Isothermal Compressibility - Top Dog Engineer

Back to: General Petroleum Engineering

In petroleum engineering everything has compressibility including rocks, oil, and water. It’s not just gases anymore. The concept of compressibility shows up in several reservoir engineering applications including 1) material balance and 2) reservoir simulation, therfore you should try to master this concept. In fact compressibility is the main driving force behind the primary production phase of an oil and gas reservoir. The concept of isothermal compressibility can be difficult to understand but I’m hopeful the perspective I present in this discussion will help clear up any doubts or confusion regarding the subject.

In the academic world of petroleum engineering we are often thrown the isothermal compressiblity equation which looks like this:

(1) $\begin{equation*} c= -\frac{1}{V}\frac{\partial V}{\partial P} \vert _T \end{equation*}$

where $c$ is the isothermal compressibility coefficent, $V$ is volume, and $P$ is pressure. There’s no tangible item to associate it with. It’s just left as that. But if go into the mechanical engineering relm, it makes a lot more sense because they actually show you where it comes from. Let’s go back to solid mechanics for my MechE’s out there. Compressibility is directly related to bulk modulus so we will start with this concept first. Bulk’s modulus (B) is determined from volumetric compression of a substance or material and is defined as the resistance of a substance to change volume under a confining pressure load. The higher the bulk modulus value, the harder it is to compress the substance. The figure below illustrates how a material reacts under a confined compression load.

The way bulk’s modulus is related to pressure is very similar to how Young’s modulus (E) and Hooke’s law are related for the tensile loading of a elastic specimen. Consider a rod placed under a tensile load and a fluid placed under a pressure load in a confined space:

Looking at the loading situations above they are completely different, however, it is clear that relations are very similar. Bulk’s modulus ( $B$ ) can be treated like Young’s modulus ( $E$ ), strain ( $\epsilon$ ) can be treated like volumetric strain ( $\epsilon_v$ ), and stress ( $\sigma$ ) can be treated like pressure ( $\triangle P$ ). Further, bulk’s modulus can be extended by Hooke’s law by the following relation:

(2) $\begin{equation*} \triangle P = B\frac{\triangle V}{V_o} \end{equation*}$

where $\triangle V$ is the change in volume of the substance, whether it be compression or expansion of a fluid, and $V_o$ is the initial volume of the substance. In essence, you can think of bulk’s modulus as the 3-dimensional form of Young’s modulus because we are considering loading in three dimensions vs. one. The other cool thing is they all have similar units, which make’s it even easier to remember. The table below summarizes the analogies:

Hooke’s Law Parameter	Bulk Modulus Equivalent
Young’s modulus, $E$	Bulk Modulus, $B$
strain, $\epsilon$	Volumetric strain, $\epsilon_v$
Stress, $\sigma$	Pressure, $\triangle P$

The analogy is cool and everything, but how do we extend it to compressibility?? Well bulk modulus is the reciprocal of compressibility. The relationship between compressibilty and bulk modulus is given by the expression below:

(3) $\begin{equation*} B= \frac{1}{c} \end{equation*}$

where $c$ is the isothermal compressibility coefficient. Substituting equation 3 into the Hooke’s law relationship for bulk’s modulus (Equation 2 and rearranging it, we can develop a definition that describes reservoir depletion. The expression is given by the following:

(4) $\begin{equation*} \triangle V= cV_o \triangle P \end{equation*}$

where $\triangle V$ is the expanded volume and $\triangle P$ is the pressure drop in the reservoir. This is the most fundamental equation when it comes to describing the primary recovery phase of a reservoir. The expanded volume, $\triangle V$ which corresponds to the fluid expansion, manifests itself as production at the surface.

If you compare Equation 1 and Equation 4 they are basically the same minus the partial derivatives and the negative sign. And the only reason Equation 1 is negative is to ensure that the compressibility term is positive i.e. fluids expand when confining pressure decreases.

Although the isothermal compressibility coefficent is not always a constant value, this example provides the fundamental insight into what the compressibility means and how it applies to expanding volumes in the reservoir. If you understand the methodology used to obtain Equation 4, it will go along ways in understanding reservoir engineering applications.

Leave a Reply Cancel reply