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Subsurface Fluid Flow: The Hydrology of Geothermal Systems 57

A

B

C

FIGUre 4.4 Schematic two-dimensional slice through a fractured porous rock. Fracture A is a large aper-
ture, continuous fracture that has significant surface roughness and a highly variable fracture opening. The
darker gray masses on either fracture wall represent secondary minerals that have grown on the wall. Fracture
B is also a through-going fracture with very smooth walls and a variable aperture. Fracture set C is discontinu-
ous and would provide no permeability.

fracTure TransmissiviTy
The transmissivity of a fracture, which is the discharge through a fracture at some velocity across a
given unit aperture, can now be defined as

T = (ρ × g/μ) × (a /12) × a = (ρ × g × a )/(12 × μ). (4.6)
2
3
fr
This relationship is often referred to as the cubic law because of the dependence of the transmis-
sivity on the cube of the aperture. Hence, the overall movement of fluid through a fracture set in a
rock volume can be characterized primarily by the fracture aperture and fluid properties.
However, as is evident from the schematic diagram of fracture A in Figure 4.4, the distance
between faces can be highly variable and the definition of aperture becomes a challenge. In addi-
tion, the roughness of surfaces, when combined with the variability of the fracture aperture, can
often lead to the development of preferential flow paths. Hence, the total volume of open space in
a fracture is not necessarily representative of the volume of flowing fluid that could be moving in a
fracture at any particular instant in time.
Nevertheless, measured permeability of fractured materials provide a clear demonstration of the
importance of knowing fracture properties for any geothermal application in which significant fluid
flow rates must be obtained. In Figure 4.5, fluid flux is shown as a function of permeability and pres-
sure gradient. The permeability ranges from Table 4.1 are also shown for reference. It is clear from
the graph that the flux that can be obtained from highly fractured rocks is, at a given pressure gradi-
ent, at least two to five orders of magnitude greater than that which can be obtained from matrix flow
in fine sand. This difference has profound importance for many geothermal applications where flow
rate, and hence the amount of heat that can be made into work at a given rate, determines the eco-
nomics or efficiency of an application. The requirements for flow rates are discussed in more detail
in the chapters dealing with specific applications.
In Figure 4.6 the effect on permeability of fracture aperture and spacing is modeled. Note that
for a given fracture spacing, an order of magnitude increase in the fracture width results in a two
order of magnitude increase in the permeability. However, for a given fracture width, a decrease
in the fracture spacing by an order of magnitude only results in an order of magnitude increase in