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# Rock acoustic velocities and in-situ stress

Rock at depth is subjected to stresses resulting from the weight of the overlying strata and from locked in stresses of tectonic origin. When an opening is excavated in this rock (such as a wellbore), the stress field is locally disrupted and a new set of stresses are induced in the rock surrounding the opening. This page describes the effect of in-situ stresses on acoustic rock velocities.

## In-situ stresses

The in-situ "lithostatic" stresses are usually unequal. Such different stresses are required or faults, folds, and other structural features would never be developed. In contrast, most laboratory data are collected under equal stress or "hydrostatic" conditions. Differential or triaxial measurements are comparatively rare (e.g., Gregory, Nur and Simmons, Yin, and Scott et al.).

In a simple compacting basin with neither lateral deformation nor tectonic stresses, the vertical stress will be largest. Lateral stresses will be developed in a basin as sediments are buried and compacted but are constrained horizontally. Both uniform hydrostatic and unequal lithostatic stress conditions are shown in Fig. 1.

## Axial stress

A simple estimate of the horizontal stress, σh, can be made from the axial stress, σv, by ....................(1)

where ν is Poisson’s ratio. Calculated stresses typical for sands (ν = 0.1) and more clay-rich rocks (ν = 0.25) are also shown in Fig. 1. This basic relation (Eq. 1) is an oversimplification of actual conditions, but it does provide a useful conceptual model, and lateral stresses indeed are found to be lower in sandstones than in shaly sections in most places.

## Axial and lateral stress

From a matrix of velocities measured over axial and lateral stress conditions, velocity surfaces could be calculated for a given rock sample. Data such as those shown in Fig. 2 were fitted to a form based on that of Eq. 2 to develop Eq. 3: ....................(2)

where f is approximately linear. ....................(3)

where σe is the effective stress. Fits are usually very good even for consolidated rocks with regression factors of around 0.98.

Velocities can vary substantially over the stress field shown in Fig. 1, not only among samples but also between compressional and shear waves. Fig. 3 shows the Vp and Vs surfaces for Woodbine sandstone. Figures such as 3 demonstrate that the Vp, Vs, and Vp/Vs ratio will all be strongly dependent on the exact stress tensor at depth. Laboratory measurements under hydrostatic conditions are at best a first-order approximation.

## Nomenclature

 Vp = compressional velocity, m/s Vs = shear velocity, m/s σ = normal stress, GPa or MPa