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# Geomechanics of Compressibility: Part 3-Different Compressibility Coefficients & Their Applications

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: