- 1 Introduction
- 2 Tectonogenesis Assessment and Palinspastic Reconstruction
- 3 Paleostress Analysis
- 4 Direct Graphical Methods of Stress Inversion
- 5 References
A structural model is an important part of the geotechnical model of a mine development. The purpose of the structural model is to illustrate the location, orientation and character of major geologic structures, as well as identify the spatial distribution of features that make up the rock mass fabric (i.e minor faults, folds, foliation). The identification of patterns or spatial trends in rock mass fabric allows the selection of structural domains. 
Developing domains simplifies the structural evaluation of the block to be mined, by illustrating the interaction of major structures and rock fabric, which may derive a particular overall stress state within that block.  Comprehensive structural evaluation includes geologic structures at all scales – regional mine scale to drill core scale – with enough detail to produce a representative model that reflects the mining block’s entire geological history, formally referred to as tectonogenesis (Figure 1). Illustrating a block’s structural features and unique fault-related movement data contributes to paleostress analysis methods by identifying the controlling structural features and associated stresses.
Completing a sequenced paleostress analysis as a means to back-calculate stress through geological time is of significant value to mine sequence planning. This allows assessment of the possibility of remobilizing old structures or veins, resulting in a fault-slip burst or seismic event during mining. Paleostress analysis methods are recommended to be applied at all burst or seismic event prone mine-sites. Using the paleostress back-analysis to estimate the state of stress yields invaluable data for helping develop efficient and safe excavation sequencing during mine-planning. 
Figure 1: Block model reconstruction of geological history and associated paleostress inversion analysis results for inferred stress states based on evaluation of fault slip data. 
Tectonogenesis Assessment and Palinspastic Reconstruction
Tectonogenesis is the unraveling of a block’s geologic tectonic history. Palinspastic reconstruction is the recreation of geo-historical depositional geologic events and kinematic influences. Coupling these together allows the identification of the most influential and controlling structural features of a mining area, which plays a significant role in stability analysis.
In the past, tectonogenesis and palinspastic reconstruction have been commonly used for the resolution of geologic history within the petroleum industry, locating oil reservoirs within structurally complex geology. With sufficient research and funding in this particular industry, numerical methods have been established, allowing more rigorous modeling. This rigorous numerical modeling, however, is only effective with good understanding of the area that is being modeled, further necessitating the procedure of tectonogenesis and palinspastic reconstruction. The following steps are recommended to complete a tectogenetic assessment:
- Literature review of the area for regional and local geology, including surface geologic mapping
- Establish the chronological sequence of tectonic events, and regional and local scale deformation (i.e. faulting).
- Examine available drill core with particular attention paid to major structural feature intersections with the drill holes
- Modeling in three-dimensions the identified drill hole intersections with major structural features, and possibly project them in 3D space relative to the geological model.
- Display structural data representing rock mass fabric and major structures on stereonets to define structural domains at the regional scale.
- Isolate block models and domain stereonets to define the sequence of events at the various scales.
- Link, where possible, inferred structural paleostress with principal ground stresses. This will be a preliminary estimate until mining occurs and stress measurements are collected.
The goal of paleostress analysis is to derive the direction of slip on a fault, using basic fault data collected in the field. There are many methods of paleostress inversion, which outline different approaches to the problem, however they are all based on similar assumptions:
- Slip on a fault plane occurs in the direction of the resolved shear stress
- Individual faults do not interact – movement on one fault is independent of another
- The blocks bounded by the fault planes do not rotate
- The stress field activating the faults is time-dependent and homogeneous
Data pertaining to a particular fault includes the fault plane orientation, slip direction and sense of slip, as well as location and character of fault attributes, the sense of slip also referred to as striations, indicate the direction of fault movement (Figure 2).
Figure 2: Fault data pertaining to the paleostress tensor. 
The data pertaining to the fault movement is used to calculate the paleostress tensor, which indicate the stress orientation interpretation relative to kinematic movement of a fault.  It is important to collect as many fault data measurements as possible, as most paleostress methods are statistic. The following criteria exist to distinguish fault rock types and their relation to tectonic regimes, contributing to the reconstruction process:
- Geometrical relationships of unconformities, indicating whether separate faults are from the same event, or are superimposed.
- Qualitative aspect of fault planes, indicating similar or different geologic events (i.e. alteration, mineral coatings, weathering)
- Influence of lithology
- Size of the fault plane and relative movement
Classification of Faults
Faults are most often classified by their stress state, as developed by E.M. Anderson in the early 1900’s.  Anderson derived the belief that the magnitude of horizontal stresses (σ2 and σ3) relative to that of the vertical stress (σ1) can change, which gives rise to three main types of faults: thrust fault, normal fault and strike-slip fault (Figure 3). Principal stress directions are represented by the three unit vectors S1, S2 and S3, for their respective principal stresses σ1, σ2 and σ3. This also derives the three main tectonic regimes:
- S1 vertical: extensional tectonic regime
- S2 vertical: strike-slip regime
- S3 vertical: compressional regime
The basics of faulting and stress geometry assumes that the body of rock is homogeneous and isotropic, and that the shear surface of the fault follows Mohr-Coulomb shear failure criterion, that is, a fault occurs on the plane which intersects the failure envelope. Lineations on a slip surface indicate the movement direction, and are assumed to have the same direction and sense as the resolved shear stress on the fault plane. Striations represent the intersection of the fault surface with the S1-S3 plane. The intersection of conjugate faults defines the intermediate principal stress direction S2, and the acute angle between conjugate faults is bisected by the largest principal stress S1. The aforementioned is represented graphically by Mohr-Coulomb failure envelope (Figure 4).
Figure 3: Interpretation of faults - Anderson's "standard" relationship between stresses and ideal faults.
Figure 4: Mohr-Coulomb failure criteria representing conjugate faults and associated principal stress orientations.
Direct Graphical Methods of Stress Inversion
Numerous methods of graphical stress inversion using stereonet or coordinate geometry have been developed.  With increased success of numerical analysis, many of these methods have been coded to produce solutions in a fraction of the time.
The P-T Method
The letters P and T are used to indicate axes which are located at 45° to the nodal planes of a fault plane solution and at 90° from the intersection of the nodal planes. P and T stand for pressure and tension, respectively, and can be equated to principal stress directions σ1 and σ3 (Figure 5). This is a simple and direct representation of fault geometry and the sense of slip.
Figure 5: Characteristic earthquake focal plane solution P-T diagram.
Seismologists begin by determining the shape of the first arrival vertical component P-wave as compressional or dilational. First arrival energy received in the form of a compressional or P-wave is plotted, as it is the most reliable and least violent wave data (Figure 6).
Figure 6: Conventions for plotting focal solution mechanisms P-T "beachball" diagrams from seismic data.
It is essential that reliable lineation data is acquired alongside the back-analysis of stress states through “beachball” diagrams. For simple state, P-T axes line up orthogonally with the poles and great circles of the faults (Figure 7A). However, the geometry of P-T with respect to the fault plane great circle differs when the slip is oblique to the fault (Figure 7B).
Figure 7: Comparison focal mechanism solution for normal fault dipping at 35° towards 115° with pure dip slip movement (A) versus oblique slip towards 050° (B).
The graphical P-T technique examines a single fault in isotropic rock with a determined sense and direction of displacement. The movement plane is defined as the plane perpendicular to the fault plane which includes slip direction, which is assumed to contain the greatest and least (S1 and S3) principal stress vectors.
If the angle between the fault plane and S1 vector is approximately 30°, for example, the sense of slip in the direction of S1 and S3 directions may be determined using the following steps:
- Plot both the fault plane and its pole on an equal area stereonet projection
- Plot the slickenline on the great circle that represents the fault
- Add an arrow on the slickenside point to indicate the relative movement of the hanging wall
- Rotate the plot to find the great circle that contains the slip direction and the pole to the fault. This circle defines the movement plane direction
- Along the great circle of the movement plane, the compression P direction is plotted at 60° to the pole of the fault and 30° to the slip direction, in a sense consistent with the sense of movement
- The extension T direction is plotted at 30° to the pole of the fault and 60° to the slip direction
The Right Dihedra Method
This method is based on the concept that for any fault, the orientation of the maximum principal stress is constrained to the P quadrant and the minimum principal stress is in the T quadrant. Spatial orientation and position of quadrants is defined by orientation of the fault plan and slip movement direction. This method assumes that all faults that are active in the same stress field derive from a common state of stress, and the intersection of P and T quadrants for multiple faults should have common principal stresses. This assumption forms the basis of the procedure to define the directions of major and minor principal stresses. Plotting a fault with its complementary auxiliary plane separates the stereonet into four quadrants. Opposite quadrants together form a dihedron (Figure 8). Typically, the movement data of one fault is insufficient to back-calculate the principal stress axes. This is repeated for all other fault movement and plane pairs. Superimposing all dihedral progressively narrows down the areas of compression and extension, and the orientation of the principal stress axes is located (Figure 9).
Figure 8: The principal of the right dihedral method.
Figure 9: Reduced overlap areas defining location of principal stress axes.
Analytical Mohr Circle Solutions
The stress magnitudes and geometry of faults may be described effectively using a combination of Mohr-circle and stereonet analysis; this forms the basis of “Fault Slip Tendency” analysis. This analysis, combined with numerical modeling, allows the best possible direct back-analysis of fault data. Stress results from this may be input to conventional rock mechanics software to assist in design.
The approach is based on the simple concept that a fault slip can be measured by comparing the ratio of shear stress to normal stress on the given fault plane. Reactivation could develop where post-original faulting stress magnitudes (deviatoric states) exceeded original stress differences (Figure 10).
Figure 10: Basis for slip tendency assessment methods for evaluating fault reactivation.
Stresses (shear and normal) acting on faults that would show tendency for slip and reactivation would plot above or on the friction line (Figure 10). The first step to calculating the stress components on the plane of interest is to plot the principal stresses on the stereonet, where the plane of interest is inclined to the stress axes at angles α, β and γ. Knowing the values of these angles, it is possible to calculate the shear and normal stresses acting on the plane and the ratio between them (τ/σn), referred to as the slip tendency, Ts. This ratio is magnitude dependent and specific for a fault’s given friction angle and material properties, thus should be normalized.
- T’s = Ts / (Ts)max = τ tanφ / σn
- (Ts)max = the maximum value of Ts limited by the slope of the sliding envelope for each given structure.
Principal states of stress that correspond with the normalized slip tendency an be related to the Anderson style of faulting and the relative orientation of principal stresses. The stress shape ratio may also help define the faulting domain. The stress shape ratio is defined as:
- Φ = (σ2 – σ3)/(σ1 – σ3)
- Φ = 0 implies axial compression.
- Φ = 1 implies axial tension.
The procedure to plot a Mohr diagram (Figure 11) involves the following steps:
- Sketch a set of three Mohr circles of radii α, β and γ, so the α circle is centered at (σ2+ σ3)/2 and has radius [σ2-(σ2+ σ3)/2], the γ circle is centered at (σ2+ σ1)/2 and has radius [σ1-(σ2+ σ1)/2], and the β circle encompasses both α and γ circles, is centered at (σ1+ σ3)/2 and has radius [σ1-(σ1+ σ3)/2]
- Plot a line through the γ and β circles from σ1 on the stress axis at an inclination α measured counter-clockwise from a parallel line to the τ axis
- Repeat this for the γ circle constructing a line through the α and β circles drawn from the location of σ3 on the stress axis and extended at an inclination of γ measured clockwise from the τ axis
- The intersection of the two construction arcs defines the stress point of interest (Figure 12).
- Plotting the Mohr-Coulomb failure envelope defines the critical boundary between stable and unstable slip.
Figure 11: Procedure for compiling Mohr circle plots for determining stresses acting on a plane of given orientation with respect to principal stress axes.
Figure 12: Representation of a fault plane P in a three-dimensional Mohr diagram.
Plotting Mohr circle diagrams provides an effective way of using P-T fabric diagrams one step further for the inverse analysis. Plotting the shear stress versus the normal stress for each fault data point on a Mohr diagram helps improve the use of P-T diagrams and helps quantify the likelihood of specific faults to be reactivated given their principal stress orientations. Evaluation of the type of fault under question, however, is difficult using Mohr diagrams alone (Figure 13), as the character of a Mohr diagram can be similar for different types of faults. Because of this ambiguity, it is always recommended that both Mohr circles and P-T “beachball” dihedrals be viewed and examined together, which will define the paleostress state for the Anderson modes of faulting.
Figure 13: Comparative Mohr diagrams and various stereonet plots for classic Anderson faults.
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