Keywords

1 Introduction

From a geometric point of view, mineral deposits are a combination of various geometric images of many possible space forms. They could take forms of isolated irregular geometric bodies (e.g. lenses and stockworks), stretched irregular geometric bodies (veins) or stretched regular bodies confined by planes (blanket deposits) [1,2,3].

Diverse space forms of mineral deposits are most pronounced in the setting of mountainous terrain, which can be explained by genesis of accumulation and its tectonics. Surface irregularities have a great impact on the geometric structure of a subsoil mineral deposit, which is most noticeable in the setting of mountainous terrain. From a geometric point of view, blanket deposits have a simpler form that allows for a relatively accurate estimation of their metric characteristics and a higher quality mining.

2 Recent Research and Publications Analysis

Research suggests that improved methods of geological analysis and project planning are key to proper and rational borehole utilization. Computer-aided design systems are now more and more often used in this field [4,5,6,7,8,9,10].

So for example, in the study of [7] a point is made that building of 3D geologic visual and computational models is of great importance for mining and geological projects. A 3D model can illustrate spatial properties of boreholes and a visual model transformed into computational provides for engineering analysis. A method of building a 3D geologic model of solid mine surface powered by AutoCAD is analyzed in [9]. The article reviews optimization issues based on a geologic model and technology mining. The author of [10] suggests a new visualization method based on spatial database and graphical analysis.

A core pole provides various data about geological structure of a blanket deposit gap. Core poles reveal mechanical and structural properties of mineral deposits and help evaluate mineral reserves [11, 12].

Moreover, boring of several exploration holes on a studied gap makes it possible to estimate the deposit shape, size and volume and, as a result, its reserves. This task is especially relevant for mineral deposits found in the setting of mountainous terrain.

3 Objective Statement

Computer-aided design systems are now frequently used in geology. To improve the results of mineral deposits prospecting and outlining the authors introduce computer programs addressing respective goals, which help to obtain information about deposit parameters before completion of prospecting and exploration operations and cut down material expenses for field observations and reduce time of data collection.

The goal was to solve the following problems:

  • Research and analysis of geometric modeling of exploratory bores and methods for approximating the surfaces of mineral deposits with regular geometric shapes

  • Development of geometric algorithms for solving exploration and outlining problems of mineral deposits by means of geometric modeling based on approximation techniques by regular geometric images (geometric algorithms are considered to be a sequence of geometric operations for solving a problem expressed in symbols)

  • Investigation of the possibility of using the obtained geometric algorithms for the automated solution of exploration and outlining problems of mineral deposits

  • Development of a mathematical description of the presented graphic algorithms for solving the tasks

  • Development of algorithms for the automated solution of the main tasks of exploration and outlining of mineral deposits based on the completed mathematical description and software development

4 Main Part

Geometric modeling of boreholes is based on their approximation to the classic geometric images. From the geometric point of view, a borehole is a cylinder. So, some positional and metric problems could be solved using a core pole using descriptive geometry [13,14,15].

Suppose boreholes of 120–140 mm diameter are bored in solid rocks. Let us create a geometric model of a core pole on a complex diagram. A core pole in Fig. 1 has pronounced contact surface of materials shown as parallels prescribed by projections of ellipses.

Fig. 1.
figure 1

Strike and dip measurements for blanket deposit using a core pole. Contact surfaces – parallels.

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Suppose a mineral deposit is found between contact surfaces. With the help of descriptive geometry, we define deposit parameters.

Assume planes P and Q are contact surfaces of geological materials. Let us create diagonal lines AB||H, A1B1||H and frontals CD||V, C1D1||V in these planes. Then we form traces of contact planes: PV and PH – traces of contact plane P, QV and QH – traces of contact plane Q. We obtain angle α ─ an angle of stretch. We define angle β, an angle of dip, with the line of largest inclination. By dropping a perpendicular from plane P to plane Q we obtain t02 – reservoir capacity.

While we know the coordinates of points A, B, C, D of contact surface and core pole diameter D we are able to provide an analytical description to the geometrical algorithm for solution of this problem.

We need to find angle of dip β and angle of stretch α.

To solve a problem represented by Fig. 1 we follow these steps:

  1. 1.

    Find coordinates of \( n_{1}^{'} \), pierced by frontal trace QV of contact plane Q:

    $$ \left( {\frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }} \cdot \left( {x_{a} \frac{{y_{b} - y_{a} }}{{x_{b} - x_{a} }} - y_{a} } \right);z_{{a_{1}^{'} }} } \right) $$
    (1)
  2. 2.

    Define an equation of straight-line QV, prescribing frontal trace of contact plane Q:

    $$ z - z_{{a_{1}^{'} }} = \frac{{z_{{d_{1}^{'} }} - z_{{c_{1}^{'} }} }}{{x_{{d_{1}^{'} }} - x_{{c_{1}^{'} }} }}\left( {x - \frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }} \cdot \left( {x_{a} \frac{{y_{b} - y_{a} }}{{x_{b} - x_{a} }} - y_{a} } \right)} \right) $$
    (2)
  3. 3.

    Find vanishing point coordinates of Q plane trace on axis OX. Coordinates of point in question:

    $$ \left( {x_{a} - y_{a} \frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }} - z_{{a_{1}^{'} }} \frac{{x_{{d_{1}^{'} }} - x_{{c_{1}^{'} }} }}{{z_{{d_{1}^{'} }} - z_{{c_{1}^{'} }} }};0} \right) $$
    (3)
  4. 4.

    Define a straight-line equation of horizontal trace of QH hanging wall plane:

    $$ y = \frac{{y_{b} - y_{a} }}{{x_{b} - x_{a} }}\left( {x - x_{a} + y_{a} \frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }} + z_{{a_{1}^{'} }} \frac{{x_{{d_{1}^{'} }} - x_{{c_{1}^{'} }} }}{{z_{{d_{1}^{'} }} - z_{{c_{1}^{'} }} }}} \right) $$
    (4)
  5. 5.

    Angle of dip β is found from triangle ∆eff0:

    $$ tg\beta = \frac{{ff^{'} }}{ef} \Rightarrow \beta = arctg\frac{{ff^{'} }}{tf} $$
    (5)
  6. 6.

    We express a straight-line equation of horizontal trace of hanging wall plane PH:

    $$ y = \frac{{y_{b} - y_{a} }}{{x_{b} - x_{a} }}\left( {x - x_{a} + y_{a} \frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }} + z_{{a_{1}^{'} }} \frac{{x_{{d^{'} }} - x_{{c^{'} }} }}{{z_{{d^{'} }} - z_{{c^{'} }} }}} \right) $$
    (6)
  7. 7.

    We find stretch angle α:

    $$ \alpha = arctg\left( { - \frac{{x_{b} - x_{a} }}{{y_{b} - y_{a} }}} \right) $$
    (7)
  8. 8.

    Calculate reservoir capacity, equal to length of the segment Pxt0:

    $$ m = \left| {y_{px} - y_{t0} } \right| , $$
    (8)

where ypx and yt0 – ordinates of points Px and t0.

The developed algorithm and its mathematical formulation help to design a program for automatic solving of this problem [16,17,18,19,20].

Figure 2 is a flow diagram of automatic strike and dip measurement using a core pole when contact surfaces of materials are parallel.

Fig. 2.
figure 2

Algorithm for strike and dip measurement by contact surface on a core pole. Contact surfaces – parallel.

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When contact surfaces are non-parallel solution algorithm is the same. Suppose a middle section of a core pole between contact surfaces is where a mineral deposit crosses a borehole.

In Sect. 1 we establish a horizontal DE and frontal DF to find traces of Q plane. Largest angle of inclination on β1 plane defines a dip and direction of horizontal line and horizontal trace demonstrate its strike α1.

We find traces of P plane in Fig. 3 with horizontal AB and frontal BC. Piercing of planes P and Q characterizes pinching-out of a deposit constrained by these planes. Angle α2 defines the strike, and β2 is angle of dip.

Fig. 3.
figure 3

Strike and dip measurement for cross beds of a core pole.

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Geometric algorithm for solving of this problem:

  1. 1.

    A, B, C Є P; 1-2 ⊥ PH; 1-2 ─ line of largest inclination P; β - angle of dip of plane P;

  2. 2.

    D, E, F Є Q; 3-4 ⊥ QH; 3-4 ─ line of largest inclination Q; β1 ─ angle of dip of plane Q;

  3. 3.

    P ∩ Q = 6-7; 6-7 ─ line of deposit pinching-out; β2 dip angle of pinching-out line; o1h ⊥ 6-7; o1h = L ─ cross-cut between borehole axis and pinching-out line.

Figure 4 is a geometrical algorithm and a mathematical formulation for problem solution when contact surfaces of materials are non-parallel.

Fig. 4.
figure 4

Algorithm for strike and dip measurement by contact surface on a core pole. Contact surfaces - non-parallel.

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As a result of the research, software modules for solving the main problems of exploration and outlining of mineral deposits are presented in Fig. 5.

Fig. 5.
figure 5

Structure of software modules.

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The developed automated system for solving the main problems of exploration and outlining of mineral deposits consists of the following subsystems:

  • The input data input subsystem, responsible for input data and verification of their correctness

  • The subsystem for creating graphic images allows you to define (build) a front and a horizontal projection of the original data

  • A subsystem of calculations, in which the solution of such problems as determination of the intersection points of the bores with contours of the mineral deposit, determination of the elements of bedding along the pole core, determination of the boundaries and parameters of the deposit output to the earth’s surface are realized

5 Conclusion

To conclude, geometric modeling helps solve the problems related to deposits of minerals by methods of descriptive geometry. This makes the application of CAD system possible and reduces the costs of field observations.