Stability Analysis of Jointed Rock Slopes Based on the Universal Elliptical Disc Model

A discrete fracture network (DFN) represents the random distribution of fractures within rock masses in three dimensions and has been widely used in the stability analysis of jointed rock slopes. This study introduces a novel method for constructing three-dimensional DFN models to address the limitations of traditional circular or polygon-based models. This method employs the universal elliptical disc (UED) model, combined with the discrete element method (DEM) in 3DEC, for the probabilistic analysis of jointed rock slope stability. The paper begins with an explanation of the fundamental principles of the UED model, and continues to detail the procedures for creating a three-dimensional DFN using the UED model and for analyzing slope stability using the DEM. An open-pit mine rock slope in the USA serves as a case study to illustrate the application of the proposed method. This approach facilitates the integration of the UED model into 3DEC and offers a more realistic depiction of the three-dimensional DFN, which enhances both the accuracy and efficiency of stability analysis in jointed rock slopes.


Introduction
Owing to diverse geological, physical, and chemical processes, rock masses exhibit extensive and complex fractures and discontinuities, which play a pivotal role in determining the structural characteristics of rock masses.These fractures significantly influence the macroscopic mechanical properties, seepage behaviors, and slope stability.Accurately and quantitatively mapping these rock discontinuities poses a substantial challenge.The current methodology typically involves creating a three-dimensional (3D) discrete fracture network (DFN) based on limited measurements from visible discontinuities on rock outcrops, underlining the importance of selecting an apt model for DFN representation to ensure the reliability and effectiveness of analyses related to slope seepage and stability in the presence of rock mass discontinuities.
The task of depicting rock discontinuities via a 3D DFN has been extensively explored, with many studies utilizing the Baecher circular disc model [1] for its mathematical simplicity and computational efficiency.However, this model struggles to accurately simulate slender fractures or joints of complex shapes [2], impacting the precise evaluation of slope seepage and stability [3,4].To overcome these shortcomings, alternative models have been proposed, such as the parallelogram-based model [5], the polygon-based model [6], and the rectangle-based model [7], each offering improved flexibility in representing the intricate shapes of rock fractures.Despite these advancements, challenges and ambiguities remain in effectively portraying the complexity of rock fractures.Recognizing these limitations, Zhang et al. (2002) introduced an ellipse-based model, further adapted by Jin et al. [8], to simulate rock discontinuities.Subsequent developments [3,9] and Zheng et al. (2020) led to the creation of new universal elliptical disc (UED) models, which showcased potential in accurately representing realistic rock fractures [10].Nonetheless, the UED model has not yet been used extensively in slope engineering practice.
This study proposes a novel approach to incorporate the UED model within a 3DEC framework for constructing a more realistic 3D DFN of rock masses.The application of this methodology to an openpit mine rock slope in the USA demonstrates the vital role of the UED model in generating realistic DFNs for rock slope analysis, thereby underscoring its potential for broader adoption in slope engineering practices.

Universal elliptical disc (UED) model
The conventional Baecher disc model parameters encompass the centroid location, diameter, dip direction, dip angle, and discontinuity density.The UED model introduces three additional parameters: the major axis length, the ratio of the major to minor axis, and the rotation angle, endowing the UED model with enhanced versatility in capturing complex rock fracture geometries [3,4].The forthcoming sections of this paper discuss the principles and procedural steps of the advocated approach.

Identification and construction of equivalent fracture planes
Discontinuities within jointed rock masses are inherently three-dimensional and exhibit complex configurations.Occasionally, the vertices of a discontinuous surface may not align on a single plane, necessitating the projection of these vertices onto an optimized two-dimensional (2D) plane prior to employing the UED model for DFN construction [3], because the model assumes that the fracture surfaces are planar.
Initially, the convexity of spatial fracture polygons is determined, followed by the computation of the centroid coordinates (xm, ym, zm) of the polygons using geometric methods [11].Subsequently, the matrix H0 representing the coordinates of n vertices is calculated relative to the centroid of the polygon.To determine the optimal fitting plane, the singular value decomposition (SVD) method is employed to factorize matrix H0 as follows [3]: (1) Therefore, U is a n × n unitary matrix, V is a 3 × 3 unitary matrix, and Σ is a n × 3 diagonal matrix.The unitary matrix V is then multiplied by the unit column vector (0, 0, 1) T to obtain the normal vector n of the projected plane in 3D space.Project spatial polygons onto a fitting plane and assume that the coordinates of the n-th vertex after projection are (xn, yn, zn).
Thereafter, the geometric parameters α and β shown in figure 1 are calculated and the polygon is rotated onto the xoy plane through coordinate transformation.Figure 1 The z-axis coordinate has a certain value; therefore, it was not considered.Therefore, the 3D surface was converted into a 2D figure.
(a) Fitting of a real facture polygon using an ellipse (b) Illustration of an ellipse with an arbitrary rotation angle γ Figure 2. Fitting of a real facture polygon using an ellipse with an arbitrary rotation angle γ.

Figure 3.
Step-by-step illustration of the optimal elliptic disc fitting process.

Determination of the optimal ellipse parameters
To accurately represent the rock fractures with ellipses, the initial step involves aligning the center of a fracture polygon with the center of an ellipse of equivalent area, as illustrated in figure 2. The geometry of the ellipse is defined by several model parameters: length of major axis e, length of minor axis f, the ratio of major axis to minor axis k, the semi-latus rectum g, and the rotation angle γ.Specifically, the rotation angle γ refers to the angle between the semi-major axis above the x-axis and the positive x-axis in an anti-clockwise direction, refer to figure 2(b), and has a value between 0 and π.A computer graphics technique was subsequently applied to calculate the union and intersection areas between the fracture polygon and the optimally fitted ellipse.The key steps are outlined below: (1) Calculate the polygon area SP.
(2) Position the fracture polygon and fitted ellipse within the same coordinate space and digitize both figures.The area of the digitized ellipse, denoted as SE, corresponds to the number of pixels within the ellipse (Foley et al., 1995).
(3) Superimpose the ellipse over the fracture polygon and tally the pixels within the union area, SU, of the fracture polygon and fitted ellipse.(3) To determine the optimal ellipse, the intersection over union (IoU) metric, which is the ratio of the intersection area to the union area between the fracture polygon and fitted ellipse, was utilized and assessed as follows: According to Foley et al. [11], a threshold IoU value of 0.5 is typically used.An optimal ellipse is identified when its IoU surpasses this threshold value.The adaptive grid-based exhaustive sampling method facilitates traversal sampling across various combinations of k and γ, calculating the IoU values for all parameter combinations.Ultimately, the elliptical model parameters associated with the highest IoU value are selected.Figure 3 showcases the process of fitting the optimal elliptical disk.

Construction of optimal ellipses
The procedures outlined in the previous section are then repeated for all rock fracture planes, and the statistical distributions of e, k and γ can be obtained.In addition, using the field measurements at the rock outcrops, the statistical distributions of dip direction κ and dip angle φ can also be determined.The remaining model parameters such as the centroid coordinates of the ellipse (xm, ym, zm) followed a uniform distribution, as reported by Baecher [1].Random samples of the model parameters were generated by performing MCS.Based on the equation of a 2D ellipse, it is rotated into 3D space through a coordinate transformation.The conversion of the coordinates is presented in figure 4.
where θ denotes a parameter.Based on Equation ( 5), numerous spatial fractures are simulated.

Integration of the UED model in 3DEC
To construct realistic 3D DFNs and analyze slope stability utilizing the UED model, this study introduces a Python subroutine for integrating the UED model into 3DEC.The seven steps of the integration process are summarized as follows: (1) Generate random samples of ellipse model parameters via MCS using Equation ( 5) to simulate fractures.(2) Sequentially export the coordinates of the 20 vertices for each ellipse to a "DFN.txt"file.
(3) Employ the "geometry set s" command in 3DEC to archive the 3D ellipses.(4) Utilize the "call" command in 3DEC to import the "DFN.txt"file.
(5) Assign DFN properties to all 3D ellipses with the "DFN gimport geometry s" command.(6) Construct a rock slope model within 3DEC.(7) Integrate the DFN with the intact rock slope model using the "Jset DFN" command in 3DEC, resulting in a comprehensive jointed rock slope model.Figure 5 encapsulates the implementation strategy of the proposed method for creating DFN models of jointed rock masses employing the UED model.

Illustrative example
This section evaluates the effectiveness of the UED model in generating DFN for jointed rock masses and conducting a 3D discrete element analysis on slope stability, leveraging field measurements from an open-pit mine rock slope in the USA. Figure 6 presents the 3D slope model utilized in this investigation.To supplement the limited field measurements, statistics regarding the physical and mechanical properties of the rock masses and discontinuities, as documented by Lei and Wang [12], were incorporated, detailed in Tables 1 and 2.  From on-site geological survey data, a dataset comprising 56 fracture vertex coordinate points was compiled, establishing a fracture density of 0.1 trace/m 3 .The azimuth and UED model parameters for the fracture groups are enumerated in Table 3.The fitting of the optimal UED model follows the procedure described in Section 2.2.Furthermore, the sampling augmentation method introduced by Jiang et al. [13] is applied to expand the initial dataset, yielding 800 samples of e, k, and γ are obtained.Additionally, the dip direction and angle were sampled according to a Fisher distribution, as reported by Kemeny et al. [14].
The strength reduction method, a common approach in slope stability analyses, diminishes the shear strength of a rock mass until a critical failure state is reached.This technique, as applied within the "solve fos" command in 3DEC, facilitates the discrete element-based strength reduction analysis.Figure 7 illustrates a 3D DFN, spanning dimensions of 20 m × 20 m × 20 m generated in 3DEC using the UED model.Figure 8 displays the corresponding 3D jointed rock slope model.The safety factor for the slope was determined to be 2.67. Figure 9 exhibits the contour of horizontal displacement across a crosssection at x = 7 m.

Discussion
Compared to the Baecher disc model, the UED model offers a precise representation of jointed rock masses, establishing a robust basis for the accurate assessment of mechanical properties, seepage characteristics, and stability of rock slopes.To augment the engineering application value of the UED model, future research should extend to evaluating the impacts of various lithologies and rock bridges, thereby enhancing the resemblance between DFNs and actual rock mass fractures.

Conclusion
This study introduced a methodology for constructing DFNs utilizing the UED model, subsequently integrating these DFNs into 3DEC for the stability analysis of jointed rock slopes.The implementation procedures are detailed within this document.An open-pit mine rock slope in the USA served as a case study to demonstrate the effectiveness of the proposed method.Through the development of Python subroutines, the UED model was successfully incorporated into 3DEC, facilitating the creation of a realistic and accurate 3D DFN model.This integration significantly simplifies the application of the UED model in engineering practices, ensuring that the generated slope model closely mimics the characteristics of jointed rock slopes and offers more authentic stability evaluations of such slopes.

1 .
depicts the conversion of the coordinates.(a) Step 1 of coordinate transformation (b) Step 2 of coordinate transformation Figure Illustrations of the coordinate transformation from the 3D discontinous planes to the 2D discontinous planes.

O
After these two rounds of coordinate transformations, the coordinates of the n-th vertex of a planar polygon can be expressed as cos

O
Obtain the coordinates of the fracture vertices after planarization Calculate the area S p and digitize the fracture polygon Conduct grid-based exhaustive sampling of combinations of k and γ Construct and digitize the ellipses based on samples of k and γ Identify the most optimal combination of k and γ Calculate the union area, S U , and intersection area, S coin , of the fracture polygon and fitted ellipse, and IoU value for each sample (4) The intersection area Scoin of the fracture polygon and fitted ellipse is evaluated as follows:

4 .
(a) Step 1 of coordinate transformation (b) Step 2 of coordinate transformation (c) Step 3 of coordinate transformation Figure Illustrations of the coordinate transformation from the two-dimensional ellipse to thecorresponding ellipse in the three-dimensional space.After completing three rounds of coordinate transformation, the 2D ellipse is converted into a 3D object, and its transformation equations are formulated as follows:

Figure 5 .
Figure 5. Summary of the procedure to construct the DFN models using the UED model.

6 Figure 6 .
Figure 6.3D geometric model of an open pit mine rock slope in USA (Unit: m).

Figure 8 .
Figure 8. 3D jointed rock slope constructed using the UED model.

Table 1
Physical and mechanical parameters of the rock mass.

Table 2
Mechanical parameters of the discontinuity

Table 3
Statistics of the azimuth and UED model parameters