A New Approach to Rapid 3D Mapping of Rock Mass Structure

The 12th International Conference of

International Association for Computer Methods and Advances in Geomechanics (IACMAG)

1-6 October, 2008

Goa, India

A New Approach to Rapid 3D Mapping of Rock Mass Structure

Alparslan TURANBOY

Seydisehir Voc. Sch. of High. Educ., Selçuk University, Konya, Turkey

Erkan ÜLKER

Department of Computer Engineering, Selçuk University, Konya, Turkey

Key Words: Discontinuity, rock mass, mathematical transformation, isometric perspective, 3D mapping.

ABSTRACT: The prediction of rock mass behavior is an important task in many engineering projects, as the

behavior of rock masses can be controlled by the presence of discontinuities. Being able to map the structure

of a rock mass is crucial to understanding its potential behavior. This understanding can positively impact the

safety and efficiency of an engineering project. In this research, rock masses were mapped and analyzed using

linear mathematical transformations and isometric perspective methods to achieve meaningful three-dimensional

(3D) results. The rock mass fracture representation is based on explicit mapping of rock faces. The developed

model can improve safety and productivity through its application in the determination and analysis of rock

mass structure for geological and geotechnical assessment. Based on the methods explained here, a software

system was developed for analyzing the geometric characteristics of discontinuities in a rock mass. In this model,

discontinuities in a rock mass can be visualized both individually and in combination, and cross-sections can be

generated at any desired location. In addition, intersection lines between discontinuities can be generated as dip

direction vectors. The natural structure attained by using this developed model agrees well with field

measurements.

1 Introduction

Visualization is the task of generating and understanding images that contain important features. Visual sight

constitutes 70% of sense of object perception (Ming and Peter, 2006), which is valid in engineering applications,

where the structural reconstruction of an actual object is a required step for accurate visualization. To

comprehend, render, and reveal complex structural objects such as in open pit mines, appropriate geometric

models must be designed. Two-dimensional (2D) models, including geological maps, cross-sections, sketches of

strain and stress patterns, and stereographic nets, are used in mining operations. Normally, the set of

observations and measurements supporting these models is small in relation to the complexity of the real objects

from which they are derived. Therefore, these models need additional guidance and explanations. However,

geological modeling techniques have evolved from highly conceptual methods to practical computing methods,

which include three-dimensional (3D) approaches (Zhong et al., 2006). 3D models are accepted as promising

tools for a better comprehensive understanding of engineering object characteristics. Accurate, detailed 3D

representations of real objects are needed for successful engineering design. In addition, suitable

representations can prove to be useful as strong communication tools between technical experts and nonspecialists.

In designing large and risky engineering structures, it is advised to first conduct research on models to determine

alternative solutions before deciding on the best alternative, and in order to reveal the optimum economical

boundaries and the feasibility of the project. For these aims, one of the modeling methods used for technological

purposes is geometric modeling, in which the similarities between the model and a scaled prototype of the model

are considered. In other words, there is a fixed ratio between the corresponding points for any specific feature in a

coordinate system. In rock masses, these models are made from discontinuity intersection lines as 2D and 3D

networks. In many geomechanical models, the most commonly used geometrical features are spacing,

orientation, and trace length of discontinuities, which have been defined in detail previously (ISRM, 1978).

The discrete element method (DEM) is a widely used modeling technique in rock engineering applications. The

foundation of the DEM was first developed by Cundall (1971). Several researchers (Hocking, 1977; Williams et

al., 1985; Pande et al., 1990; Shi, 1996; Jing, 1998) then introduced additional aspects of the DEM in different

related engineering problems. The key concepts of the DEM are that the domain of interest is treated as an

assemblage of rigid or deformable blocks/particles, and that the contacts between them need to be identified and

32

continuously updated during the entire deformation/motion process and be represented by appropriate

constitutive models. The discrete fracture network (DFN) approach has been also widely used to investigate the

behavior of rock masses, especially the hydraulic behavior of fractured rock masses. This technique was

developed in the early 1980s for both 2D and 3D problems (Long et al., 1982; Andersson, 1984; Endo et al.,

1984; Smith and Schwartz, 1984; Elsworth, 1986; Dershowitz and Einstein, 1987). The DFN method considers

fluid flow and transport processes in fractured rock masses through a system of connected fractures. There are a

number of software programs derived from DEM and DFN which are available. The most representative explicit

DEM methods are the computer codes UDEC and 3DEC (by Itasca Consulting Group Inc.) for 2D and 3D

problems in rock mechanics. The most widely known DFN codes are NAPSAC (by AEA Decommissioning &

Radwaste, Harwell, UK) and FRACMAN/MAFIC (by Golder Associates, Inc.) for 2D and 3D problems in rock

mechanics. These software programs have been successfully used to simulate the behavior of rock fractured

systems. However, their application to engineering problems is still limited to simplified representations of rock

masses, and the inclusion of discontinuities is still a challenging task.

In this 3D modeling research, we aimed to provide accurate and practical information and details for engineering

applications such as rock stability, pre-design of mines, blasting, and connectivity of fluids. In the model,

geometric features of discontinuities measured with scan-line surveys on the rock mass outcrop, such as dip, dip

direction, and spacing, were used as basic geometric features of the rock mass. The main instruments for taking

data were a compass-clinometer and a tape meter. Additionally, mathematical transformations, perspective rules

and developed schemes, and mathematical approaches and equations were used for the developed model.

Moreover, linear relations were used and an isometric presentation was preferred for all illustrations.

2 The basics of the proposed 3D modelling method

The modeling sequence followed includes: a) identification of spatial positions of the discontinuity planes and the

outcrop as four types; b) mathematical definition of the discontinuity planes; c) isometric transformation and

projection of the rock structure, including discontinuity planes and intersection lines; d) generation of the

intersection lines as dip vectors; and e) developing cross-sections from the obtained 3D model.

2.1 Classification of discontinuities positions

In this study, the first step is the classification of the discontinuity orientations. The classification term is a term

used to define the reorganizing of the classical discontinuity orientations into classes. Discontinuity and rock mass

outcrop orientations may be different in nature relative to north. Because of this, two strategies can be developed

in the model. The first strategy uses four possible alternative discontinuity orientations in a coordinate system

(Figure 1(a)). In the boundary values of the discontinuity orientation, the apparent dip, dip direction, and outcrop

(YZ plane) orientations are considered. The angles used in the calculation are shown in Figure 1(b). This figure

was drawn for the first discontinuity type shown in Figure 1 (a). The second strategy uses 64 possible

discontinuity orientations according to four different possible discontinuities. (Figure 1(c)). In this figure, the

possible spatial orientations of the discontinuities are given on the XY plane.

Figure 1. Classification of the possible orientations of any discontinuity; (a) four possible general discontinuity

orientations, (b) explanation of the angular conditions and values and numeric results for the 1/1 (column/row)

type discontinuity (α: dip direction, γ: outcrop orientation, the search angle is shown with a dashed line), (c)

possible orientations for these four discontinuities relative to the north (on the XY plane).

33

The process of classifying discontinuities is the first step of our proposed model. Discontinuity planes and rock

mass outcrops can be naturally in different orientations relative to north. Because of this, the two following

strategies, which depend on each other, were developed. These strategies are related to 3D discontinuity planes

in the rock mass and 2D discontinuity traces on the visible surfaces of the rock mass, respectively. In the first

strategy, the four possible alternative discontinuity orientations in a constant coordinate system and the north

(Figure 1(a)) were constructed. In these illustrations, discontinuities perpendicular and parallel to the XZ, XY, and

YZ surfaces are neglected. The dip angle (β) (on the YZ surface), dip direction angle (α), and the outcrop

orientation angle (γ; on the XY surface) were considered as the boundary values of the discontinuity orientation.

The dip direction, outcrop orientation (in this paper, we used spatial positions of the outcrop relative to north as

the outcrop orientation), and the unknown angle (dashed line) which is used in the calculations are also shown in

Figure 1(b). Here, the outcrop surface was assumed to be vertical, and this surface was taken as the YZ surface.

This figure was generated using the first discontinuity type as an example (details of this type are also shown in

Figure 1). In the second strategy, 16 configurations for each discontinuity type (Figure 1 (c)) were developed

relative to north. Thus, a total of 64 configuration types were obtained for the four discontinuity types. Here, the

XY plane was divided into eight equal areas to assess the possible conditions. In this paper, the angular condition

and angular value definitions are proposed as a new concept.

In Figure 1(c), outcrop orientations were classified relative to north in every row as 16 configurations. In addition,

discontinuity types are classified according to the constant outcrop position in each of the 16 columns for the four

configurations. All these illustrations are on the XY surface. Derived angular conditions and values for the 1/1, 1/2,

1/3 and 1/4 (column/row) discontinuity types (Figure 1 (a and c)) are presented in Table 1. It is possible to write

three or four angular conditions and three angular values for each configuration. Consequently, a total of 192

angular values were obtained. In our method, the discontinuity type (Figure 1(a); type 1, 2, 3, or 4) is selected

first. The final angular values are then obtained by utilizing the angular conditions. For each angular condition

configuration, there is an angular value. An angular value consists of three sub-elements which are on the x, y,

and z axes. Discontinuity type 1/1 (the first type) in Figure 1(b) can be considered as an example of the geometric

approach. Only the β (dip) angle can be used on the YZ plane for the drawing of the discontinuity trace. During

this process, the three angles considered are 90°, β, 270°+β clockwise from the discontinuity trace to the x-, y-,

and z-axes, respectively. On the XY plane, the angular conditions formed should be taken into account. Here, the

x-axis was the North direction, and the required angle represented by a dashed line can be calculated by using

the outcrop orientation angle (γ) and the dip direction angle (α) which was recorded from the scan line survey.

The required angle is taken as clockwise between the discontinuity trace and the x-axis.

Table 1. The angular conditions and values for types 1/1, 1/2, 1/3 and 1/4 (column/row).

2.2 Mathematical definition of the discontinuity planes

After assigning the classes for the discontinuities, the boundary points of the discontinuity will determine the

modeling of the discontinuity in the rock block. Afterwards, the discontinuity plane is generated from these points.

The method used for this is discussed in this section. Let the angles of two straight lines be αi, βi, and γi related to

34

the x-, y-, and z-axes, respectively. The parametric equations used for calculating the intersection point between

these two straight lines are as follows:

( 1 i1 ) 2 ( i2 ) 3 x + λ * Cos α = x + λ * Cos α = x (1)

( ) ( ) 1 i1 2 i2 3 y + λ * Cos β = y + λ * Cos β = y (2)

( ) ( ) 1 i1 2 i2 3 z + λ * Cos γ = z + λ * Cos γ = z (3)

where the left parts of the equations are the parametric equality of a straight line and the remaining parts are the

parametric equality of another straight line and the coordinates of the intersecting point between two straight lines,

respectively. For every adjacent discontinuity trace as a straight line in the equation system, a λ value and a t

value should be calculated. In these equations, the first equation (Eq. 1) does not fit the solution; the software

reaches the conclusion according to Eqs. 2 and 3. In Eq. 2, one of the unknowns is given in the form of the

others, and the solution is obtained using Eq. 3. The λ value and the t value are calculated using Eqs. 4 and 5.

( ) ( )

( ) ( )

( ) ( )⎥⎦

⎤

⎢⎣

⎡

−

⎥⎦

⎤

⎢⎣

⎡

⎟ ⎟⎠

⎞

⎜ ⎜⎝

⎛ −

− −

=

i2

i1

i2 i1

i1

i1

2 1

2 1

Cos γ

Cos β

Cos β * Cos γ

* Cos γ

Cos β

z z y y

t (4)

( )

( )

( ) * t

Cos

Cos

Cos

y y

1 i

i2

1 i

2 1 ⎟

⎟⎠

⎞

⎜ ⎜⎝

⎛

β

β

+ ⎟

⎟⎠

⎞

⎜ ⎜⎝

⎛

β

−

λ = (5)

Using the obtained λ and t values, the intersection point values of the p (x3, y3, z3) can be found and recorded in a

file. In this study, the visible discontinuity traces on the outcrop and prism edges were assumed to be straight

lines. The discontinuity simulation process is based on constructing the edges which formed the discontinuity

plane in a prism. This calculation process is performed on the visible surfaces of the prism. Thus, all intersection

points between edges belonging to prism and the prism edge–discontinuity trace can be obtained using the

aforementioned equations. The discontinuity traces can be drawn on visible surface of the prism as straight lines

using the intersection points which are calculated on the prism edges. These lines are projected to the hidden

edges of the prism. The obtained four straight lines which bounded the discontinuity plane were used as a tool in

the simulation of the discontinuity plane. In addition, the intersection lines between the discontinuity planes can

also be derived using similar processes.

In this paper, an isometric transformation process was applied for all intersection points belonging to the

discontinuity trace–discontinuity trace and the discontinuity trace–prism edge. These points were used in the

software as the construction elements, as explained below.

3 The developed software

In this study, a program was developed to simulate the rock structure. The Visual Basic 6.0 programming

language was used in the software, and the inputs of the package program are the data obtained from the outcrop

of the rock mass by the scan-line method. To make a full representation of each discontinuity, the dip and dip

direction are recorded. An identity number (ID) was given to each discontinuity. Whereas dip and dip direction

data are variable, there are also constant data used to simulate the rock mass, including the dimensions of the

prism in which the rock mass is simulated, the scan-line height, and outcrop orientation (γ) angles between the

rock mass and north. An input table is formed by adding these values to the discontinuity data. The input table is

called the information system table (IST). Each row of the IST represents information related to a discontinuity,

and each column represents the qualities of the rock mass. There are five columns in the IST. These columns are

discontinuity ID, dip direction (α), dip (β), outcrop orientation (γ), and cumulative spacing. An example of the IST

which was used in the field experiment is given in Table 2.

The calculation which enables the simulation of the rock mass can then be initiated. The calculations perform the

following steps in turn. Let us consider discontinuity is ID-1. With the dip and dip direction information, the class of

ID-1 is determined (discontinuity type one, two, three, or four). By determining the class of ID-1, its line on the YZ

plane |AB| is calculated. The points which pass through the YZ plane of the line (A and B) must be determined.

The approach, which gives the intersection of two discontinuities and which is newly suggested by us, is used

here. The first discontinuity is ID-1, whereas the second discontinuity is the line RM1RM2. The intersection of the

two discontinuities A is calculated using Equations 1, 2, and 3. The second discontinuity is called the RM3RM4

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line. Point B is obtained using the suggested approach. Here, point RM1 is taken as (0, 0, 0), point RM2 is taken

as (0, Ybd, 0), point RM3 is taken as (0, 0, Zbd), and point RM4 is taken as (0, Ybd, Zbd). The same processes

are used in the calculation of points C and D. Consequently, discontinuity plane ID-1 is obtained (Figure 2). Points

A, B, C, and D are recorded for later use. After the discontinuity plane is obtained, it is tested to determine

whether it crosses over the prism edges. If there is a crossover, the discontinuity plane is trimmed with the prism

boundaries. The software can obtain all discontinuity planes in this way.

Figure 2. The simulation of discontinuity plane ID–1 in the selected prism.

The intersection of the discontinuity planes is a problem that is solved in the software. All intersections can be

detected. If there are detected intersections, the intersection lines are calculated. These detection processes are

dealt after all discontinuity planes are obtained. For instance, discontinuity ID-1 is compared with discontinuities

ID-2, ID-3,…, ID-N on the YZ plane. The plane coordinates of two different discontinuities are found on both the

YZ plane and the XY planes. For both discontinuities, the traditional mathematical approach is used to detect

whether the plane lines are parallel to the YZ plane or the XY plane. If there is no parallelism, there is an

intersection point. With the help of Equations 1, 2, 3, 4, and 5, the intersection point K, which is between the two

discontinuities on the YZ plane, is calculated. Intersection point K can be on the YZ plane of the rock mass, or it

could possibly also be outside the boundaries of the mass. For this reason, the control of point K with the mass

boundaries is also performed. Similarly, point L, which is on the XY plane, is also calculated. These processes are

performed on all planes of the rock mass except for the XZ planes. All obtained intersection points are recorded.

There is a possibility that a discontinuity plane can intersect more than one discontinuity plane. In this case, the

discontinuity is compared with all the remaining planes. These calculations increase the flexibility of the image in

the software, and each discontinuity can be examined individually.

The general flow diagram and a typical interface of the developed software are shown in Figures 3 and 4. As seen

in Figure 3, the building of the information system table includes recording data from the first phase. For the

calculation of a discontinuity plane, four intersection points are computed on the prism edges. Then, the

discontinuity plane is generated from these points. Moreover, the software can find the intersection point for any

two or more discontinuity planes and the corresponding intersection lines. Thus, the images of all discontinuity

planes are simulated.

Table 2. An example of the IST that was used in the field experiment.

Discontinuitiy Dip Dip Outcrop Cumulative Points

Prism

Dimension

number Direction Orientation Spacing (m)

(α) (β) (γ) (m) X Y Z X Y Z

1 160 113 23 2,15 0 2,15 1,5 8 35 4

2 246 55 23 2,6 0 2,6 1,5 8 35 4

3 249 115 23 9,15 0 9,15 1,5 8 35 4

4 267 110 23 10,05 0 10,1 1,5 8 35 4

5 193 52 23 12,05 0 12,1 1,5 8 35 4

6 250 127 23 17,1 0 17,1 1,5 8 35 4

7 252 132 23 20,85 0 20,9 1,5 4 35 4

8 140 100 23 26,2 0 26,2 1,5 8 35 4

9 142 102 23 27,35 0 27,4 1,5 8 35 4

10 142 92 23 28 0 28 1,5 8 35 4

11 145 102 23 29,95 0 30 1,5 8 35 4

12 198 177 23 32,2 0 32,2 1,5 8 35 4

36

Figure 3. Flow diagram of the developed software

Figure 4. A typical interface of developed software.

The developed software can also generating cross-sections throughout the 0Y axis where desired, and transform

them to orthogonal views using similar processes.

4 Case study and results

The highway rock cut at 110 km on the Konya-Antalya road in Turkey was chosen for the field experiment (Figure

5). In this region, the main formation is composed of limestone. Two primary discontinuity sets, which have

bedding and joints approximately perpendicular to each other, can be followed easily in the field with the naked

eye. The rock outcrop is relatively smooth, and discontinuities appear as fracture traces, not fracture surfaces.

The outcrop on which the data are recorded is shown in Figure 5(a), and the YZ plane of the experimental prism

and approximate discontinuity traces is shown in Figure 5(b).

The scan-line height was taken as 1.5 m. The outcrop orientation was 23°. The dimensions of the sampled prism

dimensions were selected as (x, y, z) 8×35×8 m. Then, the information system table (Table 2) was given to the

software as input, and all discontinuity planes were generated by the software.

37

The software can obtain and display several features from the selected sample prism, including all the

discontinuity planes (Figure 6(a)), all discontinuity planes with intersection lines (Figure 6(b)), the individual

intersection lines as dip direction vectors (Figure 6(c)), cross-sections throughout the 0Y axis where desired

(cross-section spacing were selected as 5 m; Figure 6(d)) and orthogonal views of the cross-sections (Figure

6(e)).

Figure 5. a) Outcrop chosen for the modeling (Konya-Antalya road at km 110 in Turkey), b) Approximate

discontinuity traces and sampled prism boundaries on the outcrop.

Figure 6. Model outputs: a) all discontinuity planes, b) all discontinuity planes with intersection lines, c) the

individual intersection lines as dip direction vectors, d) cross-sections throughout the 0Y axis where desired

(cross-section spacing were selected as 5 m), and e) orthogonal view of the cross-sections (2D view).

5 Conclusions

In this study, a rock structure was taken as a sample rectangular prism. Discontinuities were analyzed as linear

features within this prism. In the field, simple measurement instruments, including a tape measure and a

compass-clinometer, were adequate to collect the data from the exposures of rock masses. The recorded data

were used in the software directly and results were obtained easily and quickly. The primary functions of the

developed software are to: a) simulate the discontinuity network, b) obtain the intersection lines between

discontinuities, c) generate cross-sections where desired, directly from the simulated rock structure, and d)

38

provide a user-friendly interface. In spite of the sophisticated 3D model, our goal was to obtain useful, practical,

and rapid results.

Based on the results, we believe that the rock mass was adequately modeled for many practical engineering and

pre-evaluation applications. Despite assuming that the discontinuities have infinite sizes, the developed model

appears to be an improvement over other sampling methods such as photogrammetry or window sampling, and

thus the more realistic results can be obtained. By developing the present model, it is possible to analyze rock

masses for detailed engineering applications such as slope stability, fluid flow analysis, and blasting design.

6 Acknowledgement

This study has been supported by the Scientific Projects of Selçuk University (Turkey).

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# A New Approach to Rapid 3D Mapping of Rock Mass Structure

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