Pressure correction equation single phase flows

For single-phase flows, pressure is shared by three momentum equations and requires special algorithms to compute the pressure field. Most of these algorithms (discussed in the previous chapter) use one continuity equation and three momentum equations to derive pressure and/or pressure correction equations. However, for multiphase flows, there is more than one continuity equation. Answers to questions such as which continuity equation should be used to derive pressure equations are not obvious. As discussed in the previous chapter, it is customary to employ iterative techniques to solve single-phase flow equations. Such iterative techniques can, in principle, be extended to simulate multiphase flows. In practice, however, the process... 

Usual interpolation rules and definitions of velocity and pressure corrections, similar to single-phase flows (Eq. (6.29)), can be used to derive a pressure correction equation from the discretized form of the overall continuity (normalized) equation. The momentum equation for multiphase flows (Eq. (7.16)) can also be written in the form of Eq. (6.28) for single-phase flows. Again, following the approximation of SIMPLE, one can write an equation for velocity correction in terms of pressure correction,/ ... 

By far, the most often cited work is the early paper by Outmans (1963), which applied differential-equation-based filtration methods developed in chemical engineering to static and dynamic invasion in the borehole. In this single-phase flow study, where lineal flow was assumed and the applied differential pressure was completely supported by the mudcake, Outmans derived the well known Vt law, subject to the further proviso of cake incompressibility. (The effects of cake nonlinearity and compaction can be important over time e.g., see Figure 14-7.) Thus, the Vt law cannot be used when the net flow resistance offered by the formation is comparable to that of the mudcake (e.g., thin muds in permeable formations, or thick muds in very impermeable rocks). Also, the law does not apply to slimholes, where the radial geometry is important. Finally, the Vt law does not generally apply to reservoirs with two-phase, immiscible flow, or miscible flow, or both. Only under these restrictive assumptions does Outmans correctly derived law hold. 

What is the minimum number of variables to specify fully a stream A stream can be defined as the flow of material between two units in a flowsheet. The variables normally associated with a stream are its temperature, pressure, total flow, overall mole fractions, phase fractions and phase mole fractions, total enthalpy, phase enthalpies, entropy, etc. Assuming phase and chemical equilibrium, how many of those variables must be specified to completely fix the stream Without further considerations, for this case, intuition gives us the correct answer. We know without writing equations that if we specify temperature, pressure, and individual component flows, the stream is fully specified. Of course, a priori we cannot know the final state of the stream (i.e., multiphase or single phase liquid, vapor, solid, or a mixture of them). If we are interested in a stream with some specific conditions like saturated liquid, we cannot specify simultaneously pressure and temperature but pressure (or temperature) and phase fraction. A convention in process simulators is that when vapor (liquid) phase fraction is specified to zero or one, saturated conditions are assumed (bubble point or dew point). However, when vapor or liquid phase fractions are calculated, a value of one (zero) does not mean saturated conditions but that the stream is in vapor (liquid) phase. 

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