Isothermal compressibility definition

Reservoir fluids (oil, water, gas) and the rock matrix are contained under high temperatures and pressures they are compressed relative to their densities at standard temperature and pressure. Any reduction in pressure on the fluids or rock will result in an increase in the volume, according to the definition of compressibility. As discussed in Section 5.2, isothermal conditions are assumed in the reservoir. Isothermal compressibility is defined as ... 

At pressures above the bubble point, the coefficient of isothermal compressibility of oil is defined exactly as the coefficient of isothermal compressibility of a gas. At pressures below the bubble point an additional term must be added to the definition to account for the volume of gas which evolves. 

The definition of the coefficient of isothermal compressibility at pressures above the bubble point is... 

The normal procedure for estimating formation volume factor at pressures above the bubble point is first to estimate the factor at bubble-point pressure and reservoir temperature using one of the methods just described. Then, adjust the factor to higher pressure through the use of the coefficient of isothermal compressibility. The equation used for this adjustment follows directly from the definition of the compressibility coefficient at pressures above the bubble point. 

In order to introduce basic equations and quantities, a preliminary survey is made in Section II of the statistical mechanics foundations of the structural theories of fluids. In particular, the definitions of the structural functions and their relationships with thermodynamic quantities, as the internal energy, the pressure, and the isothermal compressibility, are briefly recalled together with the exact equations that relate them to the interparticular potential. We take advantage of the survey of these quantities to introduce what is a natural constraint, namely, the thermodynamic consistency. 

Preliminary to such a search we examine several thermodynamic properties of fluids at or close to criticality, that clearly show why and how fluctuations dominate under such conditions, (i) Consider first the isothermal compressibility, kj = —(dV/dP)T/V. At the critical point the isotherm dP/dV)r has zero slope thus, Ki grows indefinitely as T —> Tc. (ii) Using Eq. (1.3.13) and the definition for K one finds that (dV/dT)p = -(dV/dP)TidPldT)v = KiV dP/dT)y, wherein (dP/dT)v does not vanish. Therefore, the coefficient of thermal expansion, = i /V) dV/BT)p also grows without limit as the critical point is approached, (iii) According to the Clausius-Clapeyron equation in the form AH = T(Vg — Vi)(dP/dT), the heat of vaporization of the fluid near the critical point becomes very small, since Vg — Vi 0, whereas dP/dT remains finite. 

The pressure, being an intensive variable, can only be a function of intensive variables so that holding V constant in the partial derivative is of no importance. From the definition of the isothermal compressibility xt s p-1(dp/dP)T it follows that (dp/dN)T,v = 1 IVp2xt)- Also from the equation dE = TdS — pdV + pdN, it follows that... 

Obviously, there is a relationship between the heat capacity and something like a susceptibility, and are the isothermal compressibility and the isentropic compressibility. The common definition of compressibility and electric and magnetic susceptibility suggests that susceptibility should be defined rather as... 

Both compressibilities are intensive measurable state functions, though K7 is proportional to a class I derivative, while Kg is proportional to one of class 111. Because volume decreases with increasing pressure, these definitions contain negative signs to make the compressibilities positive. Besides PvT experiments. Kg can also be obtained from measurements of the speed of sound. The reciprocal isothermal compressibility is called the hulk modulus. 

Based on the Maxwell relations, the definitions of the isothermal compressibility (Kj or jSr), the isoentropic compressibility (Ks or jSr), and the thermal expansion coefficient (a) can be written with the In V removed as, respectively. 

Table 2.4 lists the values of ten state functions of an aqueous sucrose solution in a particular state. The first four properties (T, p, ha, b) are ones that we can vary independently, and their values suffice to define the state for most purposes. Experimental measurements will convince us that, whenever these four properties have these particular values, each of the other properties has the one definite value listed—we cannot alter any of the other properties without changing one or more of the first four variables. Thus we can take T, p, ha, and B as the independent variables, and the six other properties as dependent variables. The other properties include one (F) that is determined by an equation of state three (m, p, and Xb) that can be calculated from the independent variables and the equation of state a solution property (77) treated by thermodynamics (Sec. 12.4.4) and an optical property ( d)- In addition to these six independent variables, this system has innumerable others energy, isothermal compressibility, heat capacity at constant pressure, and so on. 

Table 8.1 Definitions and values of the major critical exponents. The quantities Kt and Cy are the isothermal compressibility and constant volume specific heat capacity respectively, piiq and Pvap are the densities of the coexisting liquid (liq) and vapour (vap) phases, T the temperature, and and the critical temperature and pressure respectively.

Here, Ag is the wavelength of light in vacuum, n is the refractive index, is the Boltzmann constant, T is the absolute temperature, 0 is the isothermal compressibility, N is the number of scattering units per unit volume, and 5 is the mean square of the anisotropic parameter of polarizabihty per scattering unit. Finally, from the definition of the turbidity t in Equation 2.12, the fight scattering loss (dB/km) is related to the turbidity t (cm ) by... 

This procedure is conceptually straightforward, as one utilizes the very definition of the isothermal compressibility and its connection to the number fluctuations. Furthermore, Eqs. (141) and (142) are very advantageous when studying quantum hard-sphere fluids, since the error bars of the pressure estimates are far more controllable than when using the virial pressure involving Fierz s term [96]. By extension, the same is expected to happen when studying quantum fluids in which very strong repulsions between the particles play a dominant role. 

The partial differentials in the above relation may be recast as follows " as wiU be established in Section 1.12 by independent arguments, one may set dS/dT)y = Cy/T, and (dS/dV)j = dP/dT)y , where Cy is the heat capacity at constant volume and composition, and where the appropriate Maxwell relation has been introduced as the second relation. Next, set up the identity dP/dT)y = — dV/dT)p ,/ dV/dP)j. also, replace the numerator and denominator by the definitions —aV and — V, where a and are the isobaric coefficient of expansion and the isothermal compressibility, respectively. Last, set dS/dni)j y = 5, as defining the differential entropy at constant temperature, volume, and composition of species i. In this revised notation, Eq. (1.10.3c) assumes the form... 

Preliminary to such a search we examine several thermodynamic properties of fluids at or close to criticality that clearly show why and how fluctuations dominate under such conditions. (1) Consider first the isothermal compressibility, k/= —(dV/dP) / At the critical point the isotherm (dP/dV)j-has zero slope thus, kj grows indefinitely as T —> (2) Using Eq. (1.3.8) and the definition for x/ one... 

Using arguments similar to the proof of Eq. (C.7), and the definitions of the thermal expansion coefficient a, isothermal compressibility kj- and adiabatic compressibility ks, prove the relations given in Eqs. (C.8)-(C.10). 

With this definition, the isothermal compressibility has a positive value corresponding to volume decreasing with increasing pressure. 

Thus, from an investigation of the compressibility of a gas we can deduce the values of its critical constants. We observe that, according to van der Waals theory, liquid and gas are really two distant states on the same isotherm, and having therefore the same characteristic equation. Another theory supposes that each state has its own characteristic equation, with definite constants, which however vary with the temperature, so that both equations continuously coalesce at the critical point. The correlation of the liquid and gaseous states effected by van der Waals theory is, however, rightly regarded as one of the greatest achievements of molecular theory. 

Again, if we consider the initial substances in the state of liquids or solids, these will have a definite vapour pressure, and the free energy changes, i.e., the maximum work of an isothermal reaction between the condensed forms, may be calculated by supposing the requisite amounts drawn off in the form of saturated vapours, these expanded or compressed to the concentrations in the equilibrium box, passed into the latter, and the products then abstracted from the box, expanded to the concentrations of the saturated vapours, and finally condensed on the solids or liquids. Since the changes of volume of the condensed phases are negligibly small, the maximum work is again ... 

---