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Subsurface Fluid Flow: The Hydrology of Geothermal Systems 55

Table 4.2
The range of hydraulic conductivities (m/s) for Various rocks
material hydraulic conductivity (low) hydraulic conductivity (high)
Clay 1.2e−13 1.2e−7
Fine sand 7.0e−7 3.0e−6
Coarse sand 5.8e−6 2.3e−5
Gravel 2.3e−5 7.4e−4
Granite 3.5e−9 3.5e−7
Slate 1.2e−13 1.2e−10

The KoZeny–carman equaTion
The factors that determine permeability were formally quantified by Kozeny (1927) and modified
by Carman (1937, 1956) through the relationship
2
κ = [n /(1 −n) ]/(5 × S ) , (4.2)
2
3
A
where n is the porosity and S is the specific surface area of the pore spaces per unit volume of solid.
A
Equation 4.2 is known as the Kozeny–Carmen equation. This equation allows the dependence of the
permeability on the porosity of a porous sample to be determined. Implicit in this relationship are all
of the factors discussed above regarding flow in porous rocks. Of particular importance for permeabil-
ity is the tortuosity of the flow path—the more tortuous the network of pores through which fluid must
flow, the lower will be the permeability. Tortuosity can be accounted for by recasting Equation 4.2 as
3
2
κ = c × T × [n /(1 −n) ]/S , (4.2a)
2
A
o
where T is the tortuosity, which is equivalent to the ratio of a straight path of length L connecting
two points to the actual path followed along some tubular route L (i.e., L/L ). The c is a constant
0
t
t
characteristic of the system. Generally, c × T = 0.2, thus reducing Equation 4.2a to Equation 4.2.
o
hydraulic conducTiviTy
A useful measure of the ability of a rock to allow fluid to flow is the hydraulic conductivity, K. The
hydraulic conductivity is the proportionality constant in Darcy’s law, (κ/μ) × specific weight, and is
expressed in units of meters/second. It is defined as the volume of fluid flowing through a specified
cross-sectional area under the influence of a unit hydraulic gradient. Examples of values for the
hydraulic conductivity are given in Table 4.2 for various types of rocks.
The hydraulic conductivity, as with the permeability, can vary with direction, the scale over
which it is considered and obviously with the type of rock. For these reasons, laboratory measure-
ments that derive a value for the hydraulic conductivity may be problematic when applied to a field
site. Caution is warranted when using such measurements, and it is for this reason that it is common
to have several measurements done on a suite of samples from a potential subsurface reservoir if a
region is of particular interest for a geothermal application.

FracTUre porosITy and permeabIlITy
Many geological materials possess a population of cracks or fractures. The mechanical processes
that cause fracturing are many: tectonic forces associated with movement along faults, slow uplift
or burial that warps rocks, and cooling or heating, to name a few (see sidebar for discussion of the
relationship between stress and fracture properties). The properties of the resulting fractures reflect