Automatic Geological Structure Reconstruction

INTRODUCTION

In this article, a new geological structure reconstruction method is described, based on information regarding the occurrence of geological horizons obtained by exploration.

A number of terms used in the description of the technique need to be determined. Under the wells in this note seem geotechnical boreholes that determine the physical and mechanical properties of soils. A borehole provides information on the vertical distribution of layers through the soil depth. The layers of materials around the drilling wells are shown by the segments in Figure 1.

Reconstruction of borehole

Fig. 1: Layers of materials revealed by boreholes

When creating the geological models in specialized software packages, some additional steps are required in most cases. For example, when reconstructing geological layers using borehole data, the grouping of layer segments is performed manually. This can be a very complicated process for sites with a large number of boreholes and layers.

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Algorithms and Mathematical Notes of Creating Cylinder Shaped Voxel Meshes

In the range of applied problems for various industries, for example, architecture, fine arts, animation (movies), meshes of complex configuration may be of great demand. Let us show, how it is possible to describe a configurationally complex computational mesh, consisted of voxels (bubble shaped), which are located on a sheet and twisted into cylinder.

Computational mesh,which is constituent of voxels, situated on a sheet and twisted into cylinder

Let us describe a mathematical model used at the design of algorithms of cylinder shaped voxel meshes.

INITIAL DATA FOR CREATING VOXEL MESH

There is a sheet with voxels (bubbles of cylinder shape) located on its surface. Then, it is assumed that voxels are situated on the sheet according to the template, demonstrated in the picture 1.

Voxels location. Template

Picture 1– Voxel location template

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The transfer of boundary conditions on an orthogonal hexahedral mesh

In the simulation of physical processes and phenomena overall there is a problem of numerical solutions of differential problems in partial derivatives. One of the methods for numerical solutions of mathematical physics equations is the approach, based on the finite difference approximation. However, the major drawback of this method is the need to use orthogonal hexahedral mesh. While solving problems in practice, sometimes a complex geometric configuration of the computational domain (for example, see Figure 1) is used, and therefore, the drawback, mentioned above, is enough critical.

Picture 1 – The example of a complex surface as a part of the computational domain

Due to the aforesaid, there is quite a natural problem of the geometrical configuration approximation of computational domain by cell faces of the given orthogonal hexahedral mesh.

Let us turn to a more detailed problem statement. Let the computational mesh in 3D space be orthogonal and hexahedral. Namely, let be a set of mesh divisions along axis arranged in ascending order, similar to it let us introduce the ordered sets of mesh divisions along and respectively. According to designations, the following set of points is the set nodes of computational mesh . Geometric configuration that requires approximation by mesh faces, is defined by triangulated surface. Let be a set of triangles. This set is a part of surface triangulation, and its quantity is .With the numerical solution of mathematical physics problem, besides geometric approximation of surface by faces of mesh cells there is a problem of adequate transfer of triangulated surface area on cells faces, involved in its approximation. Thus, by means of set of triangles it is necessary to determine the set of faces of mesh cells, that approximate the triangulated surface, and match the transferred area with each of such face.

Let us give the algorithm, which solves the set above problem

1) Let us consider the mesh, which is dual to the initial one, i.e. such a mesh the nodes of which are cells centers of the initial computational mesh. Let us store a 3D array of real numbers, elements of which correspond to the cells of dual mesh. First of all we initialize elements of array by zeros.

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3D geological modeling based on boreholes data

Introduction

3D geological modeling is a very actual issue nowadays in building development, environmental assessment of soil (variably-saturated porous medium) pollution, assessment of mineral deposits, etc. There are different approaches to solve this problem by means of modern software designed for simulation in geology [1-3]. The most frequently used method is that of reconstructing geological model. This method is based on information about the levels of geological horizons occurrence received from the results of drilling [4-6]. The implementation of this method itself may have some peculiarities.

In this article an alternative approach for 3D geological model creation is being proposed. It is based on the following:
1) Surface triangulation of site topology
2) Automatic cross-section generation
3) Segment height interpolation for each layer of geological model.
This approach allows both simplify and accelerate 3D geological model creation while maintaining acceptable 3D site building accuracy.

The proposed method consists of six basic steps described below. The following information on boreholes is considered as given data: 1) borehole coordinates; 2) seamark; 3) capacity of geological horizons.

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Partitioning a segment with optional thickening toward ends

Preface:

The initial line segment is partitioned with “obligatory” points (they should be mesh nodes), which can both be or not be points of thickening. On each part of the segment the method of subpartitioning, described below, can be applied.

Let  – be the minimum acceptable step,  – be the maximum acceptable step and let  be the length of the segment, where . Then the minimal possible number of steps equals – , and the maximal possible is – . If , then the problem statement is incorrect. Assume that .

If both edges of the segment do not require thickening, we can take a uniform partition with any number of steps   so that .

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Fast volume computing algorithm of arbitrary polyhedron in 3D space

Introduction

Computing of the volume of polyhedron in 3D space isn’t a trivial problem, but the following trivial method exists: dividing the volume into simple pyramids and counting the sum of these volumes. However, this methodology is difficult for implementation, as well as it is resource-intensive and slow. What is more, computing algorithm according to this methodology is difficult to parallelize. And taking into account trends in the field of parallel computing on GPU development, the ability to increase the rate of computing repeatedly is being lost.

In this article the algorithm for determining the volume of arbitrary geometry in 3D space in terms of fast computing is described.

Input data

Preparatory step. A rectangular grid (pos.1 on fig.1) around geometry (pos.2 on fig.1) is built using its minimum and maximum coordinates.

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