Geomechanics of Compressibility – Part II: Drained versus Undrained
Read the first part of this series here.
The Four Main Parameters: The common concept of compressibility in geomechanics has been developed to study the variations in either of bulk volume or pore volume of rocks in response to variation in confining pressure or pore pressure. Let’s talk a bit about these four elements before going forward.
The Two Volumes: The two types of volumes used in different compressibility definitions are bulk volume (Vb) and pore volume (Vp). We usually like to know variations in Vb as it shows how our operations can affect ground deformation and we are interested in variation of Vp as it shows how porosity, a crucial parameter for fluid flow analysis, changes with underground operations. Apparently, these two volumes are related and by knowing rock’s porosity, they can be easily translated to each other.
The Two Pressures: Rock’s volume tends to change if either of confining pressure (σc) or pore pressure (p) varies. Traditionally, different types of compressibility coefficients used in the industry assume that pore pressure and confining pressure can change independently (this type of rock response is called drained). At first glance, it seems that rock’s deformation is only caused by pore pressure variation during injection or production but, in reality, in-situ stresses (which in essence are the pressures confining the rock) almost always change along with pore pressure variation (see Figure 1). So, the volume change is commonly the result of changes in both pore pressure and confining pressure simultaneously (this type of rock’s response is called undrained). Due to its importance, for more clarification, let’s pause here and talk more about the difference between the drained and undrained behaviours.
Figure 1. A simple schematic showing how total stresses within and around a reservoir change by pore pressure decrease (i.e., production). Each pair of arrows pointing towards each other demonstrates a compressive state of stress change that squeezes the element in the direction of arrows while the ones pointing away from each other denote an extensile state of stress change that is trying to pull the element apart in the direction of arrows.
Drained versus Undrained Behaviour
A Geotechnical Classic: Let’s start with a classic example from the field of geotechnical engineering (Figure 2) that models the ground behaviour by using a cylinder and piston system. The fluid under the piston stands for the pore fluid and the spring mimics the soil/rock’s matrix behaviour. The fluid flow through valve represents the hydraulic conductivity (which increases with permeability of the soil/rock). While the building in Figrue 2 is constructed at the ground surface, if the draining valve is closed and there is no pathway for the fluid to escape, the applied weight of the building will be taken in parts by both of the rock’s skeleton (as effective stress) and the pore fluid (as pressure), simultaneously. This is called undrained behaviour. Now, if the fluid can escape because the draining valve is open, the excessive pore pressure disappears almost instantly and we can assume pore pressure change is negligible and, therefore, all the load is taken by the skeleton. In his case, the effective load taken by the skeleton equals to the total load applied to the rock. This is called drained behaviour. Of course, it is possible to have other conditions between these two extremes if the valve is half open.
Figure 2. This classic and great schematic tries to show the difference between drained and undrained response of soil/rock to the construction load and also the concept of effective stress as introduced by Terzaghi. In this simple model, the saturated soil/rock system is represented by a container filled with fluid and it is capped by a piston that resembles the ground surface. The load of piston can be carried by either or both of the fluid and the spring (that resembles the soil/rock skeleton). In undrained condition, the valve is closed so there is no escape path for the fluid and, so, it has to bear the load along with the skeleton. As soon as the fluid can escape through the valve, it will not take any of the construction load and all the load has to be taken by the skeleton.
A More General Definition: A more inclusive definition f0r the drained and undrained behaviour can be explained based on the concept of dependency or coupling of confining pressure (total stresses) and pore pressure. Based on this definition, if pore pressure and confining pressure are not coupled and can change independently, the rock’s behaviour is called drained while if these two are coupled and can affect each other, the rock’s behaviour is called undrained.
In Reservoir Geomechanics: As in the case of production and injection, pressure variation almost always leads to changes in total stresses as shown in Figure 1, according to the given definition, we will have an undrained behaviour.
Drained/Undrained Conditions and Compressibility Tests: In practice, most of compressibility tests used to measure rocks’ compressibility coefficients are performed by increasing confining hydrostatic pressure (i.e., an external omindirectional stress) on dry/unsaturated samples. This is a form of drained behaviour as pore pressure does not play a role in these tests. Some of the more developed drained tests try to mimic what happens in the field by letting the dry/unsaturated sample deform only in the vertical direction (i.e., uniaxial deformation).
In a more realistic version of compressibility testing, estimated in-situ stresses are initially applied to the rock sample and, while keeping the deformation uniaxial to represent common reservoirs’ behaviour, pore pressure is gradually changed to simulate injection or production. This type of test can be called an undrained compressibility test as stresses change with changing pore pressure.
Read the third part of this series here.