Institute of Geology, Chinese Academy of Sciences, Beijing 100029)
Geological body contains all kinds of geological information, which can be measured in the field or obtained in the laboratory. The spatial distribution of geological data is usually scattered and discontinuous. Therefore, different fitting functions are needed when processing geological data and simulating various geological information. In this paper, several fitting functions for simulating different geological information are proposed. In order to realize the simulation of geological body and its graphic display, a computer program is compiled. Using this program, the spatial distribution of geological data can be displayed. Mathematical simulation and graphic display of three-dimensional geological bodies are important tools in geological research, which can significantly improve the efficiency of geological work.
Keywords three-dimensional mathematical simulation; Fitting function; graphic display
1 Introduction
Various geological information can be obtained from geological bodies, such as surface topography, groundwater level, faults, joints, ground level and various geophysical and geochemical data. These data can be measured in the field or obtained through the laboratory. Geologists always do mathematical processing on these data before using them. The fitting function proposed in this paper is suitable for processing the above geological information and displaying the three-dimensional graphics of geological bodies. Some examples given in this paper show that it is an effective geological research tool to use fitting function to simulate geological bodies in three dimensions.
Three-dimensional mathematical simulation of geological bodies II
Different fitting functions are applied to different geological information to establish geological models. The geological information involved in this paper can be summarized into two types: plane and surface with spatial distribution, which have different fitting functions respectively.
2. 1 space plane fitting function
Faults and joints in geological information can be approximately regarded as spatial planes, and the positioning parameters of a fault plane in space are: ① fault measuring points PC (XC, YC, ZC); ② The dip angle α and β of the fault; ③ the extension length s of the fault along the dip.
Using the above parameters, the unit normal vector of fault plane can be obtained, as shown in figure 1. Can be expressed as
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Where: is the unit vector of three-dimensional coordinate axes x, y, Z Y and z.
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Figure 1 Schematic Diagram of Space Plane
If the point P(x, y, z) is located on the fault plane, the connecting line PPc is vertical, so the relationship between the coordinates x, y, z of any point P(x, y, z) on the fault plane is as follows.
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It is often necessary to judge the relationship between two points in space and the position of fault plane in geological simulation. For example, whether they are located on the same side or both sides of the fault plane. If they are located on both sides of the fault plane, where does their connecting line intersect with the fault plane? The following formula provides an analysis method for this judgment. Assuming that points D 1 (X 1, Y 1, Z 1) and D2(x2, y2, z2) are located on both sides of the fault plane (Figure 1), then the intersection point P0(x0, y0, z) between the connecting line and the fault plane.
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Among them:
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U represents a parameter with D 1 as the starting point, D 1D2 as the unit length, and D 1 as the positive value in the direction of D2. Therefore, when u < 0 or u > 1, D 1 and D2 are on the same side of the fault plane; When 0 < u < 1, D 1 and D2 are on both sides of the fault plane, and the intersection of the connecting line and the fault plane is P0 (x0, y0, z0); If u=0 or u= 1, then D 1 or D2 is located on the fault plane; When the denominator of Equation (5) is zero, the connecting line of D 1 and D2 is parallel to the fault plane. Finally, when z0 is less than ZC-s sin β, it means that its intersection point p has crossed the actual fault plane, as shown in figure 1.
Spatial plane can also be used for the distribution of joints in rock mass, which requires statistical methods to group joints and calculate the average bulk density, joint circle radius, dip, dip angle and its variance of each group of joints. Here, it is assumed that each joint surface is a disk model. Then the spatial distribution of these discoid joints in the designated geological body can be obtained by random sampling from the above parameters.
2.2 Spatial surface fitting function
Spatial surface fitting function can be used to describe the spatial distribution of surface topography, groundwater level, ground elevation and various geophysical and geochemical data. Assuming that the value Ai representing a certain physical quantity is measured at the midpoint i(xi, yi, zi) of geological body, Ai will have an influence on the points (x, y, z) around it, and its influence value W can generally be described as
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Where: r is the influence radius, and its influence function w gradually decreases with the increase of ri; When ri=0, w = ai; when ri=R, w = 0；; And when ri=0 and r, dW(ri)/dri=0. Spatial surface function can be divided into single-valued surface, multi-valued surface and ground surface.
(1) single-valued surface fitting function
A single-valued surface in this paper refers to a surface function with only one corresponding function value at any point P(x, y) on the datum plane (z=0). Its value can represent the elevation of the surface at point P, so it can be used to describe the spatial distribution of surface fluctuation, groundwater level, loose layer and so on. Suppose a set of measurement data from a single-valued geological surface, yi, zi(i= 1, 2, …, n), that is, the surface passes through these n measurement points in space. The fitting function W2 can be obtained by the formula (6) by superposing different ri and Ai values n times.
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Among them, in order to determine the function influence value Ai at each measuring point, N linear equations need to be established to solve.
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Among them, if Ai is substituted into equation (7) and obtained from equation (8), then equation (7) is a spatial curved surface passing through the above N measuring points, which is smooth and continuous everywhere.
(2) Multi-valued surface fitting function
It refers to the surface function with corresponding function value at any point P(x, y, z) in geological space. At this time, when the function value is specified as a constant, the independent variables x, y, z, y and z of the function describe a spatially distributed equivalent surface. Multi-valued surface function can be used to describe the distribution law of various physical and geochemical data in geological bodies. Suppose a physical measurement obtains a set of measurement data xi, yi, zi, ui(i= 1, 2, …, n) of n points; , Yi, Zi are the spatial coordinates of the measuring point, and ui is the measured value of the physical quantity corresponding to the measuring point, similar to equations (7) and (8). The multivalued surface function and corresponding linear equation used to solve Ai can be expressed as follows.
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formula
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Under the constraint of formula (10), the function W3 will be equal to the measured values of all measuring points, and it will be smooth and continuous everywhere in the process of spatial change, so formula (9) can be used to describe the magnitude distribution law of various geophysical quantities in geological space.
(3) the fitting function of stratum surface
Strata are arranged according to their sequence in geological body, and each stratum has its corresponding thickness h. Equations (9) and (10) can also be used to describe the distribution of strata in geological space, but they should be modified accordingly. Because the ground plane is the interface between different layers, it is necessary to pass through the thickness of the corresponding layer from one ground plane to its adjacent ground plane. In the process of function fitting, all layers need to be numbered from top to bottom, and different function constants should be specified for different layers to describe them. Assuming that there are L strata in the geological body, there are L- 1 strata, as shown in Figure 2. The thickness of the same stratum may vary from place to place. If the average value is taken in the following calculation, the function constant Vi of layer I can be expressed as
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It can be seen from the formula (1 1) that the function value Vi of each layer is related to the formation thickness Hj, and the Vi values of different layers are discontinuous.
In order to fit all levels, we need a set of measurement level positioning data with n points, yi, zi, k(i= 1, 2, …, n). Where yi and zi are the coordinates of the ground plane space measuring point, and k is the number of the plane where the measuring point I is located, so the left end of the formula (10) can be changed to the Vk value of the kth ground plane.
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When the Ai in formula (9) is obtained by formula (12), when W3(x, y, z)=Vk, formula (9) describes the distribution of the kth layer in geological space.
Fig. 2 Schematic diagram of stratigraphic sequence
The ground plane may be dislocated by the fault plane when passing through the fault, as shown in Figure 3, so it is necessary to modify the corresponding function to make it locally discontinuous. This can be achieved by modifying ri in Formula (9) and rij in Formula (12) as follows.
Fig. 3 schematic diagram of dislocation along the ground plane of the fault
Suppose there are points P(x, y, z), Pi(xi, yi, zi) and Pj(xj, yj, zj) in the space. If the judgment result of Formula (5) shows that P and Pi are located on both sides of the fault plane, and the intersection of their connecting lines and the fault plane is calculated as P0(x0, y0, z0) by Formula (4), then they are located on the actual fault plane.
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Similarly, if Pi and Pj are located on both sides of the fault, and the intersection P0(x0, y0, z0) between their connecting lines and the fault plane is located on the actual fault plane, then rij in the formula (12) is replaced by the following formula.
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In which: zb=zc-s.sinβ.
xb=xc-(zc-zb).cotβ cosα
yb=xc-(zc-zb).cotβ.cosα
Ak is the fault dislocation coefficient, which is used to control the dislocation displacement of the bedding plane along the fault plane. When ri and rij in formulas (9) and (12) are replaced by formulas (13) and (14), discontinuity will automatically occur when the ground plane passes through the fault.
3 graphic display of 3D geological model
The graphic display program compiled in this paper can display the distribution of various geological information in geological bodies through a series of fitting functions. This program can display the cross-sectional view of any section in space and the three-dimensional view of any block.
3. 1 profile
The positioning parameters of the profile in the three-dimensional space (Figure 4) are: ① the plane coordinates of the starting point and ending point of the profile line P 1(x 1, y 1) and p2 (x2, y2); ② The bottom height of the profile is z1; ③ The width of the contour v0④ The inclination angle of the side is β.
Fig. 4 schematic diagram of cross-sectional spatial positioning
Using the above parameters, the length u0, the top height z0 and the angle α between the section line and the X axis can be calculated.
u0 =[(x2-x 1)2+(y2-y 1)2] 1/2。
z2=z 1+V0 sinβ
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Therefore, any point P(u, v) in the local coordinate system with U0 and V0 as the plane coordinate axes in the section can be converted into the coordinate values x, y, z corresponding to P(x, y, z) in the overall coordinate system through the positioning parameters.
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Because the independent variables of the above-mentioned various fitting functions are expressed by global coordinates X, Y (single-valued surface) or X, Y, Z, using the formula (16), we can calculate various corresponding fitting function values at any point P(u, v) of the cross section for graphic display.
3.2 Three-dimensional stereogram
The three-dimensional map can display any local block in the computing domain in a three-dimensional way. The spatial positioning parameters of this plot (Figure 5) are: ① the coordinates of the origin x0, y0 and z0 of this plot; ② elevation ZL at the top of the plot; ③ The block length is xl and the width is YL; ④ The included angle between XL block and X axis is α.
Fig. 5 schematic diagram of local block space positioning
In graphic display, it is necessary to project a three-dimensional block onto the viewing plane, so the projection formula must be given. In this paper, the parallel projection formula is adopted, which is as follows:
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Where: u is the abscissa of the observation plane; V is the ordinate of the observation plane, which is in the same direction as the total coordinate 2; A is the azimuth of the observation plane; β is the apparent inclination angle of the apparent plane; αx = arctan(sinαcosβ/cosα)； αy=arctan(cosαcosβ/sinα). Using different α and β values, we can observe the block from different directions, calculate various geological information of the visible surface of the block, and display it as the corresponding profile method.
Four examples
It is assumed that the specified geological body contains geological information such as surface topography, groundwater level, faults, random joints, ground and intrusions. The following five examples show the distribution of different information in three-dimensional blocks. Finally, two parts are randomly cut out in the corresponding block for display.
Example 1 is an overall perspective view of the designated block, including information of surface topography, groundwater level, weathered loose layer, faults and ground level. Block length xl= 100, width yl= 100, origin coordinate x0=0, y0=0 and z0=0, as shown in Figure 6.
Figure 6 3D Block 1
Example 2 shows the random joint distribution in the above plot and the intrusion in the lower left corner, as shown in Figure 7.
Fig. 7 three-dimensional block 2
Example 3 shows the information distribution on each slope after the block in example 1 is excavated at five levels, as shown in Figure 8.
Fig. 8 3D block 3
Example 4 is the display result after adding the intruder in Example 3, as shown in Figure 9.
Fig. 9 three-dimensional block 4
Example 5 shows the spatial contour distribution of a geophysical quantity in a designated block, as shown in figure 10.
Figure 10 3D Block 5
The following is a sectional view. Figure 1 1 is a vertical section taken along the diagonal of the block after selecting two kinds of information in Figure 6 and Figure 7, namely, x 1=0, y 1=0 to x2= 100 and y2= 100.
Figure 1 1 part 1
Figure 12 Section Figure 2
5 conclusion
Three-dimensional geological simulation of measured data and fitting function is a powerful tool in geoscience research, which can be used for processing and graphic display of various geological information.
Geological data used for simulation can be divided into spatial plane data and spatial curved surface data. Spatial surfaces can be divided into single-valued surfaces, multi-valued surfaces and ground surfaces. In this paper, the corresponding fitting function is given for each kind of information data.
At the same time, the graphic display of three-dimensional information is discussed, and the display methods of cross-sectional view and three-dimensional view are given Finally, examples of five perspective views and two cross-sectional views are given. These figures show that satisfactory results can be obtained by processing and displaying geological information with the method and program provided in this paper.