Estimation of the distribution of fractures by sizes and orientations based on data on fracture traces

Abstract

For modeling a fractured rock mass, it is necessary to have information about the geometric characteristics of the fractures - their sizes, orientations, and numbers.
As a result of geological surveys and observations during mining operations, data are obtained on the number and orientation of fracture traces.
This leads to the tasks of restoring the spatial pattern of the fracture distribution on surfaces or through boreholes. The tasks that actually arise here are tomography tasks. This work is dedicated to their mathematical formulation and reduction to classical problems of finding the inverse Radon transform.
In this work, when considering the tasks of finding the distribution of fractures by orientation alone, under a fracture we will understand a section of a flat surface, having an arbitrary shape.
In solving the problem of finding the joint distribution of fractures by size and orientation, we will consider the fractures to be disc-shaped. Assuming, for example, elliptical fractures makes the problem unsolvable. This is because an elliptical fracture is defined by five parameters: the orientation of the plane, the direction of the main axes, and their magnitudes. Therefore, the distribution function of such fractures by shapes and orientations is a function of five variables.
On the other hand, the distribution function of fracture traces by sizes and orientations is already a function of four variables - the direction of the intersecting plane and the size and direction of the trace there. Therefore, the task of finding the distribution of fractures for elliptical fractures, generally speaking, is not solvable unambiguously, which is why disc-shaped fractures are assumed.