Page 84 - Reservoir Geomechanics

68 Reservoir geomechanics

for very small contact areas. It is clear in Figure 3.5b that pore fluid pressure does not
affect shear stress components, S ij .
Empirical data have shown that the effective stress law is a useful approximation
which works well for a number of rock properties (such as intact rock strength and the
frictional strength of faults as described in Chapter 4), but for other rock properties,
the law needs modification. For example, Nur and Byerlee (1971) proposed an “exact”
effective stress law, which works well for volumetric strain. In their formulation

σ ij = S ij − δ ij αP p (3.10)
where α is the Biot parameter

α = 1 − K b /K g
and K b is drained bulk modulus of the rock or aggregate and K g is the bulk modulus of
the rock’s individual solid grains. It is obvious that 0 ≤ α ≤ 1. For a nearly solid rock
with no interconnected pores (such as quartzite),

lim α = 0
φ→0
such that pore pressure has no influence on rock behavior. Conversely, for a highly
porous, compliant formation (such as uncemented sands)

lim α = 1
φ→0
and pore pressure has maximum influence. Figure 3.5c shows measured values of the
Biot parameter for two materials: a compliant unconsolidated sand in which α is high
and a dry, well-cemented sandstone in which α has intermediate values (J. Dvorkin,
written communication). In both cases, α decreases moderately with confining pressure.
Hofmann (2006) has recently compiled values for α for a wide range of rocks.
Thus, to consider the effect of pore fluids on stress we can re-write equation (3.3)as
follows

(3.11)
S ij = λδ ij ε 00 + 2Gε ij − αδ ij P 0
such that the last term incorporates pore pressure effects.
The relation of compressive and tensile rock strength to effective stress will be dis-
cussed briefly in Chapter 4.With respect to fluid transport, both Zoback and Byerlee
(1975) and Walls and Nur (1979)have shown that permeabilities of sandstones con-
taining clay minerals are more sensitive to pore pressure than confining pressure. This
results in an effective stress law for permeability in which another empirical constant
replaces that in equation (3.10). This constant is generally ≥1 for sandstones (Zoback
and Byerlee 1975) and appears to depend on clay content (Walls and Nur 1979). More
recently, Kwon, Kronenberg et al.(2001)have shown that this effect breaks down in
shales with extremely high clay content. For such situations, permeability seems to