Read the first part of this series here.

Read the second part of this series here.

As discussed, the common concept of compressibility in geomechanics has been developed to study the changes in either of bulk volume (Vb) or pore volume (Vp) of rocks in response to variation in confining pressure (σc) or pore pressure (p). Also, I explained the concept of coupling between pore pressure and confining pressure and the fact that in drained conditions, effects of these two parameters on volume changes are uncoupled from each other though this assumption is not valid for most of problems in reservoir geomechanics.

However, assuming an uncoupled condition, it is possible to define four different types of compressibility coefficients to relate the two mentioned pressures to the two named volumes (Zimmerman, 1991) as listed below. In these definitions, one of pore pressure or confining pressure is assumed to remain unchanged while the other varies.

1- Bulk volume compressibility Coefficient (Cbc):

Figure 1. A plate tectonic model that uses compressibility coefficient to calculate variation in density induced by change in confining pressure (Nikolaeva et al., 2011)

Bulk volume compressibility coefficient (Cbc) equals to the change in the bulk volume of rock (Vb) with respect to the variation in the confining pressure (σc) while the pore pressure (p) is held unchanged:

`Cbc= (-1/Vb) (∂Vb/∂σc ) `

where p=constant.

Cbc is usually used in large-scale tectonic modeling and also in wave propagation analysis. In tectonic modeling, this parameter is implemented to account for the dependency of rock compressibility (usually in high temperatures) to tectonic forces. In the case of wave propagation problems, wave velocities are closely dependent on the rock’s matrix compressibility (though it is usually stated in terms of other elastic parameters such as bulk modulus).

Probably a major importance of Cbc is the fact that it is analogous to the compressibility of non-porous media and so it can be compared to the compressibility of different solids and fluids.

Figure 1. A plate tectonic model that uses compressibility coefficient to calculate variation in density induced by change in confining pressure (Nikolaeva et al., 2011)

2- Pseudo-bulk compressibility Coefficient (Cbp):

Figure 2. Evidence of ground surface subsidence around a well in Baytown in the Harris-Galveston District, Texas (source: www.aquaheroes.pbworks.com )

This type of bulk volume compressibility coefficient (Cbp), also called ‘pseudo-bulk compressibility coefficient’ quantifies the change in bulk volume of the rock (Vb) with respect to variation in the pore pressure (p) while the confining pressure (σc) is held unchanged:

`Cbp=(-1/Vb) (∂Vb/∂p) `

where σc=constant.

Cbp is useful for heave/subsidence calculations induced by pore pressure change during production or injection. Several cases of such deformations have been documented in the histories of underground water extraction and hydrocarbon production. Some of the famous examples are San Joaquin Valley in California with 9m of subsidence between 1935 and 1977, Wilmington oil field in Long Beach ,California with 8.8m of subsidence between 1932 and 1965, Ekofisk oil field in North Sea with 8.5m of subsidence between mid 1970s and 2004, Wairakei geothermal field in the News Zealand with 14m of subsidence between 1950 and 1997, and Maracaibo Lake in Venezuela with 7m of subsidence between 1926 and 2004.

Figure 2. Evidence of ground surface subsidence around a well in Baytown in the Harris-Galveston District, Texas (source: www.aquaheroes.pbworks.com )

3- Formation compaction Coefficient (Cpc)

Figure 3. When it comes to settlement of buildings, there is nothing more famous than the leaning tower of Pisa with its uneven settlement. (source: Wikipedia)

This pore volume compressibility coefficient (Cpc) which is also called ‘formation compaction coefficient’ equals to the change in pore volume of the rock (Vp) with respect to the variation in the confining pressure (σc) while pore pressure (p) is held unchanged:

`Cpc=(-1/Vp) (∂Vp/∂σc) `

where p=constant.

Cpc is used in subsidence (settlement) calculations induced by external loadings such as construction at the ground surface. Foundation settlement is an inevitable consequence of construction that needs to be controlled by geotechnical engineers. Almost all of us are familiar with the consequences of large and especially uneven settlement of foundations that can lead to ranges of effects from trivial to devastating on buildings and infrastructures. Probably the most famous case of foundation settlement is the leaning tower of Pisa that has made it an attraction for the tourists but there are several other famous examples around the world.

Figure 3. When it comes to settlement of buildings, there is nothing more famous than the leaning tower of Pisa with its uneven settlement. (source: Wikipedia)

4- Effective pore compressibility Coefficient (Cpp)

Figure 4. Results of fluid flow simulations using stream line method comparing cumulative oil production with and without compressibility effects (Source: Osako and Datta-Gupta, 2007).

Pore volume compressibility (Cpp), also called ‘effective pore compressibility’, equals to the variation in pore volume of the rock (Vp) with respect to the change in the pore pressure (p) while the confining pressure (σc) is unchanged:

`Cpp=(-1/Vp) (∂Vp/∂p) `

where σc=constant

Cpp is frequently used in modeling of fluid flow in reservoirs and aquifers. Almost all the fluid flow simulations can take this effect in consideration. This parameter can become a critical parameter in less consolidated rocks where compaction acts as an important drive mechanism for hydrocarbon production.

Figure 4. Results of fluid flow simulations using stream line method comparing cumulative oil production with and without compressibility effects (Source: Osako and Datta-Gupta, 2007).

5. Uniaxial Compressibility Coefficient (Cbu)

In petroleum geomechanics, it is common to assume reservoir’s deformations during production and injection to be uniaxial and in vertical direction. This assumption is not far from reality in many deep reservoirs that have a relatively small thickness compared to their lateral extension. In these cases, total vertical stress (which equals to the weight of overburden) does not change significantly as a result of pressure change.

In a uniaxial compressibility test, uniaxial pore volume compressibility (Cbu) is defined for a condition that the sample is not allowed to have lateral deformations during the test.

`Cbu=(-1/H) ∂H/∂p`

where lateral strain=0 and H in this equation is the sample’s height .

When Rock Behaves Elastically …

It is no secret that assuming elastic behaviour for rocksis not totally credible specially for large pressure changes and also in unconsolidated rocks. Nevertheless, it has been very common in the industry to assume an elastic behaviour for rocks due to its simplicity and also availability of elastic data from different sources (field tests, logs, seismic). None of these reasons, however, could give a green light to use such simplifications in rock behaviour without enough due diligence.

In cases where the rocks behave elastically, bulk compressibility of rocks (Cbc) is simply the inverse of its elastic bulk modulus (K). For an isotropic rock, this can be written as:

`Cbc =1/K=E/[3(1-2v)]`

where E and v are Young’s modulus and Poisson’s ratio of the rock, respectively.

Similarly, uniaxial bulk compressibility of rocks (Cbu) is the inverse of constrained elastic modulus (also called P-wave modulus) of rocks (M):

`Cbu =1/M=E(1-v)/[(1+v)(1-2v)]`

Relations Between Different Compressibility Coefficients

The following equations have been simply (and wrongly) suggested based on the relation between different volumetric components of rocks:

```
Cbp =φCpp+(1-φ)Cm
Cbc =φCpc+(1-φ)Cm
```

where φ is rock porosity and Cm is the compressibility of rock matrix (or grains in granular rock). As Zimmerman (1991) discussed, these equations have no theoretical basis and physical support.

Zimmerman (1991) showed that, assuming the validity of elastic behaviour, the following equations are also valid between compressibility coefficients measured at different applied pressure conditions:

```
Cbp = Cbc-Cm
Cpp = Cpc-Cm
Cpc =Cbp / φ=(Cbc-Cm)/φ = [Cbc-(1+φ)Cm]/φ
```

Once more, note that the oversimplification of mechanical rock behaviour and its compressibility using elastic parameters may lead to conclusions that are far from reality.