Graph theory-based moisture and ion transmission behavior analysis method under internal connected crack network of cement-based material

Abstract

The invention belongs to the field of material science and engineering, and particularly relates to a graph theory-based method for analyzing moisture and ion transmission behaviors under an internal connected crack network of a cement-based material, which comprises the following steps of: 1, preprocessing internal crack network data of a cement-based material sample, and extracting geometrical characteristics of cracks; 2, a topological model of the cracks is constructed, nodes represent crack intersection points, edges represent crack channels, and the weights of the edges are calculated according to a penetration capacity formula; step 3, key indexes of connectivity, shortest path, average path length and global efficiency of the crack network are calculated through a graph theory analysis method, and key paths and nodes in the network are identified; and 4, simulating moisture transport and ion diffusion behaviors by combining a Darcy law and a Fick diffusion law, and quantifying the transmission efficiency and diffusion characteristics of the crack network. By combining a physical transmission model and topological characteristic analysis, the water flow and ion diffusion rules are disclosed accurately.

Description

Technical Field

[0001] The present invention belongs to the field of materials science and engineering, and specifically relates to the study of water and ion transport behavior under an internal connected crack network of cement-based materials, and is particularly suitable for the analysis and mechanism revelation of water penetration and ion diffusion processes in complex crack networks. Background Art

[0002] Crack networks are widely present in water-stabilized macadam, geotechnical materials and other porous media, and their geometric characteristics and connectivity have a decisive influence on the transport behavior of water and ions. In cement-based materials, the distribution and connectivity of crack networks directly determine the laws of water flow and ion diffusion, thereby affecting the long-term stability and durability of the materials. However, current research methods are mostly based on the assumption of macroscopic homogeneity, and there are still significant deficiencies in the study of complex crack networks.

[0003] In the prior art, the research methods of fracture networks are mainly focused on geometric modeling and physical transport simulation. The study of water flow based on Darcy's law and the study of ion diffusion based on Fick's diffusion law are commonly used theoretical frameworks, but these methods usually assume that the medium is a homogeneous continuous structure, which makes it difficult to effectively describe the heterogeneity and connectivity characteristics of the fracture network. The structural characteristics (such as length, width, direction and intersection position) and topological characteristics of complex fracture networks play a key role in transport behavior, while traditional methods lack quantitative descriptions of these characteristics. In addition, fracture connectivity and percolation effects, as important factors affecting the transport of water and ions, are still lacking in systematic research, especially the mechanism of the change in transport efficiency under the critical connectivity state has not been fully revealed.

[0004] At the same time, the identification of key paths and bottleneck paths in existing fracture network research is mostly limited to geometric analysis, and fails to combine physical transmission models and topological characteristics to comprehensively evaluate the key nodes and channels of the fracture network. This makes it difficult to accurately and comprehensively draw conclusions on transmission efficiency, ion diffusion range, and key path characteristics in the study of the permeability behavior of complex fracture networks.

[0005] In summary, the existing research methods have significant deficiencies in the following aspects: first, they fail to comprehensively combine the geometric characteristics and topological structure of the fractures to accurately model the transmission behavior; second, there is a lack of systematic research on the connectivity and percolation effect of the fracture network, especially the lack of clear quantification of the impact of critical connectivity on transmission efficiency in complex networks; third, the evaluation and identification methods of key paths, nodes and bottlenecks in the fracture network are relatively simple, making it difficult to achieve multi-level analysis of the transmission behavior of complex networks. To this end, the present invention proposes a complex fracture network transmission behavior analysis method based on graph theory, which accurately reveals the laws of water flow and ion diffusion by combining physical transmission models with topological characteristic analysis, and provides a new theoretical framework and technical support for fracture network research. Summary of the invention

[0006] The purpose of the present invention is to provide a method for analyzing the water and ion transport behavior in a connected crack network inside a cement-based material based on graph theory, and to accurately reveal the laws of water flow and ion diffusion by combining a physical transport model with topological characteristic analysis.

[0007] The technical solution adopted by the present invention is as follows:

[0008] The method for analyzing the water and ion transport behavior of the internal connected crack network of cement-based materials based on graph theory includes the following steps:

[0009] Step 1: First, preprocess the internal crack network data of cement-based material specimens to extract the geometric characteristics of the cracks;

[0010] Preferably, step 1 includes the following specific steps:

[0011] Step 101: Sample Preparation

[0012] Use 3D CT scanning technology to obtain the geometric parameters of the crack, pre-process the sample, and cut the sample size according to the specifications of the scanner; mark the coordinate points on the sample surface;

[0013] Step 102: 3D CT Scan

[0014] Use industrial CT scanning technology to perform three-dimensional scanning of the sample to obtain the internal structure data of the crack;

[0015] Step 103: Image reconstruction

[0016] The original projection data obtained by scanning is reconstructed into a three-dimensional image through a filtered back-projection algorithm to generate a three-dimensional grayscale distribution map of the crack and the surrounding matrix; the grayscale value of each pixel reflects the density difference of the material, and the data is corrected, including removing artifacts, optimizing contrast and resolution operations;

[0017] Step 104: Crack region segmentation

[0018] Firstly, a threshold value T is set based on the significant difference in grayscale between cracks and matrix materials using the threshold segmentation method. Pixels with grayscale values ​​lower than T are marked as crack areas, while pixels with grayscale values ​​higher than T are marked as matrix areas. Then, the segmented crack pixel groups are divided into independent crack bodies using the three-dimensional connectivity analysis algorithm.

[0019] Step 105: Extract crack geometry parameters

[0020] The crack length is calculated by the linear distance along the crack main axis, which can be obtained by principal component analysis (PCA):

[0021]

[0022] Among them, λ1 is the maximum eigenvalue of the crack point cloud covariance matrix;

[0023] The width of the crack W kd Obtained through direct observation under a microscope;

[0024] The crack directions are extracted by principal component analysis:

[0025] D=(cosθ x ,cosθ y ,cosθ z )

[0026] where θ x ,θ y ,θ z are the angles between the crack principal axis and the x, y, and z coordinate axes, respectively;

[0027] The crack density ρ is defined as the number of cracks per unit volume, which can be expressed by the formula ρ = N / V tj Calculate, where N is the total number of cracks, V tj is the sample volume; finally, the intersection position of the crack is extracted (x i ,y i ,z i ), record the three-dimensional coordinates of the endpoints and intersection points of each crack.

[0028] Step 2: Construct a topological model of the crack, where nodes represent crack intersections, edges represent crack channels, and edge weights are calculated based on the permeability formula;

[0029] Preferably, it is characterized in that: in the step 2, the topological model of the crack network is constructed by abstracting the geometric structure of the three-dimensional crack into a graph theory model; first, the crack intersection position (x i ,y i ,z i ) and the geometric characteristics of the crack channel, the topological structure of the crack network is established as a weighted undirected graph; the specific construction process includes the following steps:

[0030] Step 201: Define Nodes

[0031] The intersection or endpoint of the crack is defined as the node of the topological model, denoted by V; the three-dimensional spatial position of each node is represented by the coordinates (x i ,y i ,z i ) is determined; the intersection data comes from the intersection extraction result after the crack area is segmented, and the number of nodes is equal to the total number of crack intersections;

[0032] Step 202: Define the edges

[0033] The crack channel is defined as an edge in the topological model, denoted by E; each edge connects two nodes v i and v j , corresponding to the actual connectivity of the crack in three-dimensional space, the geometric properties of the edge include the length L and the direction vector D ij =(x j -x i ,y j -y i ,z j -z i ), the direction is calculated by the coordinate difference of the two nodes;

[0034] Step 203: Calculate edge weights

[0035] The weight of each edge is W ij It is used to describe the transmission capacity of the crack channel, taking into account the geometric and physical characteristics of the crack. The calculation formula is:

[0036]

[0037] Where k: permeability coefficient of fracture material; A = W kd H: cross-sectional area of ​​the crack, W kd is the crack width, H is the crack height; L: the length of the crack, indicating the physical distance between the two intersection points;

[0038] Step 204: Generate adjacency matrix

[0039] According to the definition of nodes and edges, the adjacency matrix A of the crack network is generated, where the matrix element A ij Defined as:

[0040]

[0041] Step 205: Topology model structuring

[0042] The nodes and edges are organized into a weighted undirected graph model, which contains the following data:

[0043] Node set V = {v1,v2,...,v n}; edge set E = {e ij}, each edge has a weight W ij and geometric parameters.

[0044] Step 3: Calculate the connectivity, shortest path, average path length and key indicators of global efficiency of the crack network through graph theory analysis methods, and identify the key paths and nodes in the network;

[0045] Preferably, in step 3, the specific analysis steps include the following aspects:

[0046] Step 301: Connectivity analysis

[0047] Connectivity analysis is used to determine the connectivity of nodes and edges in the crack network, identify the connected components of independent regions in the crack network, and the range of the maximum connected region; the specific steps are as follows:

[0048] Step 3011: Calculation of connected components: Using a breadth-first search (BFS) or depth-first search (DFS) algorithm, starting from any node, traverse all connected nodes, and identify the nodes and edges of each connected component;

[0049] Step 3012: Number of connected components C: Count the number of independent connected regions C in the crack network, that is, the number of unconnected subgraphs in the graph;

[0050] Step 3013: Maximum connected subgraph proportion Pmax: Calculate the proportion of the number of nodes in the maximum connected area to the total number of nodes. The formula is:

[0051]

[0052] Among them: |Vmax|: the number of nodes in the maximum connected subgraph; |V|: the total number of nodes in the graph;

[0053] Step 302: Shortest path analysis

[0054] The shortest path analysis is used to evaluate the optimal transport path of water or ions from the starting point to the end point in the crack network; the specific steps are as follows:

[0055] Step 3021: Shortest path definition: two nodes v in the crack network i and v j The shortest path length d ij Defined as the path with the minimum weight and minimum value:

[0056]

[0057] Where P: the set of edges included in the path; W k : The weight of each edge on the path, indicating the transmission capacity of the crack channel;

[0058] Step 3022: Calculate the shortest path of the entire network: Calculate the shortest path lengths between all node pairs in the crack network, and store the results as the shortest path matrix D, where D ij Represents node v i and v j The shortest path between

[0059] Step 303: Average path length analysis

[0060] The average path length is used to evaluate the average transmission distance between any two nodes in the crack network, and the formula is:

[0061]

[0062] Where: N: total number of nodes in the graph; d ij : Node v i and v j The shortest path length between

[0063] Step 304: Network efficiency analysis

[0064] The network efficiency is used to quantify the overall transport capacity of water and ions in the crack network, reflecting the global transport performance of the network structure; the formula is:

[0065]

[0066] Where: E: network efficiency;

[0067] Step 305: Node importance analysis

[0068] Node importance analysis is used to identify the key nodes in the crack network that contribute most to the overall transmission performance. The indicators include the following:

[0069] Betweenness centrality BC(v): The betweenness centrality of a node v represents the importance of the node on the shortest path and is defined as:

[0070]

[0071] Where: st : the number of shortest paths from node f to node g; σ st (v): The number of these shortest paths that pass through node v.

[0072] Step 4: Combine Darcy’s law and Fick’s diffusion law to simulate the water transport and ion diffusion behaviors and quantify the transmission efficiency and diffusion characteristics of the crack network.

[0073] Preferably, in step 4, the simulation of water and ion transport in the crack network is based on physical transport theory, and the influence of crack structure on water flow and ion diffusion behavior is analyzed by graph theory model; the specific simulation process includes the following steps:

[0074] Step 401: Water transport simulation

[0075] The simulation of water transport is based on Darcy's law, which describes the law of water seepage in the crack channel; the flow rate Q of each crack in the crack network ij It is expressed as:

[0076] Q ij =W ij ·Δh ij

[0077] Among them, Q ij : From node v i and v j Moisture flow rate; Δh ij : The water head difference at both ends of the crack;

[0078] According to the topological model of the fracture network, the total flow of water transport is calculated through the following steps:

[0079] Edge flow calculation: Based on the W of each crack ij and head difference Δh ij , calculate the flow rate Q of all cracks ij ;

[0080] Total flow calculation: The flow of all edges in the network is accumulated to obtain the total water transport of the entire fracture network:

[0081]

[0082] Critical path identification: Identify the main channels and bottlenecks of water transport by analyzing the paths with the largest flow;

[0083] Step 402: Ion transport simulation

[0084] The simulation of ion transport is based on Fick's diffusion law, which describes the diffusion process of ions in the crack network; the change of ion concentration is controlled by the following partial differential equation:

[0085]

[0086] Where, C: ion concentration; t: time; D ksxs : Diffusion coefficient, which indicates the diffusion rate of ions; The second spatial derivative of ion concentration;

[0087] In the fracture network, after discretizing the diffusion equation, the change in ion concentration at each node can be expressed as:

[0088]

[0089] in: Node v at time t i Ion concentration; N(i): related to node v i The set of connected neighbor nodes; L ij : Node v i and v j The length of the crack between

[0090] The specific steps of ion diffusion simulation are as follows:

[0091] First: Initial condition setting: set the initial ion concentration at the nodes in the network and set the concentration of other nodes to zero;

[0092] Then, the diffusion process is calculated: According to the discretized diffusion equation, the concentration of each node is updated step by step. until a steady state is reached;

[0093] Then, the diffusion range is evaluated: the concentration distribution of ions in the crack network in steady state is analyzed, and the coverage and diffusion rate of ion diffusion are calculated;

[0094] Finally, key node identification: Based on the concentration distribution, identify the key nodes and channels that have the greatest impact on diffusion behavior;

[0095] Step 403: Joint simulation

[0096] The main steps of joint simulation include:

[0097] First, flow field calculation: Calculate the flow field distribution in the fracture network according to Darcy's law to obtain the flow velocity of each fracture;

[0098] Convection-diffusion equation: Based on the flow field, considering the convection effect, the diffusion equation is corrected to:

[0099]

[0100] where v ls is the flow rate;

[0101] Then the time-stepping simulation is performed: in each time step, the flow velocity distribution of water is calculated first, and then the concentration distribution of ions is updated according to the flow velocity, and the iteration is repeated until the simulation is completed;

[0102] Then, joint property evaluation: comprehensive analysis of flow velocity and ion concentration distribution in the fracture network to evaluate the impact of water flow on ion diffusion range and rate;

[0103] Step 404: Simulation results analysis

[0104] The simulation results analysis mainly focuses on the following aspects:

[0105] Traffic distribution: Count the traffic Q of all cracks in the network ij distribution, identifying the channels with the highest transmission efficiency;

[0106] Ion concentration distribution: At steady state, plot the concentration distribution of ions in the fracture network, marking high concentration areas and diffusion boundaries;

[0107] Critical Paths and Nodes: Based on flow and concentration data, identify the critical paths and nodes in the network that are most important for overall transport behavior;

[0108] Transport efficiency evaluation: Calculate the total flow and diffusion coverage of the entire fracture network to quantify the transport performance of the network;

[0109] Step 405: Critical Path Identification

[0110] Through flow constraint or transmission efficiency maximization algorithm, the most important transmission path in the crack network is identified; the cumulative transmission contribution rate R of the critical path is calculated:

[0111]

[0112] In the formula, Q k represents the flow rate of fracture channel k, reflecting the transmission capacity of the channel in the fracture network; Pcritical represents the critical path set, that is, the path set that contributes most to the transmission efficiency.

[0113] The technical effects achieved by the present invention are:

[0114] The present invention innovatively combines graph theory with physical transmission models, and proposes a transmission behavior analysis method suitable for complex fracture networks. By accurately extracting fracture geometric parameters (such as length, width, direction, density and intersection position) through three-dimensional CT scanning technology, and modeling the fracture network as a weighted topological graph model, the present invention can comprehensively quantify the connectivity, critical path and bottleneck node characteristics of the fracture network. Compared with traditional methods, the present invention has significant advantages in systematically revealing the percolation effect in connected fracture networks and its regulatory mechanism on water flow and ion diffusion behavior. Through Darcy's law and Fick's diffusion law combined with graph theory analysis, the present invention can accurately simulate transmission efficiency and diffusion laws, especially in complex three-dimensional heterogeneous fracture networks, showing efficient computing power and excellent applicability, which not only solves the shortcomings of the existing technology in the quantification of critical connectivity, percolation threshold and transmission law, but also provides theoretical support and technical guarantee for the study of fracture transmission behavior in road engineering, geotechnical engineering and underground reservoirs.

[0115] The present invention is based on the study of the water and ion transport behavior in the complex crack network in cement-based materials, combined with graph theory and physical transport model, by abstracting the crack network into a graph model, quantifying its topological characteristics, connectivity and transport efficiency, and further revealing the influence mechanism of crack geometric characteristics and percolation effect on water and ion transport behavior. Specifically, the method first uses three-dimensional CT scanning technology to obtain the geometric parameters of the crack (such as length, width, direction and density), and accurately extracts the intersection and channel position of the crack through image processing algorithm. Based on principal component analysis (PCA), the main axis direction of the crack is calculated to quantify its spatial distribution law in the sample. Subsequently, the crack network is modeled as a weighted undirected graph, in which the node represents the intersection or end point of the crack and the edge represents the crack channel. The weight of the edge is calculated by comprehensively considering the geometric characteristics of the crack (such as length, width, cross-sectional area) and physical properties (such as permeability coefficient), thereby accurately describing the transmission capacity of the crack. On this basis, the topological characteristics of the crack network are analyzed using graph theory methods, and key indicators such as connectivity, shortest path, average path length and global efficiency are calculated. Combined with the physical transport model, the water flow and ion diffusion process are simulated to reveal the key paths, bottleneck nodes and percolation effects of the crack network on the regulation of transmission efficiency. Finally, by quantifying the relationship between crack geometry and transmission behavior, this invention provides a scientific basis and theoretical support for the study of water and ion transmission laws in complex crack networks in cement-based materials. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] Figure 1 It is a flow chart of a method for analyzing the water and ion transport behavior of a cement-based material internally connected crack network based on graph theory of the present invention;

[0117] Figure 2 It is a flow chart of crack network data processing in the present invention;

[0118] Figure 3 It is a flow chart of topological model construction in the present invention. DETAILED DESCRIPTION

[0119] In order to make the purpose and advantages of the present invention more clearly understood, the present invention is specifically described below in conjunction with embodiments. It should be understood that the following text is only used to describe one or several specific embodiments of the present invention, and does not strictly limit the scope of protection of the specific claims of the present invention.

[0120] like Figure 1-Figure 3 As shown, the method for analyzing the water and ion transport behavior of the internal connected crack network of cement-based materials based on graph theory includes the following steps:

[0121] Step 1: First, preprocess the internal crack network data of cement-based material samples to extract the geometric characteristics of the cracks, such as length, width, and intersection points.

[0122] As Figure 2 shown, Step 1 includes the following specific steps:

[0123] Step 101: Specimen preparation

[0124] Use three-dimensional CT scanning technology to obtain the geometric parameters of the crack, preprocess the sample, cut the sample size according to the scanner specifications to ensure uniform crack distribution; mark coordinate points on the sample surface; for example, paint or spray marking for subsequent image registration and correction;

[0125] Step 102: Three-dimensional CT scanning

[0126] Use industrial CT scanning technology to perform three-dimensional scanning on the sample to obtain the internal structure data of the crack; during the scanning process, first fix the sample on the rotating table, and obtain the tomographic projection images of the sample through multi-angle X-ray scanning; the resolution of the scanner is usually set between 0.01 mm and 0.1 mm, and the energy of the X-ray is selected between 100 kV and 300 kV according to the sample material to ensure complete capture of crack details;

[0127] Step 103: Image reconstruction

[0128] The original projection data obtained by scanning is reconstructed into a three-dimensional image through the filtered back-projection algorithm (FBP) to generate a three-dimensional gray-scale distribution map of the crack and the surrounding matrix; the gray value of each pixel reflects the density difference of the material, and the crack usually appears as a low-gray area; in order to improve the quality of the reconstructed image, correct the data, including removing artifacts, such as ray hardening artifacts and ring artifacts, optimizing contrast and resolution operations, so that the crack area is clearer in the image;

[0129] Step 104: Crack area segmentation

[0130] First, use the threshold segmentation method. According to the significant difference in gray values between the crack and the matrix material, set a threshold T, mark the pixels with gray values lower than T as the crack area, and the pixels higher than T as the matrix area; the specific segmentation process can be expressed by a binary formula: when the pixel gray value I(x,y,z) < T, the pixel is marked as a crack; otherwise it is marked as the matrix; then, use the three-dimensional connectivity analysis algorithm to divide the segmented crack pixel groups into independent crack bodies; these volume-connected pixel groups are considered to belong to the same crack structure; in order to remove noise interference and non-physical small crack areas, a minimum volume threshold can be set to剔除 areas with volumes smaller than this threshold;

[0131] Step 105: Extract crack geometric parameters

[0132] The crack length is calculated by the linear distance along the crack main axis, which can be obtained by principal component analysis (PCA):

[0133]

[0134] Among them, λ1 is the maximum eigenvalue of the crack point cloud covariance matrix;

[0135] The width of the crack W kd Obtained through direct observation under a microscope, the specific method is to select a characteristic section of the crack under a microscope and use a measuring tool to determine the average value of the section width;

[0136] The crack directions are extracted by principal component analysis:

[0137] D=(cosθ x ,cosθ y ,cosθ z )

[0138] where θ x ,θ y ,θ z are the angles between the crack main axis and the x, y, and z coordinate axes respectively; the specific calculation process includes the following steps: first, the covariance matrix of the three-dimensional coordinates of each crack point cloud is calculated, and the maximum eigenvector of the covariance matrix is ​​the main axis direction vector of the crack; then, the main axis direction vector is normalized to obtain the direction cosine value of the crack;

[0139] The crack density ρ is defined as the number of cracks per unit volume and can be expressed as ρ = N / V tj Calculate, where N is the total number of cracks, V tj is the sample volume; finally, the intersection position of the crack is extracted (x i ,y i ,z i ), record the three-dimensional coordinates of the endpoints and intersection points of each crack.

[0140] like Figure 3 As shown, step 2: construct a topological model of the crack, where the nodes represent crack intersections, the edges represent crack channels, and the edge weights are calculated according to the permeability formula;

[0141] In step 2, the topological model of the crack network is constructed by abstracting the geometric structure of the three-dimensional crack into a graph model for subsequent analysis and simulation; first, the crack intersection position (x i ,y i ,z i ) and the geometrical properties of the crack channel, such as length L, width W kd , Direction Dfx The topological structure of the crack network is established as a weighted undirected graph. The specific construction process includes the following steps:

[0142] Step 201: Define Nodes

[0143] The intersection or endpoint of the crack is defined as the node of the topological model, denoted by V; the three-dimensional spatial position of each node is represented by the coordinates (x i ,y i ,z i ) is determined; the intersection data comes from the intersection extraction result after the crack area is segmented, and the number of nodes is equal to the total number of crack intersections;

[0144] Step 202: Define the edges

[0145] The crack channel is defined as an edge in the topological model, denoted by E; each edge connects two nodes v i and v j , corresponding to the actual connectivity of the crack in three-dimensional space, the geometric properties of the edge include the length L and the direction vector D ij =(x j -x i ,y j -y i ,z j -z i ), the direction is calculated by the coordinate difference of the two nodes;

[0146] Step 203: Calculate edge weights

[0147] The weight of each edge is W ij It is used to describe the transmission capacity of the crack channel, taking into account the geometric and physical characteristics of the crack. The calculation formula is:

[0148]

[0149] Where k: permeability coefficient of fracture material; A = W kd H: cross-sectional area of ​​the crack, W kd is the crack width, H is the crack height, which can be estimated by microscope or sample thickness; L: the length of the crack, which represents the physical distance between the two intersection points;

[0150] Step 204: Generate adjacency matrix

[0151] According to the definition of nodes and edges, the adjacency matrix A of the crack network is generated, where the matrix element A ij Defined as:

[0152]

[0153] Step 205: Topology model structuring

[0154] The nodes and edges are organized into a weighted undirected graph model, which contains the following data:

[0155] Node set V = {v1,v2,...,v n}; edge set E = {e ij}, each edge has a weight W ij and geometric parameters such as L, D ij .

[0156] Step 3: Calculate the connectivity, shortest path, average path length and key indicators of global efficiency of the crack network through graph analysis methods, and identify the key paths and nodes in the network;

[0157] In step 3, the crack network characteristic analysis aims to quantify the structural characteristics and transmission performance of the crack network, providing a basis for further simulation of water and ion transport; the specific analysis steps include the following aspects:

[0158] Step 301: Connectivity analysis

[0159] Connectivity analysis is used to determine the connectivity of nodes and edges in the crack network, identify the connected components of independent regions in the crack network, and the range of the maximum connected region; the specific steps are as follows:

[0160] Step 3011: Calculation of connected components: Using a breadth-first search (BFS) or depth-first search (DFS) algorithm, starting from any node, traverse all connected nodes, and identify the nodes and edges of each connected component;

[0161] Step 3012: Number of connected components C: Count the number of independent connected regions C in the crack network, that is, the number of unconnected subgraphs in the graph;

[0162] Step 3013: Maximum connected subgraph proportion Pmax: Calculate the proportion of the number of nodes in the maximum connected area to the total number of nodes. The formula is:

[0163]

[0164] Among them: |Vmax|: the number of nodes in the maximum connected subgraph; |V|: the total number of nodes in the graph;

[0165] Step 302: Shortest path analysis

[0166] The shortest path analysis is used to evaluate the optimal transport path of water or ions from the starting point to the end point in the crack network; the specific steps are as follows:

[0167] Step 3021: Shortest path definition: two nodes v in the crack network iand v j The shortest path length d ij Defined as the path with the minimum weight:

[0168]

[0169] Where P: the set of edges included in the path; W k : The weight of each edge on the path, indicating the transmission capacity of the crack channel;

[0170] Step 3022: Calculate the shortest path of the entire network: Calculate the shortest path lengths between all node pairs in the crack network, and store the results as the shortest path matrix D, where D ij Represents node v i and v j The shortest path between

[0171] Step 303: Average path length analysis

[0172] The average path length is used to evaluate the average transmission distance between any two nodes in the crack network, and the formula is:

[0173]

[0174] Where: N: total number of nodes in the graph; d ij : Node v i and v j The shortest path length between

[0175] Step 304: Network efficiency analysis

[0176] The network efficiency is used to quantify the overall transport capacity of water and ions in the crack network, reflecting the global transport performance of the network structure; the formula is:

[0177]

[0178] Where: E: network efficiency;

[0179] Step 305: Node importance analysis

[0180] Node importance analysis is used to identify the key nodes in the crack network that contribute most to the overall transmission performance. The indicators include the following:

[0181] Betweenness centrality BC(v): The betweenness centrality of a node v represents the importance of the node on the shortest path and is defined as:

[0182]

[0183] Where: st : the number of shortest paths from node f to node g; σst (v): The number of these shortest paths that pass through node v.

[0184] Step 4: Combine Darcy’s law and Fick’s diffusion law to simulate the water transport and ion diffusion behaviors and quantify the transmission efficiency and diffusion characteristics of the crack network.

[0185] In step 4, the simulation of water and ion transport in the crack network is based on physical transport theory, and the influence of crack structure on water flow and ion diffusion behavior is analyzed through a graph theory model; the specific simulation process includes the following steps:

[0186] Step 401: Water transport simulation

[0187] The simulation of water transport is based on Darcy's Law, which describes the law of water seepage in the crack channel; the flow rate Q of each crack in the crack network ij It is expressed as:

[0188] Q ij =W ij ·Δh ij

[0189] Among them, Q ij : From node v i and v j Moisture flow rate; Δh ij : The water head difference at both ends of the crack;

[0190] According to the topological model of the fracture network, the total flow of water transport is calculated through the following steps:

[0191] Edge flow calculation: Based on the W of each crack ij and head difference Δh ij , calculate the flow rate Q of all cracks ij ;

[0192] Total flow calculation: The flow of all edges in the network is accumulated to obtain the total water transport of the entire fracture network:

[0193]

[0194] Critical path identification: Identify the main channels and bottlenecks of water transport by analyzing the paths with the largest flow;

[0195] Step 402: Ion transport simulation

[0196] The simulation of ion transport is based on Fick's Law, which describes the diffusion process of ions in the crack network; the change of ion concentration is controlled by the following partial differential equation:

[0197]

[0198] Where, C: ion concentration; t: time; D ksxs : Diffusion coefficient, which indicates the diffusion rate of ions; The second spatial derivative of ion concentration;

[0199] In the fracture network, after discretizing the diffusion equation, the change of ion concentration at each node can be expressed as:

[0200]

[0201] in: Node v at time t i Ion concentration; N(i): with node v i The set of connected neighbor nodes; L ij : Node v i and v j The length of the crack between

[0202] The specific steps of ion diffusion simulation are as follows:

[0203] First: Initial condition setting: set the initial ion concentration Csource at some nodes in the network, for example, the source node sets the initial ion concentration Csource; the concentration of other nodes is set to zero;

[0204] Then, the diffusion process is calculated: According to the discretized diffusion equation, the concentration of each node is updated step by step. Until a steady state is reached; that is, the concentration change tends to zero;

[0205] Then, the diffusion range is evaluated: the concentration distribution of ions in the crack network in steady state is analyzed, and the coverage and diffusion rate of ion diffusion are calculated;

[0206] Finally, key node identification: Based on the concentration distribution, identify the key nodes and channels that have the greatest impact on diffusion behavior;

[0207] Step 403: Joint simulation

[0208] In complex fracture networks, water transport and ion diffusion are often coupled. The flow of water promotes the diffusion of ions, so joint simulation is required. The main steps of joint simulation include:

[0209] First, flow field calculation: Calculate the flow field distribution in the fracture network according to Darcy's law to obtain the flow velocity of each fracture;

[0210] Convection-diffusion equation: Based on the flow field, considering the convection effect, the diffusion equation is corrected to:

[0211]

[0212] where v ls is the flow rate;

[0213] Then the time-stepping simulation is performed: in each time step, the flow velocity distribution of water is calculated first, and then the concentration distribution of ions is updated according to the flow velocity, and the iteration is repeated until the simulation is completed;

[0214] Then, joint property evaluation: comprehensive analysis of flow velocity and ion concentration distribution in the fracture network to evaluate the impact of water flow on ion diffusion range and rate;

[0215] Step 404: Simulation results analysis

[0216] The simulation results analysis mainly focuses on the following aspects:

[0217] Traffic distribution: Count the traffic Q of all cracks in the network ij distribution, identifying the channels with the highest transmission efficiency;

[0218] Ion concentration distribution: At steady state, plot the concentration distribution of ions in the fracture network, marking high concentration areas and diffusion boundaries;

[0219] Critical Paths and Nodes: Based on flow and concentration data, identify the critical paths and nodes in the network that are most important for overall transport behavior;

[0220] Transport efficiency evaluation: Calculate the total flow and diffusion coverage of the entire fracture network to quantify the transport performance of the network;

[0221] Step 405: Critical Path Identification

[0222] Through flow constraints or transmission efficiency maximization algorithms, the most important transmission paths in the crack network, such as critical paths, are identified; the cumulative transmission contribution rate R of the critical path is calculated:

[0223]

[0224] In the formula, Q k represents the flow rate of fracture channel k, reflecting the transmission capacity of the channel in the fracture network; Pcritical represents the critical path set, that is, the path set that contributes most to the transmission efficiency.

[0225] The present invention innovatively combines graph theory with physical transmission models, and proposes a transmission behavior analysis method suitable for complex fracture networks. By accurately extracting fracture geometric parameters (such as length, width, direction, density and intersection position) through three-dimensional CT scanning technology, and modeling the fracture network as a weighted topological graph model, the present invention can comprehensively quantify the connectivity, critical path and bottleneck node characteristics of the fracture network. Compared with traditional methods, the present invention has significant advantages in systematically revealing the percolation effect in connected fracture networks and its regulatory mechanism on water flow and ion diffusion behavior. Through Darcy's law and Fick's diffusion law combined with graph theory analysis, the present invention can accurately simulate transmission efficiency and diffusion laws, especially in complex three-dimensional heterogeneous fracture networks, showing efficient computing power and excellent applicability, which not only solves the shortcomings of the existing technology in the quantification of critical connectivity, percolation threshold and transmission law, but also provides theoretical support and technical guarantee for the study of fracture transmission behavior in road engineering, geotechnical engineering and underground reservoirs.

[0226] The present invention focuses on studying the water and ion transport behaviors under the interconnected crack network inside cement-based materials. By constructing a topological model of the crack network and combining physical transmission models (such as Darcy's law and Fick's diffusion law), the dynamic characteristics of water flow and ion diffusion in connected cracks are deeply simulated. The present invention reveals the regulatory mechanism of crack geometric characteristics (such as length, width, direction), connectivity and percolation effect on transmission efficiency, and provides scientific support for accurately understanding the laws of water and ion transport inside cement-based materials. This technology can be used to optimize the design of the impermeability performance of cement-based materials, improve their durability and long-term stability, play an important role in road engineering construction and maintenance, and lay the foundation for in-depth research on the influence mechanism of crack network on transmission behavior.

[0227] The above is only a preferred embodiment of the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principles of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention. The structures, devices and operating methods not specifically described and explained in the present invention shall be implemented according to the conventional means in the art unless otherwise specified and limited.

Claims

1. A graph-theory-based method for analyzing the water and ion transport behavior of a connected crack network inside a cement-based material, characterized by: The following steps are involved: Step 1: First, preprocess the internal crack network data of cement-based material specimens to extract the geometric characteristics of the cracks; Step 2: Construct a topological model of the crack, where nodes represent crack intersections, edges represent crack channels, and edge weights are calculated based on the permeability formula; Step 3: Calculate the connectivity, shortest path, average path length and key indicators of global efficiency of the crack network through graph theory analysis methods, and identify the key paths and nodes in the network; Step 4: Combine Darcy’s law and Fick’s diffusion law to simulate the water transport and ion diffusion behaviors and quantify the transmission efficiency and diffusion characteristics of the crack network.

2. The method for analyzing the water and ion transport behavior of an internally connected crack network of cement-based materials based on graph theory according to claim 1 is characterized in that: The step 1 comprises the following specific steps: Step 101: Sample Preparation The geometric parameters of the cracks were obtained using 3D CT scanning technology, the samples were pre-processed, and the size of the samples was cut according to the specifications of the scanner; Mark coordinate points on the sample surface; Step 102: 3D CT Scan Use industrial CT scanning technology to perform three-dimensional scanning of the sample to obtain the internal structure data of the crack; Step 103: Image reconstruction The original projection data obtained by scanning is reconstructed into a three-dimensional image through a filtered back-projection algorithm to generate a three-dimensional grayscale distribution map of the crack and the surrounding matrix; the grayscale value of each pixel reflects the density difference of the material, and the data is corrected, including removing artifacts, optimizing contrast and resolution operations; Step 104: Crack region segmentation Firstly, a threshold value T is set based on the significant difference in grayscale between cracks and matrix materials using the threshold segmentation method. Pixels with grayscale values ​​lower than T are marked as crack areas, while pixels with grayscale values ​​higher than T are marked as matrix areas. Then, the segmented crack pixel groups are divided into independent crack bodies using the three-dimensional connectivity analysis algorithm. Step 105: Extract crack geometry parameters The crack length is calculated by the linear distance along the crack main axis, which can be obtained by principal component analysis (PCA): Among them, λ1 is the maximum eigenvalue of the crack point cloud covariance matrix; The width of the crack W kd Obtained through direct observation under a microscope; The crack directions are extracted by principal component analysis: D=(cosθ x ,cosθ y ,cosθ z ) where θ x ,θ y ,θ z are the angles between the crack principal axis and the x, y, and z coordinate axes, respectively; The crack density ρ is defined as the number of cracks per unit volume and can be expressed as ρ = N / V tj Calculate, where N is the total number of cracks, V tj is the sample volume; finally, the intersection position of the crack is extracted (x i ,y i ,z i ), record the three-dimensional coordinates of the endpoints and intersection points of each crack.

3. The method for analyzing the water and ion transport behavior of an internally connected crack network of cement-based materials based on graph theory according to claim 1, characterized in that: In step 2, the topological model of the crack network is constructed by abstracting the geometric structure of the three-dimensional crack into a graph theory model; first, the crack intersection position (x i ,y i ,z i ) and the geometric characteristics of the crack channel, the topological structure of the crack network is established as a weighted undirected graph; The specific construction process includes the following steps: Step 201: Define Nodes The intersection or endpoint of the crack is defined as the node of the topological model, denoted by V; the three-dimensional spatial position of each node is represented by the coordinates (x i ,y i ,z i ) is determined; the intersection data comes from the intersection extraction result after the crack area is segmented, and the number of nodes is equal to the total number of crack intersections; Step 202: Define the edges The crack channel is defined as an edge in the topological model, denoted by E; each edge connects two nodes v i and v j , corresponding to the actual connectivity of the crack in three-dimensional space, the geometric properties of the edge include the length L and the direction vector D ij =(x j -x i ,y j -y i ,z j -z i ), the direction is calculated by the coordinate difference of the two nodes; Step 203: Calculate edge weights The weight of each edge is W ij It is used to describe the transmission capacity of the crack channel, taking into account the geometric and physical characteristics of the crack. The calculation formula is: Where k: permeability coefficient of fracture material; A = W kd H: cross-sectional area of ​​the crack, W kd is the crack width, H is the crack height; L: the length of the crack, indicating the physical distance between the two intersection points; Step 204: Generate adjacency matrix According to the definition of nodes and edges, the adjacency matrix A of the crack network is generated, where the matrix element A ij Defined as: Step 205: Topology model structuring The nodes and edges are organized into a weighted undirected graph model, which contains the following data: Node set V = {v1,v2,...,v n }; edge set E = {e ij }, each edge has a weight W ij and geometric parameters.

4. The method for analyzing the water and ion transport behavior of an internally connected crack network of cement-based materials based on graph theory according to claim 1, characterized in that: In step 3, the specific analysis steps include the following aspects: Step 301: Connectivity analysis Connectivity analysis is used to determine the connectivity of nodes and edges in the crack network, identify the connected components of independent regions in the crack network, and the range of the maximum connected region; the specific steps are as follows: Step 3011: Calculation of connected components: Using a breadth-first search (BFS) or depth-first search (DFS) algorithm, starting from any node, traverse all connected nodes, and identify the nodes and edges of each connected component; Step 3012: Number of connected components C: Count the number of independent connected regions C in the crack network, that is, the number of unconnected subgraphs in the graph; Step 3013: Maximum connected subgraph proportion Pmax: Calculate the proportion of the number of nodes in the maximum connected area to the total number of nodes. The formula is: Among them: |Vmax|: the number of nodes in the maximum connected subgraph; |V|: the total number of nodes in the graph; Step 302: Shortest path analysis The shortest path analysis is used to evaluate the optimal transport path of water or ions from the starting point to the end point in the crack network; the specific steps are as follows: Step 3021: Shortest path definition: two nodes v in the crack network i and v j The shortest path length d ij Defined as the path with the minimum weight: Where P: the set of edges included in the path; W k : The weight of each edge on the path, indicating the transmission capacity of the crack channel; Step 3022: Calculate the shortest path of the entire network: Calculate the shortest path lengths between all node pairs in the crack network, and store the results as the shortest path matrix D, where D ij Represents node v i and v j The shortest path between Step 303: Average path length analysis The average path length is used to evaluate the average transmission distance between any two nodes in the crack network, and the formula is: Where: N: total number of nodes in the graph; d ij : Node v i and v j The shortest path length between Step 304: Network efficiency analysis The network efficiency is used to quantify the overall transport capacity of water and ions in the crack network, reflecting the global transport performance of the network structure; the formula is: Where: E: network efficiency; Step 305: Node importance analysis Node importance analysis is used to identify the key nodes in the crack network that contribute most to the overall transmission performance. The indicators include the following: Betweenness centrality BC(v): The betweenness centrality of a node v represents the importance of the node on the shortest path and is defined as: Where: st : the number of shortest paths from node f to node g; σ st (v): The number of these shortest paths that pass through node v.

5. The method for analyzing the water and ion transport behavior of an internally connected crack network of cement-based materials based on graph theory according to claim 1, characterized in that: In step 4, the simulation of water and ion transport in the crack network is based on physical transport theory, and the influence of crack structure on water flow and ion diffusion behavior is analyzed through a graph theory model; the specific simulation process includes the following steps: Step 401: Water transport simulation The simulation of water transport is based on Darcy's law, which describes the law of water seepage in the crack channel; the flow rate Q of each crack in the crack network ij It is expressed as: Q ij =W ij ·Δh ij Among them, Q ij : From node v i and v j The moisture flow rate; Δh ij : The water head difference at both ends of the crack; According to the topological model of the fracture network, the total flow of water transport is calculated through the following steps: Edge flow calculation: Based on the W of each crack ij and head difference Δh ij , calculate the flow rate Q of all cracks ij ; Total flow calculation: The flow of all edges in the network is accumulated to obtain the total water transport of the entire fracture network: Critical path identification: Identify the main channels and bottlenecks of water transport by analyzing the paths with the largest flow; Step 402: Ion transport simulation The simulation of ion transport is based on Fick's diffusion law, which describes the diffusion process of ions in the crack network; the change of ion concentration is controlled by the following partial differential equation: Where, C: ion concentration; t: time; D ksxs : Diffusion coefficient, which indicates the diffusion rate of ions; The second spatial derivative of ion concentration; In the fracture network, after discretizing the diffusion equation, the change of ion concentration at each node can be expressed as: in: Node v at time t i Ion concentration; N(i): related to node v i The set of connected neighbor nodes; L ij : Node v i and v j The length of the crack between The specific steps of ion diffusion simulation are as follows: First: Initial condition setting: set the initial ion concentration at the nodes in the network and set the concentration of other nodes to zero; Then, the diffusion process is calculated: According to the discretized diffusion equation, the concentration of each node is updated step by step. until a steady state is reached; Then, the diffusion range is evaluated: the concentration distribution of ions in the crack network in steady state is analyzed, and the coverage and diffusion rate of ion diffusion are calculated; Finally, key node identification: Based on the concentration distribution, identify the key nodes and channels that have the greatest impact on diffusion behavior; Step 403: Joint simulation The main steps of joint simulation include: First, flow field calculation: Calculate the flow field distribution in the fracture network according to Darcy's law to obtain the flow velocity of each fracture; Convection-diffusion equation: Based on the flow field, considering the convection effect, the diffusion equation is corrected to: where v ls is the flow rate; Then the time-stepping simulation is performed: in each time step, the flow velocity distribution of water is calculated first, and then the concentration distribution of ions is updated according to the flow velocity, and the iteration is repeated until the simulation is completed; Then, joint property evaluation: comprehensive analysis of flow velocity and ion concentration distribution in the fracture network to evaluate the impact of water flow on ion diffusion range and rate; Step 404: Simulation results analysis The simulation results analysis mainly focuses on the following aspects: Traffic distribution: Count the traffic Q of all cracks in the network ij distribution, identifying the channels with the highest transmission efficiency; Ion concentration distribution: At steady state, plot the concentration distribution of ions in the fracture network, marking high concentration areas and diffusion boundaries; Critical Paths and Nodes: Based on flow and concentration data, identify the critical paths and nodes in the network that are most important for overall transport behavior; Transport efficiency evaluation: Calculate the total flow and diffusion coverage of the entire fracture network to quantify the transport performance of the network; Step 405: Critical Path Identification Through flow constraint or transmission efficiency maximization algorithm, the most important transmission path in the crack network is identified; the cumulative transmission contribution rate R of the critical path is calculated: In the formula, Q k represents the flow rate of fracture channel k, reflecting the transmission capacity of the channel in the fracture network; Pcritical represents the critical path set, that is, the path set that contributes most to the transmission efficiency.

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