The expansion

Figure 5.3 gives an example of a combustion diagram recorded during knocking conditions. This is manifested by intense pressure oscillations which continue during a part of the expansion phase. 

Reservoirs containing low compressibility oil, having small amounts of dissolved gas, will suffer from large pressure drops after only limited production. If the expansion of oil is the only method of supporting the reservoir pressure then abandonment conditions (when the reservoir pressure is no longer sufficient to produce economic quantities of oil to the surface) will be reached after production of probably less than 5% of the oil initially in place. Oil compressibility can be read from correlations. 

The expansion of the reservoir fluids, which is a function of their volume and compressibility, act as a source of drive energy which can act to support primary producf/on from the reservoir. Primary production means using the natural energy stored in the reservoir as a drive mechanism for production. Secondary recovery would imply adding some energy to the reservoir by injecting fluids such as water or gas, to help to support the reservoir pressure as production takes place. 

Figure 8.1 shows how the expansion of fluids occurs in the reservoir to replace the volume of fluids produced to the surface during production. 

Solution gas drive occurs in a reservoir which contains no initial gas cap or underlying active aquifer to support the pressure and therefore oil is produced by the driving force due to the expansion of oil and connate water, plus any compaction drive.. The contribution to drive energy from compaction and connate water is small, so the oil compressibility initially dominates the drive energy. Because the oil compressibility itself is low, pressure drops rapidly as production takes place, until the pressure reaches the bubble point. 

The primary drive mechanism for gas field production is the expansion of the gas contained in the reservoir. Relative to oil reservoirs, the material balance calculations for gas reservoirs is rather simple the recovery factor is linked to the drop in reservoir pressure in an almost linear manner. The non-linearity is due to the changing z-factor (introduced in Section 5.2.4) as the pressure drops. A plot of (P/ z) against the recovery factor is linear if aquifer influx and pore compaction are negligible. The material balance may therefore be represented by the following plot (often called the P over z plot). 

When a molecule is isolated from external fields, the Hamiltonian contains only kinetic energy operators for all of the electrons and nuclei as well as temis that account for repulsion and attraction between all distinct pairs of like and unlike charges, respectively. In such a case, the Hamiltonian is constant in time. Wlien this condition is satisfied, the representation of the time-dependent wavefiinction as a superposition of Hamiltonian eigenfiinctions can be used to detemiine the time dependence of the expansion coefficients. If equation (Al.1.39) is substituted into the tune-dependent Sclirodinger equation... 

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

Long-range forces are most conveniently expressed as a power series in Mr, the reciprocal of the intemiolecular distance. This series is called the multipole expansion. It is so connnon to use the multipole expansion that the electrostatic, mduction and dispersion energies are referred to as non-expanded if the expansion is not used. In early work it was noted that the multipole expansion did not converge in a conventional way and doubt was cast upon its use in the description of long-range electrostatic, induction and dispersion interactions. However, it is now established [8, 9, 10, H, 12 and 13] that the series is asymptotic in Poincare s sense. The interaction energy can be written as... 

We now define the tliree-dimensional vectors, Kand fl, consisting of the coefficients of the Pauli matrices in the expansion of p and//, respectively ... 

It suffices to carry out one such experiment, such as the expansion or compression of a gas, to establish that there are states inaccessible by adiabatic reversible paths, indeed even by any adiabatic irreversible path. For example, if one takes one mole of N2 gas in a volume of 24 litres at a pressure of 1.00 atm (i.e. at 25 °C), there is no combination of adiabatic reversible paths that can bring the system to a final state with the same volume and a different temperature. A higher temperature (on the ideal-gas scale Oj ) can be reached by an adiabatic irreversible path, e.g. by doing electrical work on the system, but a state with the same volume and a lower temperature Oj is inaccessible by any adiabatic path. 

For example, the expansion of a gas requires the release of a pm holding a piston in place or the opening of a stopcock, while a chemical reaction can be initiated by mixing the reactants or by adding a catalyst. One often finds statements that at equilibrium in an isolated system (constant U, V, n), the entropy is maximized . Wliat does this mean ... 

Using the leadmg temis in the expansion and the identification of the connnon temperature T, one obtains... 

Note that in this nonnalized probability, the properties of the reser >oir enter the result only through the common equilibrium temperature T. The accuracy of the expansion used above can be checked by considering the next temi, which is... 

The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

This leads to the third virial coefficient for hard spheres. In general, the nth virial coefficient of pairwise additive potentials is related to the coefficient7) in the expansion of g(r), except for Coulombic systems for which the virial coefficients diverge and special teclmiques are necessary to resiim the series. 

Only the first temi in this expansion is shown. It is identical to the last temi shown in the equation for> 2(/ ), which is the coefficient of in the expansion of tlie cavity function y(r). 

The expansion of the perturbation w(r 2> tenns of inultipole potentials (e.g. dipole-dipole, dipole-... 

Assume that the free energy can be expanded in powers of the magnetization m which is the order parameter. At zero field, only even powers of m appear in the expansion, due to the up-down symmetry of the system, and... 

Iiifomiation about the behaviour of the 3D Ising ferromagnet near the critical point was first obtained from high- and low-temperatnre expansions. The expansion parameter in the high-temperatnre series is tanli K, and the corresponding parameter in the low-temperatnre expansion is exp(-2A ). A 2D square lattice is self-dual in the sense that the bisectors of the line joining the lattice points also fomi a square lattice and the coefficients of the two expansions, for the 2D square lattice system, are identical to within a factor of two. The singularity occurs when... 

The previous seetion showed how the van der Waals equation was extended to binary mixtures. However, imieh of the early theoretieal treatment of binary mixtures ignored equation-of-state eflfeets (i.e. the eontributions of the expansion beyond the volume of a elose-paeked liquid) and implieitly avoided the distinetion between eonstant pressure and eonstant volume by putting the moleeules, assumed to be equal in size, into a kind of pseudo-lattiee. Figure A2.5.14 shows sohematieally an equimolar mixture of A and B, at a high temperature where the distribution is essentially random, and at a low temperature where the mixture has separated mto two virtually one-eomponent phases. 

Here the coefficients G2, G, and so on, are frinctions ofp and T, presumably expandable in Taylor series around p p and T- T. However, it is frequently overlooked that the derivation is accompanied by the connnent that since. . . the second-order transition point must be some singular point of tlie themiodynamic potential, there is every reason to suppose that such an expansion camiot be carried out up to temis of arbitrary order , but that tliere are grounds to suppose that its singularity is of higher order than that of the temis of the expansion used . The theory developed below was based on this assumption. 

This equation may be solved by the same methods as used with the nonreactive coupled-channel equations (discussed later in section A3.11.4.2). Flowever, because F(p, p) changes rapidly with p, it is desirable to periodically change the expansion basis set ip. To do this we divide the range of p to be integrated into sectors and within each sector choose a (usually the midpoint) to define local eigenfimctions. The coiipled-chaimel equations just given then apply withm each sector, but at sector boundaries we change basis sets. Let y and 2 be the associated with adjacent sectors. Then, at the sector boundary p we require... 

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