Effective Stress: Pore Fluid Pressure Ruins Everything

Did you know that porous materials mechanical behavior is controlled not by the applied total stress, but rather by the so-called effective stress? Not only natural geological materials (soils and rocks), but man-made materials such as concrete, as well as human and animal organic materials (skin and bones) will deform and eventually rupture once the effective stress state moves past their failure limit.

Before we dive into failure behavior description, it is important to understand, and properly apply, the concept of effective stress. Karl von Terzaghi (1925), while studying the mechanical behavior of soils, concluded that an external mechanical load (stress) applied to a soil mass would produce displacements (strain) and the external load would be totally transmitted to the skeleton (matrix) of the soil. It is very intuitive to understand that under dry conditions, the pore space of any material is filled with air and no pressure other than the ambient is present. However, under saturated conditions, where the pores are filled with a fluid (e.g., water), a pressure within the pore space will be generated as a function of the fluid column, the fluid density and the gravitational acceleration. In this scenario, Terzaghi’s effective stress principle states that any external (total) stress applied to a porous material will be distributed between the solid skeleton and the pore fluid. In other words, all quantifiable changes in volume-changing stress results from a change in the so-called effective stress. In mathematical terms, the effective stress principle is expressed as:

where σ’ is effective stress, σ is total stress, and Pp the pore fluid pressure. As a rock mechanics convention, compressive stresses are considered positive. The negative sign is there because the pore pressure serves to lessen the volume-changing stress, i.e., the presence of a fluid in the pores bears part of the total stress, so partially unloading the solid matrix from normal stresses.

The understanding and mastering of this concept are so critical that a great effort is typically put by geotechnical, geological, mining and rock mechanics engineers to properly determine pore fluid pressure values. Such values are used to assess the stress state of geological materials and establish their failure conditions.

For quite a while, the concept of effective stress was broadly applied to soil mechanics but did not receive much attention from the rock mechanics community, as rocks are much more competent than soils. It was not until the 60s, after a couple of catastrophic rock failure events happened, that it became evident that the concept of effective stress was as important to rocks as it was to soils. In 1959, the foundation of the Malpasset concrete arch dam in France collapsed as result of sliding of a block wedge (Figure 1) along pre-existing clay-filled discontinuities. ?The failure happened during the first filling of the reservoir after a heavy rainfall. Although there are some other theories on the contributing factors, it becomes intuitive to associate a raise in pore fluid pressure along the discontinuities (as the dam was filled) with a decrease in the effective normal stress on the discontinuities, resulting in shear failure (slide). The unexpected collapse of the Malpasset dam left behind a considerable loss of life.

The failure of a hard rock (gneiss) like in this case brought up the importance of pore pressure and effective stress in rock mechanics. Not only in the traditional sense of pores, but also in open structural discontinuities such as joints, faults, and even bedding planes, which are usually present in rocks.

A subsequent and similar case took place in the Vajont dam, Italy, in 1963 (Hoek, 2006). A mega tsunami, caused by a landslide on the slope of the Mount Toc during the initial filling of the reservoir, created a 100 m high wave above the top of the dam wall, flooding the Longarone town and killing about 2500 people. Here, the slopes of Mount Toc consisted of limestones and claystone beds steeply dipping toward the reservoir. When the bedding was covered by the raising reservoir waters, the increase in pore pressure reduced the effective stress and led to the ultimate slope rock failure.

The concept of pore pressure similarly applies to the deep subsurface. Therefore, the effective stress becomes a critical component along the process of building a geomechanical model and understanding rock failure as it has an impact in key risks like wellbore instability, fault reactivation or induced seismicity.? Perhaps, the most immediately related geomechanics component is pore pressure. All pore pressure models use the concept of effective stress to derive results from either seismic velocities or wireline logs. In this process, the first step is always to determine the overburden (total stress), and then using porosity proxies (density, sonic or resistivity logs) to estimate the associated effective stress. The difference between total and effective stresses yields the pore pressure value (Chilingar et al., 2002; Mouchet & Mitchell, 1989).?

We will further explore these concepts and their impact in subsurface geomechanics in future posts.

References

Chilingar, G.V.; Serebryakov, V.A.; Robertson Jr., J.O. (2002): Origin and Prediction of Abnormal Formation Pressures. Developments in Petroleum Science, 50. Elsevier Science B.V. Amsterdam.

Hoek, E. (2006): Practical Rock Engineering. Undergraduate and Graduate Courses Notes. https://static.rocscience.cloud/assets/resources/learning/hoek/Practical-Rock-Engineering-Full-Text.pdf

Martin, P. (2013): Ruines du Barrage de Malpasset avec la coupe à droite du "dièdre" suite à l'arrachement de la roche. https://commons.wikimedia.org/wiki/File:PM.Malpasset.jpg

Mouchet, J.P.; Mitchell, A. (1989): Abnormal Pressures while Drilling. Editions Technip, Paris.

Terzaghi K. (1925): Erdbaumechanik auf BodenphysikalischerGrundlage, Deuticke: Leipzig