Failure Models

Geological observations (e.g. Chapter 2) in sedimentary basins and laboratory measurements (e.g. Ay ling, 1992) show that the assumption that the rocks behave elastically under crustal conditions is rarely true. The observation that most rocks contain faults and fractures on a variety of scales, clearly shows that they have

experienced stress states which lie outside their elastic lim its and leads to the suggestion that rock stress magnitudes are governed by the shear strength o f the rocks. In a normal faulting regime, where G\ is the vertical stress, the magnitude o f Gh w ill be limited by how much differential stress the rock can withstand i.e. its shear strength. I f the rock is assumed to be in a state o f incipient failure and the Mohr-Coulomb failure criterion is used, Ch is given (Perkins, 1976; Katahara, 1996) by :

Where (}) is the internal friction angle and Cu is the cohesive strength o f the rock. Internal friction angles and cohesive strengths can be determined experimentally or can be

obtained from empirical relationships w ith other more easily measured rock properties such as porosity and clay content (e.g. Plumb, 1994).

The Mohr-Coulomb criterion does not account for curved failure envelopes (i.e. the failure envelope described by equation 3.22 plots as a straight line on a Mohr-Coulomb diagram) and also predicts that failure, and therefore minimum stress magnitude, is independent o f the intermediate stress. Katahara (1996) points out some lim itations o f this model by stating that in fact the intermediate stress can have a role in failure and also that real failure envelopes are curved.

I f the rock has cracks o f a favourable orientation (it is assumed that the pre-existing cracks have no cohesive strength) then the failure criterion in a normal fault stress regime becomes:

The states o f stress described by these equations can be represented on a M ohr diagram (Figure 3.13) where the normal stresses are effective stresses (o ') and the shear stresses

(t) are those acting on the plane whose normal makes an angle p w ith G i. Because there

is no cohesion across the failure surface, the effective stress law is the simple Terzaghi law (Thiercelin and Plumb, 1991):

c ' = a - p

Failure models predict the magnitude o f the minimum principal stress (03) from a knowledge o f the magnitude o f the maximum principal stress (Gi) or vice versa. Generally in sedimentary basins, the only stress magnitude which can be easily

determined without direct stress measurement is the vertical stress. Therefore, when the friction angles and pore pressure are known, in normal faulting regimes, Gh can be estimated from a knowledge o f the vertical stress. In reverse faulting regimes, Gh can be

determined from a knowledge o f the vertical stress. In strike slip regimes, where Qi and Ü3 are both horizontal stresses, the failure model is o f limited use without further assumptions, or some stress measurements.

r \

GO

GO

Cu

Normal Stress On

Figure 3.13 Mohr-Coulomb failure envelopes for the case o f a rock with no cohesion (solid straight line) and the case where the rock has a cohesion, Cu, (dashed straight line). The states of stress (for a normal faulting stress regime) corresponding to failure o f the above rocks are also shown.

The main restriction o f this model is that, in order to apply it confidently, some

independent observations about the present stress regime are needed. Such observations could be from stress measurements or from active faulting. Stress measurements would generally be restricted to a small area and a small depth range, so application o f the failure model would have to assume that the stress regime is the same in the area in which the measurements were taken and the area in which the model is to be applied. Observations o f active faulting tend to be made over a wider area, although focal mechanisms are generally from depths below sedimentary basins. Inferring a particular stress regime at one depth from observations at another may be risky. One reason why this is so is because horizontal and vertical effective stresses change by different amounts

during pore pressure change (e.g. Teufel et al., 1991) thus pore pressure changes can easily change the stress regime.

The Mohr-Coulomb failure criterion has been evaluated using data from hydraulic fracturing stress measurements and laboratory determined rock properties by Thiercelin and Plumb (1991), using the same rocks and stress measurements as the evaluation o f the elastic models (3.4.1). The data for the evaluation come from east Texas where recent fault movements have occurred and the formations show extensional and shear fractures indicating a normal faulting regime. Thiercelin and Plumb (1991) concluded that the failure model gives good predictions o f Gh, and also predicts the stress contrast between sandstone and shale (with shales showing higher values o f stress). The best predictions are made by using internal friction angles obtained by measuring peak strength values under non zero confining pressure as opposed to zero confining pressure.

In the study made by Thiercelin and Plumb, 1991, both the uniaxial strain model (particularly incorporating transverse isotropy) and the Mohr-Coulomb failure model both give reasonably good predictions o f the stress state as measured by hydraulic fracturing. This has been found in a number o f other studies which have been summarised by Katahara (1996) who suggests that the reason for this agreement between two fundamentally different models is the empirical equivalence o f sincj) and 1- 2v. Taking typical values for these two parameters: v = 0.25, and (|) = 30°, both sin(|) and l-2 v give a value o f 0.5 and so would both predict the same horizontal stress magnitude for a given vertical stress.

Lithologie stress contrasts, w ith higher values o f Gh in shales than sands, are also predicted by both models. This means that shales are weaker and more deformable than sands in that they can support less shear stress (lower (|)) and expand laterally ( if

unconfmed) more than sands in response to a given vertical stress and thus have a higher

A particular problem with the failure model, and the reason perhaps that it is not widely used in the oil industry, is a lack o f knowledge o f the friction angles and how to predict them from logs,