Page 276 - Geothermal Energy Systems Exploration, Development, and Utilization

252 5 Geothermal Reservoir Simulation
stress definition

s
σ = σ − α b p l
with the Biot coefficient, then it can be expressed as a function of this effective
pressure,
k = f (σ)

Several experiments determined the effective pressure coefficient. (Morrow et al.,
1986) and (Bernabe, 1987) found that a value of 1 was the limiting value for
crystalline rocks; (Zoback and Byerlee, 1975) and (Al-Wardy and Zimmerman,
2003) found that the effective stress coefficient depends on the clay content.
Often the effective pressure coefficient is taken to be constant, yielding a linear
expression for effective pressure, but (Kwon et al., 2001) mentioned that it may
itself be a function of effective stress or microstructural changes in pore structure.
It has to be taken into account that the pore geometry parameters cannot be directly
measured under pressure. Therefore, the permeability has to be correlated to other
rock properties such as porosity or formation factor (Bernabe, 1988), which can be
measured under pressure.
The dependency of permeability on pore structure was investigated by various
studies: A general overview can be found in Bear (1972); hydraulic radius models
(Kozeny, 1927; Carman, 1937) (for soils and porous rocks); geometric parameters
determined by a fractal approach (Pape et al., 2000); and, mercury injection (Katz
and Thompson, 1987). For the hydraulic radius model the equation

nD 2
k = h
6
is used with porosity and hydraulic radius as parameters. For the second perme-
ability calculation the equation
2
k = 155n + 37315n + 630(10n) 10
can be used for Rotliegend sandstones of northeast Germany and was calibrated at
several core sample measurements. By means of measured porosity changes due
to effective pressure changes, and also a more general definition of permeability
(Bear, 1972) can be used (Figure 5.3):
k = f (σ)f (n)L 2 (5.12)

(see ‘‘Nomenclature’’ for parameter definitions). The reference length can be the
hydraulic radius or particle size as well. The often used porosity function
3
f (n) = n /(1 − n) 2 (5.13)
(Bear, 1972) can be directly calculated by means of the laboratory results. The
geometry term depends on effective pressure itself, but the influence of the
geometry term on permeability is small in comparison to the porosity function.