Solid matrix partition by fracture networks

The geometrical properties of the matrix blocks formed by a random fracture network are investigated numerically, for a wide range of fracture shapes and for fracture densities ranging from the dilute limit to well above the threshold where the material is entirely partitioned into finite blocks. The main block characteristics are the density and volume fraction, the mean volume and surface area, and their number of faces. In the dilute limit, general expressions for these characteristics are obtained, which provide a good approximation of the numerical data for any fracture shape. In the dense regime, most properties are governed by power laws, which involve two fitted exponents independent of the fracture shape. The shape factors identified in the dilute limit remain relevant for dense networks and can be used to formulate a general model for the block characteristics, valid up to the total matrix fracturation. The transition density when this occurs is determined. It can also be used to account for the fracture shape effects in a very simple and fairly accurate general model. Beyond the transition density, the block characteristics converge as expected toward those in the space tesselation by infinite planes.

Figure 1

Possible configurations of connected components in a 2D fracture network. Ten connected components $C_1$ to $C_{10}$ of oriented surface elements are shown. The finite blocks delimited by $C_3$, $C_4$, and $C_6$ and the infinite block between $C_8$ and $C_9$ are shaded. Some internal wings are indicated (W).

Figure 2

Blocks formed by fractures networks with ${\rho }^{\prime }=7.5$ (a) and 10 (b). Single blocks in systems of 20-gons with ${\rho }^{\prime }=50$ (c) and 60 (d); the block in (c) is formed by 203 fractures and has $N_2=453$ sides, $V_b=6.29$, $S_e=43.9$, $S_i=110.2$, $S_b=154.1$; the block in (d) is formed by 185 fractures and has $N_2=336$ sides, $V_b=4.13$, $S_e=30.5$, $S_i=77.7$, $S_b=108.2$.

Figure 3

(a) Single block made by 13 fractures, from a network of 20-gons with ${\rho }^{\prime }=50$; it possesses 11 external faces and contains internal wings; $V_b=0.0115$, $S_b=0.549$, $S_e=0.389$, and $S_i=0.160$; the white lines indicate the positions of the three cross sections. (b–d) Cross sections of this block by $xy$ planes at $z=0.254$, 0.334, and 0.458, which illustrate some features occurring for finite fractures such as the concave character and the existence of internal wings.

Figure 4

Schematic representation of various regimes of solid matrix fragmentation by fracture networks. Block volume fraction ${\varphi }_b$ ($•$) and mean block volume $\langle V_b\rangle$ ($\blacksquare$) as functions of the dimensionless density ${\rho }^{\prime }$ for networks of 20-gons. The three main regimes are indicated by numbers: (1) dilute, (2) dense, and (3) concentrated regimes. $\mathbf{T}$ is a transition zone from partial to complete fragmentation of the solid matrix.

Figure 5

Ratios of the measured and modeled block characteristics in the dilute limit, as functions of the shape factor $\eta$. The numerical data from Table 3 are normalized by the models (10), (11), and (13). Symbols correspond to $\langle d_e\rangle$ (blue $▾$), $\langle M_1\rangle$ (magenta $▴$), ${\langle S_b\rangle }^{1/2}$ (red $•$), and ${\langle V_b\rangle }^{1/3}$ (black $\blacksquare$). Solid (resp. broken) lines join the data for regular polygonal (resp. rectangular) fractures. The labels indicate the fracture shape (see Table 1).

Figure 6

Block density ${\rho }_b$ as a function of the fracture density $\rho$ (a). Solid lines are power law fits. Normalized dimensionless density $24{\rho }_b^{\prime }/{\eta }^2$ as a function of ${\rho }^{\prime }$ (b). The thick solid line corresponds to (36). The thin solid and broken lines correspond to (32a) for 20-gons and 4-rectangles, respectively. The ratio $24{\rho }_b^{\prime }/{\eta }^2{\rho }^{\prime 4+a_{\rho }}$ as a function of $\eta$ (c) and of ${\rho }^{\prime }$ (d). Data in panels (a), (b), and (d) are for 20-gons (black, $•$), hexagons (green, $▸$), squares (red, $\blacksquare$), triangles (blue, $▴$), 2-rectangles (brown, $▾$), and 4-rectangles (magenta, $◂$). Data in (c) are for ${\rho }^{\prime }\to 0$ (red, $•$), ${\rho }^{\prime }=4$ (black, $\blacksquare$), ${\rho }^{\prime }=10$ (blue, $▴$), ${\rho }^{\prime }=30$ (green, $⧫$). The labels in (c) indicate the fracture shape (see Table 1).

Figure 7

(a) Block volume fraction ${\varphi }_b$ as a function of ${\rho }^{\prime }$. Symbols are the numerical data with the conventions of Fig. 6. Broken and solid lines correspond for each fracture shape to result (16) in the dilute limit and to power law fits in the dense regime, respectively. (b) Same data normalized by $C_{\varphi b}$ in (37). The solid line corresponds to ${\rho }^{\prime 4+a_{\varphi }}$.

Figure 8

Transition density ${\rho }_1^{\prime }$ as a function of $\eta$. Solid symbols correspond to (38) with $C_{\varphi b}$ obtained in the dilute limit (black $•$) or by power law fits of ${\varphi }_b$ in the dense regime (blue $▴$), and to (48) with $C_{\varphi S}$ resulting from power law fits of ${\varphi }_S$ in the dense regime (red $▾$). Thin solid (resp. broken) lines join data for regular polygons (resp. rectangles). Thick blue and red broken lines correspond to the models (17), (38), and (48), respectively, and the thick solid black line to model (52). Black open symbols ($\circ$) and broken line correspond to the data and numerical fit of Ref. [22] for ellipses. Labels indicate the fracture shape (see Table 1).

Figure 9

Mean block properties as functions of ${\rho }^{\prime }/{\rho }_1^{\prime }$. Symbols correspond to the fracture shape, with the conventions of Fig. 6. (a) ${\varphi }_b$ (solid symbols) and ${\varphi }_S$, ${\varphi }_{Si}$ (open symbols). The lines correspond to models (39), (49), and (50b). (b) Number of faces $\langle N_2\rangle$. (c) Number fraction $n_i$ (open symbols) and volume fraction ${\psi }_i$ (solid symbols) of blocks with internal surfaces. (d) Number fraction $n_{cc}$ (open symbols) and volume fraction ${\psi }_{cc}$ (solid symbols) of concave blocks. (e) Normalized caliper diameter $P\langle M_1\rangle /A$. The solid line corresponds to (54). (f) Number of blocks $\langle N_{bc}\rangle$ per cluster, minus 1. Black and red lines correspond to the total number of blocks in the samples, for 20-gons and squares.

Figure 10

(a) Average block volume $\langle V_b\rangle$ for various polygon shapes as functions of ${\rho }^{\prime }$. Conventions are the same as in Fig. 6. The marks on the left axis correspond to the volumes in the dilute limit. Solid lines are power-law fits, and broken lines are the asymptotes (40), for 20-gons, squares, and 4-rectangles. (b) Numerical data for $\langle V_b\rangle$ divided by (42) as functions of $\eta$; same conventions as in Fig. 6; in addition, solid (resp. broken) lines join the data for regular polygonal (resp. rectangular) fractures.

Figure 11

The average surface areas $\langle S_b\rangle$ (a) and $\langle S_i\rangle$ (b), as functions of ${\rho }^{\prime }$. Same conventions as in Fig. 6. The solid lines correspond to power-law fits. The broken lines in (a) correspond to the asymptotes (44). The broken line in (b) has a slope $-3$. The marks on the left axis correspond to $\langle S_b\rangle$ in the dilute limit. $\langle S_b\rangle$ normalized by (46) (c) and $\langle S_b\rangle$ normalized by (51) (d), as functions of $\eta$. In (c) and (d), conventions are the same as in Fig. 6; in addition, solid (resp. broken) lines join the data for regular polygonal (resp. rectangular) fractures.

Figure 12

Volume-weighted average normalized by the number-weighted average of the block volume ${\langle V_b\rangle }_V/\langle V_b\rangle$ (a) and of the block surface area ${\langle S_b\rangle }_V/\langle S_b\rangle$ (b), as functions of ${\rho }^{\prime }/{\rho }_1^{\prime }$. The symbols correspond to the fracture shape, with the same conventions as in Fig. 6. Marks on the left axis are the results in the dilute limite (see Table 3). Broken lines correspond to the results for large densities in (56).