Permeability of isotropic and anisotropic fracture networks, from the percolation threshold to very large densities

The asymptotic behaviors of the permeability of isotropic fracture networks at small and large densities are characterized, and a general heuristic formula is obtained which complies with the limiting behaviors and accurately predicts the permeability of these networks over the whole density range. Theses developments are based on extensive numerical calculations and on theoretical arguments inspired by the examination of the flow distribution in the fractures at large densities. Then, the results are extended to anisotropic networks with a Fisher distribution of the fracture orientations, to polydisperse networks, and to fractured porous media. Finally, guidelines are provided for the practical evaluation of the required parameters from typical field data. A summary of the results is given in Table III.

Figure 1

The permeability $K^{\prime }$ as a function of the sample size $L^{\prime -2.22}$ for ${\rho }^{\prime }=2.39$, 2.50, 2.60, 2.70, and 2.80 [(a) bottom to top] and ${\rho }^{\prime }=2.80$, 2.89, 3.00, 3.49, and 3.99 [(b) bottom to top]. The lines are the least square fits of the data.

Figure 2

The permeability $K^{\prime }$ as a function of ${\rho }^{\prime }-{\rho }_c^{\prime }$ for networks of hexagonal fractures. Dots are size dependent data which are used when ${\rho }^{\prime }\le 4$ to determine the permeabilities of infinite networks before ($\circ$) and after ($•$) correction for the discretization effects.

Figure 3

The permeability $K^{\prime }$ in networks of hexagons (black, $•$), squares (magenta, $\blacksquare$), triangles (blue, $▾$), and 6-rectangles (red, $⧫$) as a function of (a) $\Delta {\rho }^{\prime }$ and of (b) ${\rho }^{\prime }$. The solid lines correspond to the power law (13b) and the broken line in (a) corresponds to (13c).

Figure 4

(a) The numerical data (symbols) and the model (16) (solid lines) for the permeability $K^{\prime }$ of fracture networks as functions of ${\rho }^{\prime }$. (b) The deviation $K_S^{\prime }-K^{\prime }$. The symbols and colors are associated with the fracture shapes: hexagons (black, $•$), squares (magenta, $\blacksquare$), triangles (blue, $▾$), and 6-rectangles (red, $⧫$). In (b), the thin curves are quadratic fits in terms of $1/{\rho }^{\prime }$ and the thin broken lines and thick horizontal bars on the right correspond to (17) and (29b), respectively.

Figure 5

The permeability $K^{\prime }$ as a function of $\Delta {\rho }^{\prime }$. The symbols represent the numerical data (same conventions as in Fig. 4). The black curve corresponds to (19).

Figure 6

Hexagonal fracture taken from a network with ${\rho }^{\prime }=60$. Intersection traces (thick blue lines) and triangular mesh (thin red lines) for (a) ${\delta }^{\prime }=1/8$. The fields $W^{\prime }$ measured in meshes with ${\delta }_M^{\prime }=1/4$ (b), 1/8 (c), 1/16 (d), and 1/32 (e). Sketches of the triangular regions $\widehat{OMV}$ of hexagonal and square fractures shown in Fig. 7f.

Figure 7

Local flow statistics measured in fracture networks with ${\rho }^{\prime }=10$ [(a), (b)], 30 [(c), (d)], 40 [(e), (f)], and 60 [(g), (h)]. The fractures are hexagonal, except in (e) and (f) where they are square. Maps of $\langle W^{\prime }\rangle /{\langle W^{\prime }\rangle }_c$ are shown in the left column, with the color scale shown in (i). The white lines are at a distance $\overline{s}$ from the contour. The pdfs of $W^{\prime }/{\langle W^{\prime }\rangle }_c$ measured in the same networks at the points $O$ (black), $G$ (blue), $H$ (green), and $M$ (red) are shown in the right column.

Figure 8

Profiles of $1-W^{\prime }/{\langle W^{\prime }\rangle }_c$ along the segment $OM$ in the networks of Fig. 7 as functions of the distance $h/\overline{s}$ from $M$. Data are for: ${\rho }^{\prime }=10$ (black), 30 (blue), 40 (green), and 60 (red).

Figure 9

The deviation $K_S^{\prime }-K^{\prime }$ as a function of (a) ${\rho }^{\prime }$ and the permeability $K^{\prime }$ as a function of (b) ${\rho }^{\prime }-{\rho }_c^{\prime }$. The symbols are the numerical data for hexagons (black, $•$), squares (magenta, $\blacksquare$), triangles (blue, $▾$), and 6-rectangles (red, $⧫$). The solid lines are (19a) with the coefficients (31).

Figure 10

Permeability of anisotropic networks. (a) $K_{\perp }^{\prime }$ and (b) $K_{\parallel }^{\prime }$ as functions of ${\rho }^{\prime }$. (c) $K_{\perp }^{\prime }/{\psi }_{\perp }$ and $K_{\parallel }^{\prime }/{\psi }_{\parallel }$ as functions of ${\rho }^{\prime }$. (d) $\Phi K_{\perp }^{\prime }/{\psi }_{\perp }$ and $\Phi K_{\parallel }^{\prime }/{\psi }_{\parallel }$ as functions of ${\rho }^{\prime }-{\rho }_c^{\prime }$. Data are hexagons ($\circ$), squares ($\square$), triangles ($\nabla$), and 4-rectangles ($☆$), with $\kappa =10$ (blue), 50 (magenta), and 200 (red). Open and solid symbols in (c) correspond to $K_{\parallel }^{\prime }/{\psi }_{\parallel }$ and $K_{\perp }^{\prime }/{\psi }_{\perp }$, respectively. The curve in (d) corresponds to (19).

Figure 11

Discretization effects. (a) The permeabilities $K^{\prime }$ obtained along the $x$, $y$, and $z$ axes in 10 realizations of hexagonal fracture networks with ${\rho }^{\prime }=20$ and $L^{\prime }=6$, as functions of ${\delta }_M^{\prime }$ (thin blue dots and lines). The thick dots and line correspond to the overall average. (b) The permeabilities $K^{\prime }$ obtained in hexagonal fracture networks with ${\rho }^{\prime }=50$ and $L^{\prime }=4$, in average over the $x$, $y$, and $z$ axes and two realizations, as functions of the mesh resolution. The data correspond to triangulations with ${\delta }_M^{\prime }=1/4$, 1/8, and 1/16 ($•$), to two successive subdivisions of the meshes with ${\delta }_M^{\prime }=1/4$ ($\blacksquare$), and to the subdivision of the meshes with ${\delta }_M^{\prime }=1/16$ ($\square$). The solid lines are extrapolations based on ${\delta }_M^{\prime }=1/4$ and 1/8, and on ${\delta }_M^{\prime }=1/8$ and 1/16. The broken line is the extrapolation based on the subdivisions of the meshes with ${\delta }_M^{\prime }=1/4$.

Figure 12

The length ${\Delta }^{\prime }$ [see Eq. (A1)] as a function of ${\rho }^{\prime }$ for networks of hexagons ($•$), squares ($\blacksquare$), triangles ($▾$), and 6-rectangles ($⧫$). The solid lines are the fits (A2).

Figure 13

Tessellation of a fracture by the traces of intersections with other fractures. (a) General sketch with inner ($A_I$, shaded) and border ($A_B$, white) regions. (b) and (c) Illustration of the notations. The symbols denote polygon vertices $v_v$ ($\blacksquare$), intersections of the contour with traces $v_b$ ($\square$), trace intersections $v_i$ ($•$), and trace endpoints $v_e$ ($\circ$).