Application of Curtain Grouting for Seepage Control in the Dongzhuang Dam: A 3D Fracture Network Modeling Approach

Abstract

This study presents a 3D fracture network modeling approach for designing curtain grouting systems in building foundations, utilizing geological mapping data from the Dongzhuang Project. A one-dimensional Markov chain model is applied to simulate the transitions in fracture density, while fracture orientation and size are characterized using Fisher and statistical distribution models. To enhance the prediction accuracy, a correction method is introduced to refine the transition matrices. The model’s reliability is validated using tunnel wall fracture data and borehole detection, demonstrating strong agreement in both trend and magnitude. In under 100,000 simulations, when the allowable absolute error is set to 1, the optimal accuracy can reach 80%. Reliability analysis confirms the robustness of the approach, with 99.91% of predictions within a ± 2 error margin. The final fracture network model effectively captures spatial heterogeneity and fracture penetration across various foundation layers; the spatial distribution density index of fractures can provide a reference basis for optimizing the layout of impermeable curtains in complex geological conditions. This integrated modeling approach offers a reliable tool for improving grouting strategies in building foundation projects and other civil infrastructure.

1. Introduction

Curtain grouting is a critical technique for controlling water seepage in the construction and maintenance of building foundation structures, particularly in rock mass environments. The fracture network forms preferential pathways for grout flow, where connectivity creates directional diffusion paths—a phenomenon that becomes particularly pronounced in zones with high fracture density, ultimately impacting the construction of impermeable curtain grouting. The success of curtain grouting is significantly influenced by the distribution of rock fractures [1,2]. Due to the complex nature of geological structures, fully understanding fracture distribution remains challenging. Engineers often rely on geological investigations and practical experience to characterize these fractures [3]. A reliable estimation and characterization of fracture distribution is essential for optimizing grouting efficiency, improving sealing performance, and ensuring the long-term stability of building foundations.

The distribution of rock fractures involves determining parameters such as fracture density, orientation, and size. Common methods for acquiring these parameters include borehole data and geological sketches [4,5]. In geotechnical engineering, borehole measurements remain the most reliable technique for directly revealing the integrity of subsurface rock layers on a large scale [6,7]. Some scholars have employed geostatistical theory combined with discrete fracture network modeling to determine fracture density and other parameters, ultimately constructing a three-dimensional fracture network model for the Beishan spent fuel reprocessing site [6]. Their modeling approach provides valuable methodological guidance for this study. Borehole data are primarily used to estimate fracture density, while other parameters often rely on surface outcrops and geological sketches. However, statistical errors may arise when density and other parameters are obtained from separate fracture sources. By processing fracture traces from tunnel walls, it is possible to reduce potential errors and provide additional geological data that can improve the grouting strategy.

Fracture networks can be modeled based on the aforementioned data. Both discrete fracture network (DFN) modeling and block-based methods are commonly used to generate the distribution of rock fractures in the grouting zone. DFN modeling was first introduced by Baecher, initially using a disk model [8,9] and later expanded by researchers to polygon models [10,11] and elliptical models [12,13]. The DFN modeling process mainly consists of two steps: generating fracture locations through a point process and assigning attributes such as orientation and size to the fractures at those locations [14,15]. This method utilizes digital photography and statistical distributions to create 3D natural fracture networks, which can effectively describe rock fractures [16]. However, natural fractures form under a dynamic mechanical self-organization process, where ruptures and fractures occur at all scales [17]. Under in situ stress fields, they may develop complex topologies, such as cross-cutting, intersection, branching, termination, bending, spacing, and clustering [18,19]. Simplified DFN models cannot fully describe fracture mechanics, and although useful for statistical analysis, they may not accurately represent correlations between fracture parameters, leading to less reliable predictions of geomechanics data [20]. The discrete fracture network (DFN) modeling approach employs statistical methods to predict fracture parameters, characterized by its inherent randomness, which can limit the reliability of data for specific locations. In contrast, modeling methods that integrate geostatistical techniques utilize predictive approaches to estimate fracture parameters in unknown regions. These methods are underpinned by stronger theoretical foundations and yield results that are statistically more robust and persuasive. By leveraging spatial correlations and geological data, geostatistical approaches enhance the accuracy and reliability of fracture parameter predictions. Block-based methods, including geostatistics and multiple-point geostatistics, have been used to model fracture networks. These models, combined with geostatistics, can represent correlations between certain parameters, such as Kriging and Markov chains [21,22]. Recent advancements have shown that incorporating shear strength predictions in anisotropic materials significantly enhances the accuracy of fracture network models, which is crucial for optimizing grouting strategies in complex geological conditions [23].

The objective of this study was to investigate fracture distributions to optimize grouting processes. This study presents a methodology for estimating fracture distribution along grouting galleries in building foundation projects based on geological mapping data. The research methodology comprises two main components: (1) the application of a one-dimensional Markov chain method to estimate fracture density in simulated grouting galleries, incorporating reliability analysis for parameter determination; and (2) the integration of the Baecher disk model with discrete fracture network (DFN) modeling to develop a three-dimensional fracture network model. The proposed methodology was validated through a case study utilizing geological data from the Dongzhuang hydropower station, demonstrating its effectiveness in fracture network modeling for grouting gallery applications in building foundation structures.

2. Methodology

Current modeling approaches that integrate geostatistics with discrete fracture network (DFN) methods require extensive geological data as a foundation. However, some scholars have noted that Markov chain models can be effectively applied with limited data, thereby enhancing their practical utility. Additionally, combining Markov chains with DFN modeling enables the consideration of both the correlation and stochasticity of fracture development. Based on these advantages, this study proposes the following modeling workflow. The workflow for fracture network modeling used in this study is illustrated in Figure 1. The process begins with the input of raw geological data, which is subsequently processed through three key steps: (1) data collection and classification, (2) parameter determination, and (3) reliability analysis and modeling. In Step 1, rock fracture data is gathered from geological mapping based on excavation progress units. The K-means++ algorithm is employed to classify the dataset, with the elbow method used to determine the optimal number of clusters. This classification results in Data Set 1. In Step 2, the first-order Markov property is tested for each fracture group to define the length scale of each fracture data unit. A 1D Markov chain model is then applied to simulate the training dataset, producing diagonal correction coefficients for the transition matrix in each fracture density group. The optimized transition matrices for the four groups are then utilized to predict fracture density in the target interval. Meanwhile, fracture orientation and size parameters are determined using distribution models, leading to the formation of Data Set 2. In Step 3, the predicted fracture densities from Step 2 are evaluated, followed by a reliability assessment of the 1D Markov chain model. To further refine the model, the Monte Carlo method is used to generate additional fracture parameters. This ultimately leads to the establishment of the fracture network model, forming Data Set 3, which serves as the basis for further modeling. A detailed explanation of the methods utilized in this workflow is provided in the following sections.

2.1. First-Order Markov Chain Property Test for Fracture Data Processing

The application of Markov chains in stratigraphic variation simulation requires that unit attribute state transitions follow the first-order Markov property [24]. Since these transitions can be described by Markov chains of different orders (an aspect often overlooked in most studies), some researchers have introduced methods for determining the optimal order within coupled Markov chain theory [25]. Here, we select the simulation unit scale L based on the first-order Markov property test theory. It follows the following two steps.

The likelihood ratio test statistic η_t−1,t is calculated by [25]:

$${\eta }_{t-1,t}=2{\displaystyle {\sum }_{i,\dots ,l=1}^m{T}_{ij,\dots ,kl}\left(\ln \frac{T_{ij,\dots ,kl}}{T_{ij,\dots ,k}}-\ln \frac{T_{j,\dots ,kl}}{T_{j,\dots ,k}}\right)}$$

(1)

where T_ij,…,kl represents the number of occurrences of a state transition sequence S_i→S_j→⋯→S_k→S_l over t consecutive steps in which S_k denotes the attribute of the geological unit.

Then, the calculation of the likelihood ratio test statistic η_k,q for any order is as follows [25]:

$${\eta }_{k,q}={\eta }_{k,k+1}+{\eta }_{k+1,k+2}+\cdots +{\eta }_{q-1,q}$$

(2)

The critical value is determined of the chi-square test at the 5% significance level for the corresponding degrees of freedom, and compared to assess whether it meets the critical level restriction requirement, where the degrees of freedom for η_k,q is (m_k − m_q)(m − 1).

The Akaike information criterion (AIC) method is used for evaluating the performances of the statistical models. The loss function R(k) refers to the following [26]:

$$R\left(k\right)={\eta }_{k,q}-\left(m^M-m^k\right)\left(m-1\right)$$

(3)

where m is the total number of states, M is the upper bound order of the Markov chain, and k is the order of the Markov chain being tested. The smallest R value corresponds to the most likely order k of the Markov chain.

2.2. Methods for Parameter Determination

2.2.1. One-Dimensional Markov Chain Model

The K-means++ algorithm offers advantages in speed and effectively avoids the problem of the clustering process. The use of the K-means++ algorithm requires determining the number of clusters, n; selecting an appropriate n is crucial for accurate division results. The optimal n could be determined using methods such as the silhouette coefficient and the elbow method. The elbow method, in particular, is simple in principle and highly reliable. It utilizes the sum of squared errors (SSE) to intuitively determine the optimal number of clusters, n. The formula is as follows:

$$SSE={{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{p\subseteq C_i}\left|p-m_i\right|}}}^2$$

(4)

where n represents the number of clusters, p denotes the sample points belonging to the dominant group of the i cluster, and m_i indicates the centroid of the dominant group of the i cluster.

The 1D Markov chain model is then used to simulate the training data grouped by the previously employed K-means++ clustering algorithm. The following steps outline the process for determining these correction coefficients and applying them to estimate the data in the test set [26].

(1)

The transition probabilities between different density attributes of adjacent units are calculated using the following equation [27]:

$$\begin{array}{l}P\left(\left.X_i=S_i\right|X_{i-1}=S_l,X_N=S_q\right)={\displaystyle \frac{P\left(\left.X_N=S_q\right|X_i=S_k\right)P\left(\left.X_i=S_k\right|X_{i-1}=S_l\right)}{P\left(\left.X_N=S_q\right|X_{i-1}=S_l\right)}}\\ ={\displaystyle \frac{p_{kq}^{h\left(N-i\right)}{p}_{lk}^h}{p_{lq}^{h\left(N-i+1\right)}}}=p_{\left.lk\right|q}^h, k=1,\dots ,m\end{array}$$

(5)

where X_i represents the geological unit at location i, and S_k denotes the attribute of the geological unit. The probability $p_{kq}^{h\left(N-i\right)}$ of transitioning from attribute k in the geological unit to attribute q is obtained by raising the one-step transition probability matrix to the power of N − i.

(2)

The transition of fracture density attributes from a known unit to its adjacent unit. The process repeats until the fracture density of all predicted units is determined according to the following [27]:

$${\displaystyle {\sum }_{f=1}^{k-1}{p}_{\left.lk\right|q}}<u\le {\displaystyle {\sum }_{f=1}^k{p}_{\left.lk\right|q}},k=2,\dots ,m$$

(6)

where u represents a uniformly distributed random number generated in the interval [0, 1].

Specified, the probability of occurrences of fracture density is calculated using a selected interval [26], as below:

$$P_l=P\left(Z_1=S_{k1}\right)\times P\left(Z_2=S_{k2}\right)\times \cdots \times P\left(Z_{N_Z}=S_{k{N}_Z}\right)$$

(7)

where P(Z_i = S_ki) represents the occurrence of attribute S_ki in unit i, which is calculated as N(Z_i = S_ki), and N(Z_i = S_ki) is the number of times the attribute S_ki appears in unit Z_i over N_sim simulation runs.

The optimal correction factor for fracture density adjustment is determined by comparing the calculated P_l values from different transition matrices with the correction factor corresponding to the transition matrix that yields the highest steady-state probability.

2.2.2. Distribution Model of Fracture Orientation and Size

(1) Orientation

The distribution model of orientations commonly uses the Fisher distribution [28], as follows:

$$\left\{\begin{array}{l}f\left(\phi \right)={\displaystyle \frac{p\sin \phi e^{p\cos \phi }}{2\sinh \left(p\right)}},0\le \phi \le {\displaystyle \frac{\pi }{2}}\\ {\displaystyle \frac{1}{2\pi }},0\le \phi \le 2\pi \end{array}\right.$$

(8)

where φ and θ represents the average values of the dip angles and the azimuth vectors of all fracture surfaces, respectively; p reflects the degree of data concentration and is calculated as follows [28]:

$$P={\displaystyle \frac{N-1}{R-N}}$$

(9)

where R represents the length of the central direction cosine, while N denotes the total number of fracture attitude sample points.

(2) Diameter

Some researchers assume that the diameter and the fracture trace follow the same distribution form [29]. Fracture trace has been assumed to be following lognormal [8], exponential [30], or gamma [31] distributions. A method has been proposed by researchers to estimate the mean and standard deviation of the diameter distribution in the disk model on the corresponding parameters of the fracture trace length distributions [29].

The procedure can be succinctly outlined as follows. Initially, the optimal distribution form and its parameters for trace lengths are determined utilizing the maximum likelihood estimation method complemented by the Kolmogorov–Smirnov (KS) test. Subsequently, the parameter values for various diameter distributions are computed employing the formulae presented in

Table 1. Relationship between the distribution parameters of fracture length y and diameter x (modified after Ref. [29]. 2025, Elsevier).

Distribution	μx	(σx)2
Lognormal	${\displaystyle \frac{128{\mu }_y^3}{3{\pi }^3\left({\mu }_y^2+{\sigma }_y^2\right)}}$	${\displaystyle \frac{1536{\pi }^2\left({\mu }_y^2+{\sigma }_y^2\right){\mu }_y^4-128{\mu }_y^6}{9{\pi }^6{\left({\mu }_y^2+{\sigma }_y^2\right)}^2}}$
Exponential	${\displaystyle \frac{2}{\pi }}{\mu }_y$	${\left({\displaystyle \frac{2}{\pi }}{\mu }_y\right)}^2$
Gamma	${\displaystyle \frac{64{\mu }_y^2-3{\pi }^2\left({\mu }_y^2+{\sigma }_y^2\right)}{8\pi {\mu }_y}}$	${\displaystyle \frac{[64{\mu }_y^2-3{\pi }^2\left({\mu }_y^2+{\sigma }_y^2\right)][3{\pi }^2\left({\mu }_y^2+{\sigma }_y^2\right)-32{\mu }_y^2]}{{\left(8\pi {\mu }_y\right)}^2}}$

Note: μ_y and σ_y represent the mean and variance of the trace length distribution, respectively, while μ_x and σ_x represent the mean and variance of diameter distribution, respectively.

. Ultimately, the most suitable diameter distribution form is selected through the application of Equation (9):

$${\displaystyle \frac{E\left(x^4\right)}{E\left(x^2\right)}}={\displaystyle \frac{4E\left(y^3\right)}{3E\left(y\right)}}$$

(10)

where E(y) presents the expected value of the trace length, and the other parameters similarly represent the expected values of their corresponding parameters.

2.3. Evaluation

2.3.1. Performance of Fracture Density Prediction

The performance of prediction results on the fracture density are evaluated using the coefficient of determination R², the accuracy matric ACC [1], and the confusion matrix. Both R² and ACC include the mean and maximum values when the number of simulations is set to N.

(1)

Coefficient of determination, R²

R² is commonly used to evaluate the accuracy of predicted values against actual values in regression models. It exhibits varying values and is calculated by the following:

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i-1}^N{\left(y_i-y_i^{*}\right)}^2}}{{\displaystyle {\sum }_{i-1}^N{\left(y_i-\overline{y}\right)}^2}}}$$

(11)

where N represents the total number of observations, y_i denotes the predicted value for the i observation, and $y_i^{*}$ stands for the actual value of the i observation, while $\overline{y}$ represents the mean of all observed values.

(2)

Accuracy matric, ACC

ACC metric provides an intuitive understanding of the proportion of correct predictions. It is calculated as the ratio of correctly predicted values to the total number of predictions, as follows:

$$\mathrm{ACC}={\displaystyle \frac{N_{\mathrm{true}}}{N_{\mathrm{all}}}}$$

(12)

where N_true represents the number of correctly observed values, and N_all denotes the total number of observations.

(3)

Confusion matrix

A confusion matrix is also used to evaluate the accuracy of predictions in classification problems [32]. Each row of the confusion matrix represents the total number of instances in a specific class, while each column represents the predicted instances in that class. Compared to overall accuracy, the confusion matrix provides a clearer depiction of how misclassifications are distributed across different classes. The basic form of the transition matrix is shown in Figure 2, where A_i represents the actual class, Ā_j represents the predicted class, and N_ij denotes the number of instances that actually belong to class A_i but are predicted as class Ā_j.

2.3.2. Reliability Analysis

The proportion of predicted units with absolute error between predicted and actual values within 2 as an evaluation metric are referred to as the conformation factor, ST. A threshold is set such that if the conformation factor exceeds 50%, (i.e., ST ≥ 50%), the prediction is considered valid. Thus, the proportion of valid results over N_sim simulations is defined as the reliability (RE) and used to assess the reliability of the modeling method as follows:

$$RE={\displaystyle \frac{N_{ST\ge 0.5}}{N_{\mathrm{sim}}}}$$

(13)

3. Simulation

3.1. Background

The Dongzhuang project is challenged by the distribution of fracture networks for the grouting project [33,34,35]. Situated on the northern limb of the Tangwangling syncline, the dam is constructed in monocline strata with an overall strike of NWW. Dip directions range from 190° to 210°, and dip angles vary from 30° to 55° (see Figure 3a). The site is generally free of very large regional folds or faults. Key geological features include faults, bedding shear zones, large bedding fractures, and joint fracture sets. Located in a limestone region affected by karst, the grouting scope is extensive for the bedrock impermeability curtain. The grouting gallery on the left bank area, WML3, can be taken as an example (see Figure 3b). As shown in Figure 3c, the holding rock is mainly thick with very thick layers of limestone, classified as Class II to III (Figure 4a). The rock mass is relatively fresh and intact, with a low to moderate distribution of in situ stress. The strike of the rock layers intersects at a small angle with the tunnel axis, indicating good overall rock stability. However, localized areas exhibit dissolution weathering zones (Figure 4b), tectonic fracture zones, and densely jointed zones, where the rock mass is more fractured and classified as Category IV or V. Given the presence of adverse geological conditions such as dissolution fractures, curtain grouting is necessary to address these fracture-related issues and ensure project safety. Therefore, this study employed fracture network modeling to visualize these adverse geological conditions, thereby facilitating the implementation of curtain grouting construction.

The grouting curtain in both the left and right bank areas and the dam shoulder consists of five layers of galleries. The top three layers have cross-sections of 3 m × 3.5 m, while the bottom two layers measure 4 m × 4.5 m. The elevations of these layers are 592 m, 640 m, 685 m, 751 m, and 804 m, respectively. Each grouting gallery’s curtain depth must extend 5 m below the elevation of the lower tunnel’s slab to ensure a tight connection between the upper- and lower-layer curtains.

3.2. Data Division

The elbow method was employed to analyze the aforementioned borehole fracture data, with the results illustrated in Figure 5a. It can be seen that the SSE decreases with the increasing number of clusters and does not vary much since n = 4. By plotting the density of fractures in the pole diagram (Figure 5b), it is supported that a division of 4 is reasonable. Subsequently, the rose diagrams based on the above four clusters are illustrated in Figure 6. It can be seen that the advanced mean strikes are 225°, 355°, 185°, and 55°, for sets 1 to 4, respectively. There is no obvious advanced dip presented. These results indicate that the dip directions and dip angles of fractures in the study area are widely distributed but predominantly concentrated in the SE and NW directions, Among the four orientations, fractures with a dip direction in the northwest account for a larger proportion.

3.3. Parameter Determinations

(1)

Fracture orientations

For the subsequent simulation of fracture orientations and diameters using the DFN method, we organized the above classification results into Data Set 1. This includes the range and mean values of dip direction θ and dip angle φ for each group. According to Equation (8), the mean values $\overline{\theta }$ and $\overline{\phi }$ are calculated by converting the angles into spherical coordinates, while the ranges are determined using the maximum and minimum actual values of dip direction and dip angle. Finally, the orientation for each group, the number of fractures per group, and the Fisher distribution parameter p for the four groups of fracture orientation data are calculated using Equation (9), with the final results presented in

Table 2. Fracture occurrence grouping with Fisher distribution parameters.

Set	θ Range, °	$\overline{\mathit{\theta }}$, °	φ Range, °	$\overline{\mathit{\phi }}$, °	Quantity	R	p
1	285/360	137	7/90	67	434	403	13.8
2	0/105	27	10/87	63	214	183	6.9
3	205/282	197	17/89	67	227	204	9.9
4	110/201	302	10/89	61	330	298	10.2

.

(2)

Fracture density

Figure 7 illustrates the methodology for data acquisition and unit division in fracture modeling. Figure 7a shows the spatial distribution characteristics of tunnel floor trace data, with a tunnel floor width of 4 m and a prediction section length of 20 m. The initial unit size along the tunnel length is set to 2 m, with further optimization based on first-order Markov property testing results. Figure 7b presents the simulation unit division scheme and fracture counting methodology. Fractures are counted by determining whether their midpoint lies within a small unit (counted as 1), with sequential summation until all fractures within the specified unit are accounted for. Since the counting process incorporates both roof and floor trace data, the resulting fracture counts represent three-dimensional spatial distributions. Given the known unit dimensions, fracture density is directly represented by the number of fractures per unit. As this value is a constant multiple, it does not affect prediction accuracy. In Figure 7a, Set1 and Set2 represent fracture groups with different orientations, showing statistical densities of 2 and 1, respectively, labeled as attributes S₁ and S₂. The white area in Figure 7b indicates regions with unknown fracture distributions, where unit fracture density is labeled as S_k.

Based on the unit division results described earlier, we obtained the fracture density data of small units within the grouting chamber WML-3. Analysis revealed that a few units exhibited abnormally high density values. By imposing a threshold on unit density, the complexity of density transitions between adjacent units is reduced, thereby simplifying the categorical structure of the transition probability matrix. This enhancement in data stability facilitates compliance with the assumptions of first-order Markov analysis. To eliminate the impact of these outliers on the prediction results and ensure the first-order nature of the transition results, we adopted a data correction method by setting the maximum density value for each group to 5. The rationale for selecting 5 as the upper limit is that the proportion of units with density values greater than 5 in each group is only 1%, 0%, 0%, and 0.4%, respectively. Outlier removal is a widely adopted technique in data simulation to enhance the robustness of statistical analyses. Prior to computing the transition matrix, a density threshold was established to streamline the estimation process for the prediction interval. Given the minimal proportion of high-density units within the study area, the impact of this threshold on prediction outcomes is negligible. Furthermore, as the threshold is set such that the difference between the maximum density and the peak density within small units is within 4, the effect of this adjustment on large-scale fracture statistics is effectively inconsequential. This approach ensures data stability and predictive reliability while adhering to the assumptions of first-order Markov analysis.

In this study, the optimal Markov order for unit fracture density transitions was determined by jointly evaluating the likelihood ratio test statistic against the critical value at a given significance level and the degrees of freedom, along with an analysis of the loss function. The Markov order corresponding to the minimum loss function value is regarded as the optimal order. The relevant results of this analysis are presented below. Based on the corrected data, we applied the first-order Markov property test method to analyze the fracture density data of the four groups.

Table 3. First-order Markov likelihood ratio test results.

(a) Results of the first set of likelihood ratio test statistic calculations
Statistics	Statistics value	Degree of freedom	5% Level of significance
η01	88.09	25	37.65
η02	230.58	175	206.87
η03	466.81	1075	1152.39
η12	142.5	150	179.58
η13	378.72	1050	1126.5
η23	236.22	900	970.9
η33	0	0	0
(b) Results of the second set of likelihood ratio test statistic calculations
Statistics	Statistics value	Degree of freedom	5% Level of significance
η01	85	25	37.65
η02	159.09	175	206.87
η03	264.54	1075	1152.39
η12	74.08	150	179.58
η13	179.54	1050	1126.5
η23	105.45	900	970.9
η33	0	0	0
(c) Results of the third set of likelihood ratio test statistic calculations
Statistics	Statistics value	Degree of freedom	5% Level of significance
η01	48.86	25	37.65
η02	118.15	175	206.87
η03	240.02	1075	1152.39
η12	69.29	150	179.58
η13	191.16	1050	1126.5
η23	121.87	900	970.9
η33	0	0	0
(d) Results of the fourth set of likelihood ratio test statistic calculations
Statistics	Statistics value	Degree of freedom	5% Level of significance
η01	58.58	25	37.65
η02	195.33	175	206.87
η03	394.18	1075	1152.39
η12	136.75	150	179.58
η13	335.6	1050	1126.5
η23	198.85	900	970.9
η33	0	0	0

presents the results of the first-order Markov likelihood ratio test statistic. Statistical values presented in bold in the table indicate that the tested Markov order satisfies the 5% significance level critical value under the chi-square distribution with degrees of freedom corresponding to the data in the third column. Thus, the Markov orders satisfying this critical value can be used to predict unit fracture density. As shown in Table 3, both the first and second orders can be used to describe the transition of unit fracture density, while the zero order is not suitable for describing the transition of unit fracture density.

To further determine the optimal transition order, we calculated the loss function values using the Akaike information criterion (AIC) method [36], as shown in

Table 4. AIC values corresponding to different orders (M = 3).

(a) AIC values of the first set
Order	0	1	2	3
AIC	−1683.19	−1721.28	−1563.78	0
(b) AIC values of the second set
Order	0	1	2	3
AIC	−1885.46	−1920.46	−1694.55	0
(c) AIC values of the third set
Order	0	1	2	3
AIC	−1909.98	−1908.84	−1678.13	0
(d) AIC values of the fourth set
Order	0	1	2	3
AIC	−1755.82	−1764.4	−1601.15	0

. The order corresponding to the minimum loss function value is considered the optimal transition order. Table 4 shows that, except for the third group, the optimal transition order for the remaining groups is the first order. Combined with the data in Table 3, it is evident that the zero order is not suitable for describing the transition of unit fracture density. Therefore, the optimal Markov transition order for the third group remains the first order.

A set of suitable transition coefficients k is selected from a predefined range to modify the diagonal elements of the 1D Markov transition matrix, thereby reconstructing the probability matrix. Subsequently, the reconstructed transition probability matrix is applied to simulate predictions on the known dataset. By comparing the predicted results with the actual data, the optimal correction coefficient is determined. Finally, the optimal coefficient is incorporated into the transition matrix construction, which is then utilized for predictive simulations on the test dataset. The results for the four groups of test data, shown in Figure 8, indicate a consistent increasing trend in average accuracy. Notably, this trend levels off as it approaches a value of 4, suggesting that an upper limit is reached. The best accuracy, exhibiting a fluctuating increasing trend, also displays phases of rapid and steady growth, confirming that all four groups have an upper limit for their best accuracy. The figure illustrates that as the diagonal elements of the transition probability matrix increase gradually, the prediction accuracy improves correspondingly, reaching a plateau determined by the steady-state probabilities. Moreover, when the diagonal elements approach 1, the unit transitions become predominantly governed by the states of known boundary units, resulting in negligible variation within the interval.

In addition, the occurrence probability P_l of fracture density corresponding to different k-values in 100,000 simulations is calculated using Equation (7). The results indicate that the maximum occurrence probabilities for the four fracture density datasets are 0.003, 0.039, 0.254, and 0.096, with corresponding optimal k-values of 2.1, 2.05, 4, and 3.95, respectively. Based on these findings, the transition probability matrices were revised using the aforementioned k-value adjustment coefficients, and the revised results are detailed in

Table 5. Transition probability matrix correction using best k.

(a) Correction transition number (Set 1, k = 2.1)
State	0	1	2	3	4	5
0	357	42	15	10	4	5
1	40	65.1	11	9	1	4
2	18	11	18.9	7	3	3
3	12	7	9	16.8	1	0
4	2	1	2	3	6.3	1
5	5	3	5	0	0	0
(b) Correction transition probability matrix (Set 1, k = 2.1)
State	0	1	2	3	4	5
0	0.82	0.1	0.03	0.02	0.01	0.01
1	0.31	0.5	0.08	0.07	0.01	0.03
2	0.3	0.18	0.31	0.11	0.05	0.05
3	0.26	0.15	0.2	0.37	0.02	0
4	0.13	0.07	0.13	0.2	0.41	0.07
5	0.38	0.23	0.38	0	0	0
(c) Correction transition probability matrix (Set 2, k = 2.05)
State	0	1	2	3	4	5
0	0.91	0.04	0.04	0	0	0
1	0.41	0.46	0.1	0.04	0	0
2	0.31	0.27	0.3	0.05	0.05	0.02
3	0.33	0.22	0.44	0	0	0
4	0.25	0.5	0.25	0	0	0
5	0.5	0	0.5	0	0	0
(d) Correction transition probability matrix (Set 3, k = 4)
State	0	1	2	3	4	5
0	0.93	0.04	0.02	0.01	0	0
1	0.49	0.35	0.11	0.04	0	0
2	0.24	0.17	0.56	0.04	0	0
3	0.42	0.11	0.26	0.21	0	0
4	0.67	0.33	0	0	0	0
5	0.33	0.67	0	0	0	0
(e) Correction transition probability matrix (Set 4, k = 3.95)
State	0	1	2	3	4	5
0	0.9	0.07	0.02	0.01	0.01	0
1	0.31	0.54	0.08	0.04	0.01	0.02
2	0.31	0.22	0.31	0.08	0.06	0.02
3	0.26	0.08	0.11	0.52	0.03	0
4	0.2	0.27	0.2	0.07	0.26	0
5	0.17	0.5	0.17	0.17	0	0

. Table 5a presents the transition count matrix for one set of training data, while Table 5b, Table 5c, Table 5d, and Table 5e display the transition probability matrices corresponding to four sets of training data, respectively.

(3)

Performances of predictions for fracture parameters

The prediction performance of fracture density parameters is systematically evaluated using the revised transition probability matrix. The procedure is as follows: First, four training datasets are independently predicted, followed by the integration of results. Due to the stochastic nature of the method, the number of predictions is incrementally increased until the accuracy stabilized, achieving computational efficiency while ensuring reliability. In the final prediction processing, the density upper limit within units is set to 5, and a normalization operation (adding 1 to density values) is applied to generate comparative histograms for visualizing discrepancies between actual and predicted values (see Figure 9). The results demonstrate an overall prediction accuracy of 43% for the test dataset. When allowing a tolerance of one fracture value deviation, the accuracy increases to 80%. In the prediction process, the tolerance ranges of ±1 and ±2 represent the allowable deviations in fracture counts within individual units. The establishment of these ranges is primarily based on the consideration that such deviations typically meet the requirements of practical engineering applications. Notably, when the prediction accuracy reaches 43%, the impact of these allowable errors on the overall modeling domain is generally acceptable. However, the specific tolerance values should be determined by engineering practitioners based on project-specific requirements to comprehensively evaluate the feasibility and applicability of the proposed method, under the aforementioned tolerance settings the determination coefficients (R²) are 0.14 and 0.52, respectively, indicating the method’s strong applicability for fracture density prediction at the unit scale. Further analysis reveals that the predicted trends align closely with observed data, particularly in both high- and low-density regions, confirming the model’s capability to effectively capture the underlying distribution patterns.

The confusion matrix is used to quantify the misclassification patterns using a confusion matrix (see Figure 10). The results reveal that 100% of misclassified samples are assigned to density value 1, which is strongly correlated with the high transition pro-portion (51% initially, increasing to 68% after diagonal element adjustment) from density attribute 1 to 1 in the transition count matrix. This behavior indicates that the model performs well in regions dominated by attribute 1 but exhibits significant accuracy degradation in areas with heterogeneous attribute distributions. Furthermore, since the final predictions are aggregated from four independent simulations, errors in any single simulation propagate through the superposition process, reducing overall reliability. The model achieves prediction accuracies above 30% for all density attributes; however, its generalization capability is limited for low-sample categories (e.g., densities 3 and 4). Future work should integrate multi-source fracture data and develop error compensation mechanisms to enhance robustness in complex scenarios.

3.4. Reliability Analysis of Simulation Results

As shown in Figure 11, it demonstrates the distribution of the proportion of units within the prediction intervals where the absolute deviation between actual and predicted values is less than 2, based on 100,000 simulations. The results show that when the error rate threshold is set to exceed 0.5, the model is deemed unreliable. However, the calculated reliability reaches 99.91% across all simulations. This finding validates the model’s high-precision characteristics in controlling prediction deviations and provides quantitative evidence for the stability of large-scale fracture density predictions.

3.5. Fracture Diameter

The aforementioned discrimination method is applied by employing Maximum Likelihood Estimation (MLE) and Kolmogorov-Smirnov test (K-S test) to determine the probability distribution types and parameters of four fracture trace length groups. The results demonstrates that the optimal fitting distributions for the four fracture trace length groups are log-normal distribution (Groups 1, 3, and 4) and gamma distribution (Group 2), with corresponding parameters of (μ = 1.67, σ = 0.61), (μ = 3.50, σ = 1.55), (μ = 1.75, σ = 0.57), and (μ = 1.59, σ = 0.51), respectively. As illustrated in Figure 12, the cumulative probability distribution curves of measured data exhibit remarkable consistency with the theoretical optimal distribution functions. Furthermore, the distribution characteristics of fracture diameters are calculated by integrating parameters from Table 1 (see

Table 6. Calculation results of diameter distribution and fracture trace length distribution parameters.

Set	Assume Distribution	${\mathit{\mu }}_{\mathit{x}}$	${({\mathit{\sigma }}_{\mathit{x}})}^2$	$\frac{\mathit{E}({\mathit{x}}^4)}{\mathit{E}({\mathit{x}}^2)}$	$\frac{4\mathit{E}({\mathit{y}}^3)}{3\mathit{E}(\mathit{y})}$	Recommended Distribution
1	Log-normal	2.031	0.206	5.263	5.765	Log-normal
	Exponential	1.066	1.136	13.636
	Gamma	2.026	0.216	5.253
2	Log-normal	4.033	1.727	26.939	61.761	Exponential
	Exponential	2.23	4.973	59.675
	Gamma	3.987	1.889	26.688
3	Log-normal	2.171	0.113	5.305	6.066	Gamma
	Exponential	1.112	1.237	14.838
	Gamma	2.17	0.116	5.306
4	Log-normal	1.98	0.079	4.332	4.5	Log-normal
	Exponential	1.01	1.02	12.241
	Gamma	1.979	0.081	4.331

). The analysis reveals that the four fracture diameter groups successively followed log-normal distribution, exponential distribution, gamma distribution, and log-normal distribution, with statistical parameters and goodness-of-fit metrics confirming the robustness of the model.

3.6. Fracture Network Modelling

This study adopted a uniform distribution assumption to simulate the spatial distribution of fracture locations. Considering that the scale of the simulation units is small while fractures are large-scale structures, this assumption is deemed reasonable under these conditions. Based on this assumption, the fracture parameters, including density, orientation, and diameter distributions, were summarized, and the specific data are presented in

Table 7. Parameters of four sets of fracture network distribution model.

Set	Density	Location	Orientation					Diameter
			Dip		Dip Direction		Fisher Constant	Distribution Constant	Range
			Range	Mean	Range	Mean
1	Markov chain	Uniform	7/90	67	285/ 360	137	13.8	Log-normal	0.1/15.9
								2.031/0.206
2			10/87	63	0/105	27	6.9	Exponential	0.9/17.1
								2.23
3			17/89	67	205/ 282	197	9.9	Gamma	1.3/14.8
								2.17/0.116
4			10/89	61	110/ 201	302	10.2	Log-normal	1/16.3
								1.98/0.079

. The fracture parameters were generated using the Monte Carlo simulation method, with the results shown in

Table 8. Partial prediction results for the location and shape parameters of fracture planes in Set 1.

Fracture Number	X (m)	Y (m)	Z (m)	Dip Direction (°)	Dip (°)	Diameter (m)
1	2.1	0.7	4.2	287	57	8.1
2	27.4	2.6	2.7	301	88	12.3
3	26.4	2.7	4	287	46	6.5
4	28.5	2.7	4.3	287	59	8
5	28.5	2.9	3.6	314	62	11.1
6	31.2	1.7	2.4	292	86	9.9
7	31	2	4.1	298	58	6

. A fracture network diagram was created based on these parameters, and the results are shown in Figure 13.

4. Discussion

Conventional surface trace data statistics primarily capture orientation and trace length information, while fracture density parameters often rely on borehole data. This inconsistency in parameter acquisition methods may introduce errors in predictions. To address this issue, our research utilized tunnel wall trace data as a unified data source, effectively reducing errors caused by multi-source data integration. Furthermore, we combined geostatistical methods with discrete fracture network (DFN) for fracture modeling. The findings demonstrate that the model exhibits good reliability within an acceptable error margin, particularly in capturing the variation trends of fracture density data.

Based on the proposed methodological framework for fracture characterization in curtain grouting zones (Figure 14), a systematic workflow is established through three-dimensional fracture network modeling along the grouting gallery axis. The vertical fracture density detection is implemented through borehole drilling from the tunnel floor, with 4.5 m simulation unit thickness as the fundamental parameter. The numerical model reveals that approximately 21.8% of cross-stratal fractures penetrate into adjacent units (Figure 14). However, comparative analysis between Figure 13 and Figure 14 demonstrated that the fracture density distribution in target strata primarily depends on the local network configuration rather than upper-layer extensions. Although the predictive layer exhibits higher fracture density in Figure 13, field detection results in Figure 14 show significant reduction of effective fractures penetrating through the gallery floor. The detection depth shows quantitative correlations with the maximum fracture dimensions. When encountering fractures with maximum diameters of 9 m, the calculation requires integration of one adjacent unit above and below the target layer (total 9 m thickness). For larger fractures up to 13.5 m, two adjacent units (total 13.5 m thickness) must be incorporated. This dynamic stratum selection mechanism, governed by fracture scale parameters, ensures engineering accuracy in density detection through optimized hierarchical computation.

While the proposed method represents an improvement over traditional fracture modeling approaches, several limitations should be acknowledged. First, the selection of fracture data for modeling is based on parameter completeness, without considering geological factors such as fracture genesis and formation timing. This oversimplification in data classification may affect the accuracy of modeling results. Second, the predictive model exhibits limited robustness in estimating fracture density, particularly when dealing with units characterized by high data variability. This issue necessitates further investigation with additional geological data to identify the underlying causes and improve the prediction reliability. Finally, the current modeling framework focuses exclusively on fracture density prediction. Future research should explore the integration of advanced reliability theories to extend the predictive capabilities to other critical fracture parameters (e.g., orientation, trace length), thereby enhancing the model’s overall applicability and reliability.

5. Conclusions

This study presents a fracture modeling approach for curtain grouting applications, with the Dongzhuang dam serving as a case study. The major conclusions are as follows:

(1) Fracture trace data collected from tunnel walls offered a consistent source for modeling fracture density, orientation, and size, minimizing errors from heterogeneous data sources. A one-dimensional Markov chain was used to simulate fracture density transitions, with diagonal correction improving the prediction accuracy. The model achieved 99.91% reliability under a ±2 deviation threshold.

(2) The DFN model, incorporating Fisher and multiple statistical distributions, successfully simulated spatial patterns of fracture orientation and scale. The reconstructed networks matched well with site observations, especially in capturing clustering and penetration features. The predicted fracture network showed good agreement with borehole verification data beneath the grouting galleries. The dynamic adjustment of model strata thickness based on fracture size improves the engineering applicability for curtain layout planning.

The current model does not account for the genesis and temporal evolution of fractures. Future research should integrate geological genesis and reliability-based extensions to improve parameter prediction for complex geological scenarios.