Empirical methods to quickly select an appropriate discrete fracture network (DFN) model representing the natural fracture facets

Abstract

The accuracy of the established discrete fracture networks (DFNs) has a significant impact on the effectiveness of solving engineering problems, while the accuracy is affected by the selection of the model. Concerning the natural fracture facets, the existing method based on the Accuracy Representation Index (ARI) to select the circular or elliptical disc models is relatively complex, costly and time-consuming, and is not convenient for engineers at the field site. This study proposed two new empirical methods for engineers to quickly choose an appropriate model. The triangular, rectangular and diamond-shaped fractures are chosen as the basic shapes. For the fractures with other convex shapes, their ARI can be estimated on the basis of the three basic shapes. Moreover, a validation of a real case from an open-pit mine in the USA was made, three volunteers estimated the values of ARI_c and ARI_e of 56 fractures based on the proposed empirical methods, where ARI_c and ARI_e represent the values of ARI of circular and elliptical disc models, respectively, and the absolute error (er) from the empirical estimating method is calculated. The results show that the values of er are less than 0.1 and there is no obvious difference between results from the three volunteers. It illustrates that the proposed empirical methods are effective and applicable.

Appendices

Appendix 1. A brief introduction to the Guo-procedure

Guo et al. (2020) developed a procedure to estimate the ARI of the circular and elliptical disc model with respect to the natural fracture facet. The procedure shows the process from inputting the raw data collected in the field to outputting the results, and a simplified summary of the procedure is as follows:

• 1. Constructing a new polygonal fracture plane by fitting a plane

The shapes of natural fracture facets present various polygon. The vertex coordinates of one fracture facet can be obtained by some means. Since the fracture facet is not always a plane and has a slight curvature, we should fit a new polygonal plane by the vertex coordinates based on the least square method. Moreover, the normal vector of the new polygonal plane can be easily calculated.

• 2. Three-dimensional shape planarization

The new polygonal plane is three-dimensional (3-D) in space and its dip direction and dip angle can be obtained by its normal vector. Owing to the mathematical convenience, it is very essential to convert the polygon to a two-dimensional (2-D) plane polygon, and the coordinate transfer based on dip direction and dip angle could achieve that.

• 3. Calculating the 2-D polygon area and centroid coordinates

Usually, the vertexes of the polygon are disorderly. For the convenience of calculating its area and centroid coordinates, the Convex hull algorithm method (McCallum and Avis 1979) can be used to arrange the vertices of the polygon in an orderly manner counterclockwise or clockwise. The area of the 2-D polygon plane can be calculated by the vector cross product.

• 4. Calculating the ARI of circular and elliptical disc model to represent the 2-D polygon

According to the definition of ARI, the circular or elliptical disc model has the same area and centre point as the polygon. The key to calculating ARI is to estimate the overlap area between the circle or elliptic and the polygon. Since the overlap area is an irregular figure, it is very difficult to calculate the area by an analytical solution. The Monte Carlo simulation can be used to estimate the irregular overlap area.

First, we constructed a calculating region to cover the entire area of the polygon and then calculated the area of the calculating region of S_region. Next, a number of coordinate points labelled as N were generated in this region using the Monte Carlo simulation. The estimation of the overlap area of S depended on how many coordinate points were inside the overlap region, labelled as N_in. So S could be expressed as

$$ S=\frac{N_{\boldsymbol{in}}}{N}\cdotp {S}_{\boldsymbol{region}} $$

Point in polygon strategies could be used to determine whether the coordinate point was inside this polygon. Taking one point as a starting point to make the horizontal radical line, the number of intersection points generated by the polygon and the radical line could be counted. If the number of intersection points was odd, the point had to be inside the polygon, and vice versa. Otherwise, the point was outside the polygon.

It should be noted that the estimating precision is mainly determined by N. Owing to the limitation of the computer memory and running time, N is not infinite. In consideration of the estimating precision and running time comprehensively, N = 1,000,000 is recommended.

For the circular disc model to represent the 2-D polygon fracture, it is very easy to calculate the overlap area between a circle and a polygon. However, the elliptical disc model has more parameters, such as the long-short axis length ratio k_e, and the rotation angle γ_e. For the elliptical disc model to represent the 2-D polygon fracture, the overlap area was controlled by the two parameters of k_e and γ_e. The range of rotation angle is from 1 to 180°, the long-short axis length ratio is more than 1, and the search method can be used to determine the upper limit value of k_e. After the ranges of γ_e and k_e are determined, the overlap area for each elliptical disc model with different combinations of γ_e and k_e can be calculated, and the maximum value of overlap area is what we want.

Appendix 2

Table 5 The data obtained by three volunteers based on the two empirical methods