velocity, gamma ray and density. The coefficient of internal friction was determined using the gamma relation, equation (28) in Table 4.4. Although this interval is comprised of almost 100% shale, the value ofµiobtained using equation (28) ranges between 0.7 and 0.84. Using the velocity data, the UCS was determined using equations (11) and (12) (Table 4.2, Figure 4.17a, b). While the overall shape of the two strength logs is approximately the same (as both are derived from the Vpdata), the mean vertically aver- aged strength derived using equations (11) is 1484±233 psi (Figure 4.18a) whereas that derived with equations (12) has a strength of 1053±182 psi (Figure 4.18b). Poros- ity was derived from the density log assuming a matrix density of 2.65 g/cm3and a fluid density of 1.1 g/cm3. The porosity-derived UCS shown in Figure 4.17c with equation (18) indicates an overall strength of 1878±191 psi (Figure 4.18c). It is noteworthy in this single example that there is an almost factor of 2 variation in mean strength.
However, as equation (12) was derived for the Gulf of Mexico region, it is probably more representative of actual rock strength at depth as it is was derived for formations of that particular region.
0 10 20 30 40 50 60 70 0
5 10 15 20 25 30 35 40
p (MPa)
q (MPa)
30%
34%
37%
39%
M
1 2
3
1 H Y D RO S TAT I C T E S T 2 T R I A X I A L C O M P R E S S I O N 3 T R I A X I A L E X T E N S I O N
E N D C A P C O M PAC T I O N SHEAR FA
ILURE
Figure 4.19. The Cam–Clay model of rock deformation is presented in p–q space as modified by Chan and Zoback (2002) following Desai and Siriwardane (1984) which allows one to define how inelastic porosity loss accompanies deformation. The contours defined by different porosities are sometimes called end-caps. Loading paths consistent with hydrostatic compression, triaxial compression and triaxial extension tests are shown.C 2002 Society Petroleum Engineers
J1and J2D are the first and the second invariant of the deviatoric stress tensor respec- tively. The equation of the yield loci shown in Figure 4.19 for the Cam-Clay model is given by Desai and Siriwardane (1984) as:
M2p2−M2p0p+q2 =0 (4.37)
where M is known as the critical state line and can be expressed as M=q/p.
The Cam-Clay model in p–q space is illustrated in Figure 4.19 from Chan and Zoback (2002). Note that the shape of the yield surface as described by equation (4.37) in the Cam-Clay model is elliptical. If the in situ stress state in the reservoir is within the domain bounded by the failure envelope in p–q space, the formation is not likely to undergo plastic deformation. The intersection of the yielding locus and the p-axis is defined as p0 (also known as the preconsolidation pressure) and each end-cap has its own unique p0 that defines the hardening behavior of the rock sample. The value of p0 can be determined easily from a series of hydrostatic compression tests in which porosity is measured as a function of confining pressure. Conceptually, it is easy to see why the end-caps should be roughly elliptical. Because shear stress facilitates the process of compaction and porosity loss, the mean confining pressure at which a certain end-cap is reached will decrease as shear stress increases.
0 100 200 300 400 0
300
200
100
((Sh + SH + Sv) /3) − Pp (MPa) Sv − Sh (MPa)
2 0 % 2 3 %
3 5 %
1 5 %
2 1 % 2 1 %
Figure 4.20. Compilation rock strength data for a wide variety of sandstones (different symbols) define the overall trend of irreversible porosity loss and confirms the general curvature of the end-caps to be similar to that predicted by the Cam-Clay model. After Schutjens, Hanssen et al.
(2001).C 2001 Society Petroleum Engineers
Three different loading paths are shown in Figure 4.19. Path 1 corresponds to hydro- static loading to 26 MPa (q = 0), Path 2 corresponds to a triaxial compression test (q/p=3) after loading to an initially hydrostatic pressure of 14 MPa, and Path 3 cor- responds to triaxial extension (q/p=3/2) after hydrostatic loading to 18 MPa.
In weak formations such as weakly cemented sand, porous chalk or diatomite, once loading reaches an end-cap, compaction and grain rearrangement (and eventually grain crushing and pore collapse) will be the dominant deformation modes. If the loading path reaches the shear failure line, M, slip on a pre-existing fault will occur.
An example of end-cap deformation is illustrated in Figure 4.20 for a compilation of lab tests on a wide variety of sandstones (Schutjens, Hanssen et al. 2001). The contour lines show the end-caps, which demonstrate how porosity is irreversibly lost at shear stresses less than that required to cause shear failure (see Desai and Siriwardane 1984 and Wood 1990). Note that even in the absence of shear stress (i.e. moving just along the abscissa) porosity would be irreversibly lost as the mean stress increases from initial porosities greater than 35% to as low as 21% at p∼360 MPa. With increased q, the contours that define the end-caps curve back toward the ordinate because the confining pressure required to cause a given reduction in porosity decreases a moderate amount.
In weak formations, such as chalks, grain crushing can occur at much lower pressures than those for sandstones (Teufel, Rhett et al. 1991).
Because reservoir compaction associated with depletion is an important process in many reservoirs, inelastic compaction is discussed in detail in Chapter 12. In addition to porosity loss, there can be substantial permeability loss in compacting reservoirs as well as the possibility of surface subsidence and production-induced faulting in normal faulting environments. The degree to which these processes are manifest depends on the properties of the reservoir (compaction will be an important factor in weak formations such as chalks and highly compressible uncemented sands), the depth and thickness of the reservoir, the initial stress state and pore pressure and the reservoir stress path, or change in horizontal stress with depletion (as described in Chapter 3). Wong, David et al.
(1997) demonstrated that the onset of grain crushing and pore collapse in sand reservoirs depends roughly on the product of the porosity times the grain radius. However, in uncemented or poorly cemented sand reservoirs, there will also be inelastic compaction due to grain rearrangement, which can be appreciable (Chapter 12).