x_i, y_j). Part of the energy is dissipated in the form of heat energy and acoustic emission in the process of energy transfer at every time step, and the other energy is transferred to the neighboring cells. When the number of dead cells reaches a certain proportion, the rock is considered to be unstable and the simulation is terminated.

In a 2D cellular automaton model with m × n cells, the effective bearing coefficient of rock at time t can be expressed by Equation (5) [31]:

$$D\left(t\right)=\frac{{\displaystyle \sum }{E}_{max}\left(x_i,y_j\right)-B\left(t\right)}{\alpha ·m·n}$$

(4)

where D(t) is the effective bearing coefficient of rock at time step t; ΣE_max(x_i, y_j) is the sum of failure thresholds of all cells; B(t) is the sum of failure thresholds of all dead cells; α is the maximum value of failure threshold in the model; m and n is the length and width of the model, respectively.

Assuming that the elastic modulus of the undamaged tuff is e, the elastic modulus and stiffness of the model at time t can be determined by Equations (5) and (6):

$$e\left(t\right)=e·\frac{{\displaystyle \sum }{E}_{max}\left(x_i,y_j\right)-B\left(t\right)}{\alpha ·m·n}$$

(5)

$$K\left(t\right)=\frac{e\left(t\right)·m·n}{l}$$

(6)

where e is the elastic modulus of the undamaged tuff; e(t) is the elastic modulus of the model at time t; K(t) is the stiffness of the model at time t; and l is the axial length of the tuff. In the 2D cellular automaton model, the ratio of the cross-sectional area m × n to the axial length l can usually be expressed by a constant C.

Based on the stress–strain curve obtained from the conventional uniaxial compression test in the existing research and the equivalent relationship between the elastic strain energy stored before unloading and the input energy, the stress and strain of the model at time t can be expressed by Equations (7) and (8):

$$\sigma \left(t\right)=\sqrt{2K\left(t\right){\displaystyle \sum }E\left(t\right)}$$

(7)

$$\epsilon \left(t+1\right)=\epsilon \left(t\right)+\frac{2E\left(t\right)}{\sigma \left(t\right)+\sigma \left(t+1\right)}$$

(8)

where σ(t) is the stress of the model at time t; ΣE(t) is the sum of input energy in the model until time t; ε(t + 1) is the strain of the model at time t + 1; ε(t) is the strain of the model at time t; E(t) is the input energy in the model at time t; and σ(t + 1) is the stress of the model at time t + 1.

4.2. Numerical Results and Discussion

Table 5. Input parameters for the cellular automata model.

Notation	Value	Unit	Description
m	250	1	Number of cells per row
n	250	1	Number of cells per column
a	0.2	mm	Length of each cell
v	1	μm/d	Normal erosion rate
e	45	GPa	Elastic modulus of undamaged tuff
m	5	1	Homogeneity index of rock
${\overline{E}}_{max}$	3	1	Average failure thresholds of cells
E(t)	1	1	Input energy in the model at time t
f(T)	f(T)	1	Reduction coefficient of the failure thresholds

lists the input parameters of the cellular automata model. To satisfy the requirement of computational efficiency and accuracy, the size of each cell is set to 0.2 mm, which means that the normal erosion length that is an integer multiple of 0.2 mm could be detected in the cellular automata model. Based on the erosion rate obtained from the CT scanning test in Section 3.2, the erosion rate of pores v of the cellular automata model is assumed as a constant value at 1 μm/d to simplify the calculation. The choice of a 0.2 mm cell size in conjunction with an erosion rate of 1 μm/d allows for the observation of meso-structural changes every 200 days.

The value of the elastic modulus of the undamaged tuff e, homogeneity index of rock m, and average failure thresholds of cells ${\overline{E}}_{max}$ are set according to the experimental results in the literature [42,43,44,45,46,47]. Input energy E(t) in the model at every time step represents the loading rate, which is controlled at a low value to simulate the actual loading process. The reduction coefficient of the failure thresholds f(T) can be expressed by Equation (9) according to the mechanical experiment of tuff in the pressurized seawater performed by Zhang et al. [4]:

$$f\left(T\right)=1-0.039\ln \left(1+0.975T\right)$$

(9)

where T is the seawater erosion time.

The failure process of the tuff which has been immersed in the pressurized seawater for 3 to 60 days is simulated first. The results are compared with those obtained in the experiment [4] to verify the accuracy of the model parameters, as Figure 12 shows. There is no relative error larger than 10%, so the validation of the model can be confirmed. The meso-structure and stress–strain relationship of the tuff when it is eroded by the seawater for 200, 400, 600, 800, and 1000 days are analyzed. Figure 13 shows the state of each cell in the cellular automata model at a corresponding time. The pores that are connected to four sides of the tuff have been eroded by the seawater, and the proportion of the number of dead-I cells, or the proportion of effective pores, increases from 0.1504% before erosion to 1.152% on the 1000th day. Figure 14 shows the stress–strain relationship of the tuff, in which the failure strength shows a downward trend when the erosion time increases. The decreasing amplitude is the largest in the first 200 days because the failure threshold of the cells is reduced sharply by seawater. On the whole, the maximum stress of the tuff reduces by about 27.3% from the initial state to the 1000th day, so the deterioration effect of pressurized seawater on the mechanical properties of the tuff beneath the offshore wind turbine foundations cannot be ignored in the long term.

To analyze the sensitivity of the erosion rate to the prediction results, the relationship between the erosion rate and both the proportion of effective pores and failure strength in the 1000th day is analyzed, as Figure 15 shows. As the erosion rate increases from 1.0 to 3.0 μm/d, the proportion of effective pores experiences an upward trend from 1.15% to 5.73%. It indicates the significant impact of erosion rate on the pore structure. Meanwhile, it should be noted that the slope indicating the increase in the proportion of effective pores becomes steeper with rising erosion rates. This trend suggests that a higher erosion rate results in a greater number of originally isolated pores becoming interconnected within the same immersion time. Correspondingly, the failure strength, denoted by the line with triangle markers, shows a decrease as the erosion rate increases. This trend reveals a negative correlation between the erosion rate and the strength of the sample. As the erosion rate increases from 1.0 to 3.0 μm/d, the model predicts a decrease of approximately 8.9% in failure strength. This suggests that the bedrock becomes weaker under a higher erosion rate.

In practical engineering projects, the reduced mechanical strength of bedrock suggests a need for a reinforced bearing capacity design of foundations to ensure stability and safety over the lifespan of an offshore wind farm. Existing geological classification systems used for the design of offshore wind turbine foundations, such as rock mass rating (RMR), can be adjusted to consider the deterioration effect of pressurized seawater on bedrock. Specifically, rock strength is one of the important evaluation indices in geological classification systems. It is typically assessed without considering seawater erosion in recent offshore wind turbine foundation designs. According to the predicted rock strength in the cellular automata model, a revised bedrock strength is advised to adjust the original rock mass classification and further optimize the bearing capacity design of foundations.

5. Conclusions

The bedrock beneath turbine foundations plays an important role in the stable operation of offshore wind turbines. During the long-term operational period of wind turbines, dynamic loads acting on the foundation cause the formation of pathways through the rock layers, allowing seawater infiltration into the fresh bedrock. This pressurized seawater subsequently induces erosion in the bedrock, adversely impacting the foundation’s load-bearing capacity. This study innovatively investigates the long-term meso-structural evolution of bedrock beneath offshore wind turbine foundations in pressurized seawater, which is a critical factor that has not been studied before in wind turbine foundation design. In this study, tuff samples from the marine area of an offshore wind farm in China are collected to analyze the evolutionary process of pore structure by CT scanning tests. Two-dimensional analysis of the pore structure is carried out based on CT slices. The gray histogram is used to capture the change in CT slices at different times and reveal the erosion of pores in the pressurized seawater. Three-dimensional analysis of the pore structure is performed by developing the pore model. The total pore volumes and the morphological change in each effective pore at different times are analyzed, respectively. A cellular automata model is developed to predict the meso-structural evolution and the deterioration of the failure strength of the tuff in the long term. Results indicate that the total pore volume of the tuff sample in the pressurized seawater increases over time. This growth of porosity mainly comes from the erosion of effective pores in the tuff sample. The real normal erosion rate of the effective pore is successfully calculated by extracting the real increment component of the effective pore after erosion using a sequence of Boolean operations. The proposed cellular automata model can accurately predict the meso-structural evolution of tuff and deterioration of failure strength in the long term, which can provide support for the bearing capacity design of offshore wind turbine foundations.