5.6  Hydraulic Conductivity in Fractured Rocks

William W. Woessner; Eileen P. Poeter

5.6 Hydraulic Conductivity in Fractured Rocks

Hydraulic conductivity of a fractured rock is dependent on its unfractured hydraulic conductivity (called matrix hydraulic conductivity) as well as the spacing between fractures, their aperture (width of the fracture opening), roughness of the fracture walls, and their interconnectedness.

When fracturing is extensive relative to the volume of material that a K values is assigned to, groundwater flow in the fracture network can be treated as a continuum and the hydraulic conductivity of the system can be used in Darcy’s law to estimate flow rates (Figure 41). In that case, the transmission properties of the fractured medium are assumed to behave as an equivalent porous medium represented by a hydraulic conductivity value in each primary direction.

Figure showing the conceptualization of an REV for a fractured rock system — **Figure 41 –** Conceptualizing the REV for a fractured rock system when an equivalent hydraulic conductivity is to be used to represent the system. The red circles represent sampling of progressively larger volumes of fractured rock. V₁ represents a sample of either a fracture (black line) or the solid matrix. In this example a representative elementary volume is recognized at V₃. Volumes larger that V₃ can be represented with the same equivalent property values until the volume is as large as V₅. The V₅ sample volume includes an area with different fracture conditions.

The site can be conceptualized as an equivalent porous medium as long as the smallest unit of interest exceeds the volume of the REV. Figure 42 presents a conceptual model of a fractured volume of rock. Fractured rocks are typically anisotropic because of the directional variations in fractures and joints (Figure 42). In some cases, it is quite common for K_z to be greater than K_x because fractures oriented in the near vertical direction are common and tend to have larger apertures than those that are not as steep. Vertical fractures tend to have larger apertures because they do not respond to the weight of overlying material to the same degree as horizontal or fractures occurring at less than vertical orientations.

Figure showing conceptual models of fractured media — **Figure 42 –** Conceptual models of fractured media with white, yellow and green lines representing fractures. a) A system that is extensively fractured relative to the volume of material being investigated can be represented using an equivalent-porous-medium K and Darcy’s law because groundwater flow would be similar to flow in porous media. b) A sparse fracture network cannot be well represented as an equivalent-porous-medium, so it is more appropriate to evaluate flow through individual fractures or fracture networks.

When the matrix hydraulic conductivity is minimal and groundwater is flowing through a sparse network of fractures or when groundwater transport of dissolved substances through fractures is of interest, it can be useful to determine the hydraulic conductivity of individual fractures or a fracture network (Figure 42b). For example, Snow (1968) developed an equation to estimate the equivalent hydraulic conductivity (and intrinsic permeability) of a medium with a parallel array of planar fractures (Equations 41 and 42) as shown in Figure 43. The aperture of the fractures, b, and the number of fractures per unit distance of rock face, N, are required to estimate K of the fractured system.

$\displaystyle K_{fracture\ set}=\frac{\frac{\rho g}{\mu }Nb^{3}}{12}$

(41)

where:

K_{fracture set}	=	equivalent hydraulic conductivity of a fracture set (L/T)
N	=	the number of fractures per unit distance of rock face, N = 1/spacing (s) where spacing is the average distance between fractures (1/L)
b	=	fracture aperture (L)
ρ	=	fluid density (M/L³)
g	=	gravitational constant (L/T²)
μ	=	Dynamic viscosity M/(LT)

$\displaystyle k_{fracture\ set} = \frac{Nb^{3}}{12}$

(42)

where:

k_{fracture set}

=

equivalent intrinsic permeability of a fracture set (L²)

Figure illustrating fracture aperture, b, and spacing between fractures — **Figure 43 –** Fracture aperture, b, and spacing between fractures, s. The number of fractures per unit length, N = 1/s.

In general, when conducting groundwater resource investigations, fractured rock systems are treated as equivalent pores media. When representations of flow in a fracture network are required, specialized groundwater modeling tools are applied.

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