Cusps

Keywords compressibility, primary-, secondary- and enhanced oil-recovery, drive mechanisms (solution gas-, gas cap-, water-drive), secondary gas cap, first production date, build-up period, plateau period, production decline, water cut, Darcy s law, recovery factor, sweep efficiency, by-passing of oil, residual oil, relative permeability, production forecasts, offtake rate, coning, cusping, horizontal wells, reservoir simulation, material balance, rate dependent processes, pre-drilling. 

Introduction and Commercial Application Section 8.0 considered the dynamic behaviour in the reservoir, away from the influence of the wells. However, when the fluid flow comes under the influence of the pressure drop near the wellbore, the displacement may be altered by the local pressure distribution, giving rise to coning or cusping. These effects may encourage the production of unwanted fluids (e.g. water or gas instead of oil), and must be understood so that their negative input can be minimised. 

Cusping occurs in the horizontal plane, that is the stabilised OWC does not lie directly beneath the producing well. The unwanted fluid, in this case water, is pulled towards the producing well along the dip of the formation. 

The above examples are shown for water coning and cusping. The same phenomena may be observed with overlying gas being pulled down into the producing oil well. This would be called gas coning or cusping. 

The third main application of horizontal wells is to reduce the effects of coning and cusping by changing the geometry of drainage c ose to the well. For example, a horizontal... 

For both types of orbitals, the coordinates r, 0 and cji refer to the position of the electron relative to a set of axes attached to the centre on which the basis orbital is located. Although STOs have the proper cusp behaviour near the nuclei, they are used primarily for atomic- and linear-molecule calculations because the multi-centre integrals which arise in polyatomic-molecule calculations caimot efficiently be perfonned when STOs are employed. In contrast, such integrals can routinely be done when GTOs are used. This fiindamental advantage of GTOs has led to the dominance of these fimetions in molecular quantum chemistry. 

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

This discussion will be limited to functions of one variable that can be plotted in 2-space over the interval considered and that constitute the upper boundar y of a well-defined area. The functions selected for illustration are simple and well-behaved, they are smooth, single valued, and have no discontinuities. When discontinuities or singularities do occur (for example the cusp point of the Is hydrogen orbital at the nucleus), we shall integrate up to the singularity but not include it. 

The reason the inner shell of carbon is represented by 6 primitives in this basis is that the cusp in the Is orbital is difficult to approximate with Gaussians that have no cusp. 

Fig. 7. Cusp-shaped potential, made up of parabola and a vertical wall.

The situation changes when moving on to low temperature. Friction affects not only the prefactor but also the instanton action itself, and the rate constant depends strongly on rj. In what follows we restrict ourselves to the action alone, and for the calculation of the prefactor we refer the reader to the original papers cited. For the cusp-shaped harmonic potential... 

As seen from (5.48) and (5.51), at high temperatures the leading exponential term in the expression for k is independent of rj and it displays the Arrhenius dependence with activation energy = Vq = Formally, because of the cusp, the instanton in this model never... 

Fig. 47. Arrhenius plot of diffusion coefficient for (a) H and (b) D atoms on the (110) face of a tungsten crystal at coverage degree 0.1-0.9 as indicated. The cusps on the curves correspond to the phase transition.

In order to find Eq we study first the auxiliary problem of a cusp-shaped harmonic potential with a wall placed at x = Xp (see fig. 7),... 

Curran [61C01] has pointed out that under certain unusual conditions the second-order phase transition might cause a cusp in the stress-volume relation resulting in a multiple wave structure, as is the case for a first-order transition. His shock-wave compression measurements on Invar (36-wt% Ni-Fe) showed large compressibilities in the low stress region but no distinct transition. 

The resulting stress-volume relations for the 28.5-at. % Ni alloys are shown in Figure 5.13. The cusp in the fee curve at 430 MPa (4.3 kbar) is the mean value observed for the Hugoniot elastic limit, whereas the dashed line shown for the fee alloy indicates the stress region for which some strain hardening is indicated from the stress profiles. It is readily apparent that below 2.5 GPa (25 kbar) the fee alloy shows a much larger compressibility than the bcc alloy. 

The variation of strength with angle of lamina orientation is smooth rather than having cusps that are not seen in experimental results. 

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