16.5 Hydraulic Properties of Fractures
Porosity
In most natural formations, the porosity associated with fractures is very low and is generally much lower than the matrix porosity, unless it is a purely fractured system with very little porosity in the matrix. Permeability, however, is a different story, as a fracture with even a small aperture can potentially move a lot of water.
Permeability
Discharge in a single idealized fracture with smooth walls and a constant aperture is governed by Equation (116), called the cubic law.
| [latex]Q = \frac{\rho_{w}gb^{3}i}{12\mu}[/latex] | (116) |
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where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]Q[/latex] | = | volumetric discharge per unit width of fracture plane (L2T-1) |
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| [latex]\rho_{w}[/latex] | = | density of water (ML-3) |
| [latex]g[/latex] | = | gravitational acceleration (LT-2) |
| [latex]b[/latex] | = | aperture (distance between walls of fracture) (L) |
| [latex]i[/latex] | = | hydraulic gradient ([latex]dh/dl[/latex]) (:) |
| [latex]\mu[/latex] | = | viscosity of water (ML-1T-1) |
The hydraulic conductivity of a fracture is expressed by Equation (117).
| [latex]K = \frac{\rho_{w}gb^{2}}{12\mu}[/latex] | (117) |
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This equation shows that hydraulic conductivity is proportional to the square of the aperture, and discharge is proportional to the cube of the aperture—hence the name cubic law.
This is the basis for a conceptual and mathematical understanding of fluid flow in fractures. In an idealized fracture, the hydraulic conductivity and discharge are a function of, and directly proportional to, the size of the aperture—that is, an increase of the aperture results in an increase in the hydraulic conductivity and an increase in the discharge, assuming the same gradient. In addition, the relationship between aperture and conductivity/discharge is not linear—a small increase in aperture results in a large increase in permeability and an even larger increase in discharge.
One can use the cubic law to calculate discharge and permeability of fractures. When that is done, it shows that a fracture with an aperture as small as 1 mm has permeability close to 1 m/s, which means that it can transmit a lot of water. Fractures therefore don’t have to be very large to have a big influence on a flow system.