Method for Generating a Discrete Fracture Network from Microseismic Data and its Application in Analyzing the Permeability of Rock Masses: a Case Study

Abstract

The deformation and failure process of rock masses is accompanied by the initiation, propagation and connection of fractures. The behaviors of fractures under engineering disturbance can be effectively determined by microseismic (MS) monitoring, and such information is essential for the stability analysis of rock masses. Based on moment tensor theory, the geometric properties of focal planes and the failure mechanisms of the seismic source can be determined, and this study proposes criteria for extracting the reasonable fracture from focal planes associated with different failure mechanisms and provides a detailed example. The fracture identified by the proposed criteria is called MS-derived fracture, and a formula describing the aperture is derived from the moment tensor theory and the motion characteristics of the MS-derived fractures associated with different failure mechanisms. This work chose the Shirengou Iron Mine as a case and selected an area exhibiting seepage and rock mass failure as the study area. Based on a three-dimensional noncontact discontinuity scan, a natural discrete fracture network (DFN) was generated. With the help of the proposed fracture generation method, a new DFN based on MS data was generated and is referred to as the MS-derived DFN. With the aid of Oda’s theory, the changes in the permeability value, principal direction and anisotropy degree in the study area were analyzed based on the MS-derived DFN, and the seepage channels in the study area were also determined. Therefore, the research methods in this paper could be used to better understand the changes in the permeability of rock masses based on MS data.

Appendices

Appendices

1.1 Moment Tensor Decomposition

According to the moment tensor decomposition method proposed by Knopoff and Randall (1970) and Vavryčuk (2015), the moment tensor can be decomposed according to the following equation.

$$\begin{aligned} {\mathbf{M}} & ={{\mathbf{M}}_{{\text{ISO}}}}+{{\mathbf{M}}_{{\text{DC}}}}+{{\mathbf{M}}_{{\text{CLVD}}}} \\ & ={M_{{\text{ISO}}}}{{\mathbf{E}}_{{\text{ISO}}}}+{M_{{\text{DC}}}}{{\mathbf{E}}_{{\text{DC}}}}+{M_{{\text{CLVD}}}}{{\mathbf{E}}_{{\text{CLVD}}}} \\ \end{aligned}$$

(24)

where ${{\mathbf{M}}_{{\text{ISO}}}}$ represents the isotropic component of the moment tensor, which is the center of expansion (or compression) and describes changes in the volume of the seismic source region; ${{\mathbf{M}}_{{\text{DC}}}}$ represents the pure double couple component; and ${{\mathbf{M}}_{{\text{CLVD}}}}$ represents the compensated linear vector dipole component. ${M_{{\text{ISO}}}}$, ${M_{{\text{CLVD}}}}$ and ${M_{{\text{DC}}}}$ are the ISO, CLVD and DC coordinates in the 3D source-type space, respectively, and ${{\mathbf{E}}_{{\text{ISO}}}}$, ${{\mathbf{E}}_{{\text{CLVD}}}}$ and ${{\mathbf{E}}_{{\text{DC}}}}$ are the ISO, CLVD and DC base tensors, respectively. The base tensors can be expressed as follows:

$${\mathbf{E}}_{{{\text{ISO}}}} = \left[ {\begin{array}{*{20}l} {\begin{array}{*{20}l} {\begin{array}{*{20}l} 1 \hfill & {\quad 0} \hfill \\ \end{array} } \hfill & 0 \hfill \\ \end{array} } \hfill \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {\quad 1} \\ \end{array} } & {\quad 0} \\ \end{array} } \\ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & {\quad 0} \\ \end{array} } & {\quad 1} \\ \end{array} } \\ \end{array} } \hfill \\ \end{array} } \right]$$

(25)

$${{\mathbf{E}}_{{\text{DC}}}}=\left[ {\begin{array}{*{20}{c}} 1&0&0 \\ 0&0&0 \\ 0&0&{ - 1} \end{array}} \right]$$

(26)

$${\mathbf{E}}_{{{\text{CLVD}}}}^{+}=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 2&0&0 \\ 0&{ - 1}&0 \\ 0&0&{ - 1} \end{array}} \right]\;\left( {{\text{if}}\;{M_1}+{M_3} - 2{M_2} \geqslant 0} \right)$$

(27)

$${\mathbf{E}}_{{{\text{CLVD}}}}^{ - }=\frac{1}{2}\left[ {\begin{array}{*{20}{c}} 1& \quad 0& \quad 0 \\ 0& \quad 1& \quad 0 \\ 0& \quad 0& \quad { - 2} \end{array}} \right]\;\left( {{\text{if}}\;{M_1}+{M_3} - 2{M_2}<0} \right)$$

(28)

The ${M_{{\text{ISO}}}}$, ${M_{{\text{CLVD}}}}$ and ${M_{{\text{DC}}}}$ can be expressed as follows:

$${M_{{\text{ISO}}}}=\frac{1}{3}({M_1}+{M_2}+{M_3})$$

(29)

$${M_{{\text{CLVD}}}}=\frac{2}{3}({M_1}+{M_3} - 2{M_2})$$

(30)

$${M_{{\text{DC}}}}=\frac{1}{2}({M_1} - {M_3} - \left| {{M_1}+{M_3} - 2{M_2}} \right|)$$

(31)

where ${M_{{\text{CLVD}}}}$ also considers the sign of the base CLVD tensor. If the base CLVD tensor is considered with its sign, then ${M_{{\text{CLVD}}}}$ should be calculated as the absolute value of Eq. (30) M₁, M₂ and M₃ are the eigenvalues of the moment tensor, where the relationship is M₁ > M₂ > M₃.

The moment tensor can characterize the event magnitude, fracture type (e.g., double-couple or tensile), and fracture orientation. It is necessary to solve the proportions of ISO, DC and CLVD in the moment tensor, ${C_{{\text{ISO}}}}$, ${C_{{\text{DC}}}}$ and ${C_{{\text{CLVD}}}}$ (Eyre 2015). The relative scale factors ${C_{{\text{ISO}}}}$, ${C_{{\text{DC}}}}$ and ${C_{{\text{CLVD}}}}$ can be defined as:

$$\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{C_{{\text{ISO}}}}} \\ {{C_{{\text{DC}}}}} \end{array}} \\ {{C_{{\text{CLVD}}}}} \end{array}} \right]=\frac{1}{M}\left[ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{M_{{\text{ISO}}}}} \\ {{M_{{\text{DC}}}}} \end{array}} \\ {{M_{{\text{CLVD}}}}} \end{array}} \right]$$

(32)

where

$$M=\left| {{M_{{\text{ISO}}}}} \right|+{M_{{\text{DC}}}}+\left| {{M_{{\text{CLVD}}}}} \right|$$

(33)

$$\left| {{C_{{\text{ISO}}}}} \right|+{C_{{\text{DC}}}}+\left| {{C_{{\text{CLVD}}}}} \right|=1$$

(34)

1.2 Fracture Geometric Properties

Based on the symmetry of the moment tensor and the eigenvalues of the moment tensor, the eigenvectors of the moment tensor can be expressed as follows:

$${{\mathbf{e}}_1}=\frac{{{\mathbf{n}}+{\mathbf{v}}}}{{\left| {{\mathbf{n}}+{\mathbf{v}}} \right|}},\;{{\mathbf{e}}_2}=\frac{{{\mathbf{n}} \otimes {\mathbf{v}}}}{{\left| {{\mathbf{n}} \otimes {\mathbf{v}}} \right|}},\;{{\mathbf{e}}_3}=\frac{{{\mathbf{n}} - {\mathbf{v}}}}{{\left| {{\mathbf{n}} - {\mathbf{v}}} \right|}}$$

(35)

where the relationship of ${{\mathbf{e}}_1}$, ${{\mathbf{e}}_2}$ and ${{\mathbf{e}}_3}$ is ${{\mathbf{e}}_1} \ \bot \ {{\mathbf{e}}_2} \ \bot \ {{\mathbf{e}}_3}$, the absolute value sign represents the vector size, and $\otimes$ is the symbol of vector multiplication.

The normal and motion direction of the focal plane can be deduced, as shown below:

$$\left\{ {\begin{array}{*{20}{c}} {{\mathbf{n}}=\sqrt {\frac{{{M_1} - {M_2}}}{{{M_1} - {M_3}}}} {{\mathbf{e}}_1}+\sqrt {\frac{{{M_2} - {M_3}}}{{{M_1} - {M_3}}}} {{\mathbf{e}}_3}} \\ {{\mathbf{v}}=\sqrt {\frac{{{M_1} - {M_2}}}{{{M_1} - {M_3}}}} {{\mathbf{e}}_1} - \sqrt {\frac{{{M_2} - {M_3}}}{{{M_1} - {M_3}}}} {{\mathbf{e}}_3}} \end{array}} \right.$$

(36)

The source dimension is usually expressed as the radius of the fault and is related to the corner frequency of the P-wave or S-wave. A general feature of seismic spectra that have been corrected for attenuation is that the corner frequency f_c can be related to the source dimensions through the following equation (Gibowicz and Kijko 1994; Mendecki 1997).

$${r_s}=\frac{{{K_c}{\beta _0}}}{{2\pi {f_c}}}$$

(37)

where ${\beta _0}$ is the S-wave velocity in the vicinity of the source, ${r_s}$ is the source radius, and ${K_c}$ is a constant that depends on the source model. In mine-related seismicity studies, the fault is usually modeled as a simple circular fault (Brune 1970; Madariaga 1976). According to Brune (1970), ${K_c}$ is equal to 2.34 for S-waves. Researchers have found that both the shape and the fracture size are smaller than those in the Brune model (Gibowicz and Kijko 1994). Madariaga (1976) suggested that ${K_c}$ is equal to 2.01 for S-wave and 1.32 for P-wave. Gibowicz and Kijko (1994) thought that Madariaga’s (1976) values were more practical. In this paper, P-waves are used to solve for the corner frequency f_c, so ${K_c}$ is equal to 1.32.