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The rapid development of digital mapping technologies such as terrestrial laser scanning, digital photogrammetry has allowed the exact acquisition of superficial information of discontinuities in rock exposures. Nevertheless, it remains challenging to quantify the spatial dimension and areal persistence of discontinuities due to the impossibility of direct observation of complete extent of discontinuities in three dimensions. The difficulties of characterizing individual discontinuities deterministically lead to the development of stochastic 3-D discontinuity networks. The stochastic modeling scheme are based on the statistical analysis of the measured discontinuity parameters; hence, the generated discontinuity networks represent discontinuity system in a statistical sense, and in most cases, the discontinuity traces formed by stochastic model are inconsistent with those observed at rock outcrops. To deal with this issue, this study presented a new method to simulate discontinuity network, which employs the real traces exposed on the cavern roof as the constraint condition of stochastic modeling and is described herein as semi-stochastic. The method takes discontinuities as circular discs and establishes geometrical relationship between the radius and the center of circular disc respectively for three classes of traces (i.e. trace has two ends, only one end, and neither end, visible in the sampling window). A case study with the proposed method shows that the traces yielded by the intersections of modeled discontinuity network with the cavern roof are in close agreement with actual situation. This semi-stochastic method further improves conventional stochastic modeling technique in terms of enabling a more realistic representation of rock discontinuities.

In hard rock masses, rock destruction is often manifested in the form of the local failure of the blocks formed due to fracturing. In this paper, we compare the application of block theory and three-dimensional (3D) block-cutting analysis to underground caverns. Using block theory, we can determine both the type of removable block and key block for different excavation surfaces. We can then identify ‘support-required’ blocks from the key blocks by drawing and analyzing the shapes of the maximum likely blocks (i.e. the largest blocks likely to be formed when the fractures are extended indefinitely in a definite-sized cavern). Subsequently, 3D block-cutting analysis is used to identify all the spatial blocks cut by all the finite-sized fractures in the rock mass region. Based on the geometrical information gathered about each spatial block, we carry out an analysis of the progressive failure of the block system. The simulation results allow us to explore the statistical laws governing the distribution of progressive failure blocks. This is a very important aspect which can be used to determine the spacing and length of the rock bolts required, or to check the design of the intended rock support regime. Overall, if we can understand the shortcomings of classic block theory (due to, for example, the assumptions made) and properly utilize the benefits gained from 3D block-cutting analysis, we can deal with more complicated issues. Thus, we hope to gain a more intensive and extensive understanding of block failure analysis.

There are many studies concerning hydro-mechanical coupling model based on discontinuous deformation analysis (DDA). However, verification of the coupling model and its application to practical engineering are rarely found. In this paper, we first introduced the distributed load sub-matrix of water pressure and described the calculation procedures of hydro-mechanical coupling analysis in the framework of DDA. Then, the correctness of the coupling model was verified respectively through analyses of buoyancy, deformation of a block under uniform water pressure and hydraulic fracturing. Finally, the model was applied to evaluate the sliding stability of a gravity dam foundation in China. The failure process and stability factor were calculated by two different stability evaluation methods (the strength reduction method and the overloading method), and the obtained stability factors were compared with those solved by the commonly used limit equilibrium analysis methods. The strength reduction method revealed that the rock strata of dam foundation slide with different displacements along several weak interlayers of different depths, whereas the overloading method revealed that the rock strata slide mainly along a special interlayer beneath the anti-sliding concrete stud, which was designed to be near the dam heel. The DDA method clearly shows the process of rock block failure and profoundly reveals the difference in sliding mode caused by using different stability evaluation methods.

Due to the complexity of geometry and the difficulty of mesh discretization of 3D (three-dimensional) blocks cut by complexly distributed fractures, explicitly considering arbitrary fracture network in fractured porous medium (FPM) flow analysis is very challenging for various numerical methods. In this study, we developed a FPM flow model by taking full advantage of numerical manifold method (NMM) with independent covers. With the independent covers, arbitrarily-shaped 3D blocks identified by block-cutting analysis can be directly used as basic computational elements. Along the boundaries of the divided blocks, fractures elements are generated according to the fractures’ apertures. Therefore, it is able to handle very complicated fracture network in 3D flow analysis without need to subdivide 3D blocks into computational meshes. In order to refine the meshes, we introduced artificial fractures with same material properties as surrounding rock into a fracture network, without need to coordinate with the shapes of the blocks. We demonstrated our new model on different 2D examples. At last, we applied our model to 2D and 3D examples with complexly distributed fractures, and achieved reasonable results. The results show that our model is very powerful to analyze fluid flow in arbitrarily and complexly fractured rock mass in 3D.

Geological solid modeling has been widely studied and applied in geological analysis relevant to rock engineering, mining, geophysics, geo-hazard assessment, and geographic information systems (GIS). The discontinuities and other faces that may be planes or surfaces need to be fully considered in an attempt to yield solids formed by the mutual intersection and cutting between faces in practical geological modeling. As one kind of geological solid modeling approach, 3D block cutting analysis can simultaneously identify tens of thousands of solids (blocks) from a complicated 3D discontinuity network without user intervention. However, this analysis can only address planes and build flat blocks. In this paper, we propose a method of 3D curved block cutting analysis. The corresponding algorithm is presented completely, with some key processes of analysis thoroughly accounted for, such as solving the surface-to-surface intersection, analyzing the composition of vertices, and analyzing relevant loops. Our proposed method fully exploits the advantages of topology and computational geometry. The 3D curved blocks can be accurately constructed and represented with minor computation and memory. Analysis examples showed that several hundreds of curved blocks can be automatically constructed within one minute using a microcomputer. This method is of significance for the development of geological solid modeling and can provide a reference for related studies.

Summary Finite element generation of complicated fracture networks is the core issue and source of technical difficulty in three-dimensional (3-D) discrete fracture network (DFN) flow models. Due to the randomness and uncertainty in the configuration of a DFN, the intersection lines (traces) are arbitrarily distributed in each face (fracture and other surfaces). Hence, subdivision of the fractures is an issue relating to subdivision of two-dimensional (2-D) domains with arbitrarily-distributed constraints. When the DFN configuration is very complicated, the well-known approaches (e.g. Voronoi Delaunay-based methods and advancing-front techniques) cannot operate properly. This paper proposes an algorithm to implement end-to-end connection between traces to subdivide 2-D domains into closed loops. The compositions of the vertices in the common edges between adjacent loops (which may belong to a single fracture or two connected fractures) are thus ensured to be topologically identical. The paper then proposes an approach for triangulating arbitrary loops which does not add any nodes to ensure consistency of the meshes at the common edges. In addition, several techniques relating to tolerance control and improving code robustness are discussed. Finally, the equivalent permeability of the rock mass is calculated for some very complicated DFNs (the DFN may contain 1272 fractures, 633 connected fractures, and 16,270 closed loops). The results are compared with other approaches to demonstrate the veracity and efficiency of the approach proposed in this paper.

Locating all spatial blocks cut by an arbitrary three-dimensional discrete fracture network (DFN) within a rock mass volume is a basic issue in many research fields related to fractured rock masses. In this paper, analysis procedures based on both the oriented rule and the closed rule are described, followed by a description of a proposed method for block progressive failure analysis. Lastly, two engineering cases in which the proposed method is implemented are presented. The results show that the identified blocks may be extremely complicated and may even be composed of thousands of loops (block faces consist of loops); block progressive failure analysis is extremely useful and efficient in determining block-reinforcement measures. Overall, the 3-d block cutting analysis is important progress in block theory and has a large potential for application in fractured rock masses.

The orientation, trace length, spacing, and location of probabilistic discontinuities in rock masses are randomly developed. Thus, the shape, size, and location of blocks cut off by these probabilistic discontinuities are accordingly stochastic. It is difficult, or even impossible, to determine the volume and location of the blocks using the block theory proposed by Goodman and Shi (1985) . Stochastic block analysis (SBA) is capable of identifying three-dimensional (3-D) stochastic blocks from a randomly developed discontinuity network (discrete fracture network). However, in practice, 3-D blocks are not identified well in simulated fracture networks and so the use of SBA is seldom encountered. In this paper, the procedures involved in stochastic block identification are first outlined. The concept and calculation of overlaying area and ratio are then introduced. Then, the stochastic block identification results are used to explore the statistical distribution of the block size and overlaying ratio. Subsequently, the laws governing development of the stochastic blocks were elucidated. The results show that the block size has a negative exponential distribution and the overlaying ratio follows a Γ distribution. The overlaying ratio increases as the trace length to spacing ratio increases. We further outline, for the first time, approaches to determine block support measures by analyzing the characteristics of the statistical distributions of the stochastic blocks. Block support issues relating to a practical underground plant were also studied. The lengths and anchor forces and spacings of the rock bolts were quantitatively determined according to the results of a statistical analysis of the stochastic blocks. Statistical analysis of stochastic blocks is of great significance in understanding the development characteristics of the stochastic blocks and in quantitatively determining block support measures.

Summary It is necessary to consider a great number of arbitrarily developed fractures in applications involving realistic 3-d fracture network flow models of rock masses. In order to model the fluid flow in a complicated arrangement of discrete fracture networks (DFNs), two core issues have to be solved. Firstly, how does one identify the connection relationships between fractures in the 3-d arbitrary fracture network? Secondly, how can one calculate numerically the fluid flow in arbitrarily-shaped 2-d domains? This paper first proposes that the boundaries of all enclosed blocks form flow pathways. All enclosed blocks can be identified using a 3-d block cutting method. The boundaries of the blocks are composed of loops which are formed by intersecting lines between all faces (fractures and other surfaces). Therefore, the connection relationships between fractures can be determined according to the linkages of the loops. Accordingly, the linkages of the finite element nodes between different faces can also be determined. On the other hand, the fluid flow occurring in the fractures can be treated as a 2-d continuous flow within the arbitrarily shaped loops in the fracture’s local coordinates. We propose that these loops can be meshed into triangles using a method of simplex triangulation of the arbitrarily shaped domain. Then, according to the linkages between nodes, the global conductivity matrix can be assembled and the solution of the equations governing flow can be derived. Three cases are used to validate the model. Finally, an analysis is made of a practical engineering example (with 8914 lines of intersection and 4188 loops) which shows that the proposed flow model is practicable and can deal with complicated DFNs.