Implicit-pressure, explicit-saturation

To model the measured transient foam displacements, equations 2 through 12 are rewritten in standard implicit-pressure, explicit-saturation (IMPES) finite difference form, with upstream weighting of the phase mobilities following standard reservoir simulation practice (10). Iteration of the nonlinear algebraic equations is by Newton s method. The three primitive unknowns are pressure, gas-phase saturation, and bubble density. Four boundary conditions are necessary because the differential mass balances are second order in pressure and first order in saturation and bubble concentration. The outlet pressure and the inlet superficial velocities of gas and liquid are fixed. No foam is injected, so Qh is set to zero in equation... 

Abstract A newly developed numerical simulator of two-phase flow using three-dimensional finite element method is presented in this paper. It is described that the fundamental simultaneous equations, the deduction to implicit pressure explicit saturation formulation and their finite element discretization method. Furthermore, its practical application to the numerical simulation project of predicting Horonobe natural gas product is also introduced. 

These formulations were solved using second-order accurate implicit schemes in the work just presented that is, our approach was implicit pressure, implicit saturation. This is in contrast to the popular implicit pressure, explicit saturation codes used in the industry, which are only conditionally stable. (The von Neumann stability of both implicit and explicit schemes was considered in Chapter 20.) This so-called IMPES scheme, in addition to its stability problems, yields undesirable saturation oscillations and overshoots that are often fixed by upstream (that is, backward) differencing of spatial derivatives. But this... 

Finite Volume Methods The finite volume method, when the permeabihty tensor is diagonal in the selected coordinate system, approximates the pressure and saturation functions as piecewise constant in each grid block. The flux components are assumed constant in their related half-cells. Thus when two cells are joined by a face, the related component of flux is assumed to be the same each side of the face. The balance laws are invoked separately on each grid block, and are discretised in time either by an explicit or fully implicit first order Euler scheme or other variant as discussed in the previous subsection. 

We will find in 9.1 that, for a pure substance in two-phase equilibrium, only one property is needed to specify the intensive state in (8.2.20) we have used temperature. However, even after we set a value for the subcritical temperature, (8.2.20) remains implicit in three unknowns the vapor pressure P plus the molar volumes of the liquid and vapor phases, and v. To close the problem we need another equation, typically, a PvT equation of state that relates P to both saturated volumes at Ae specified T. Therefore, we must choose an equation of state that is sufficiently complicated that it bifurcates and provides multiple roots for the volume over some range of states. Such equations of state are explicit in the pressure [P = P(T, v) and then we would compute 9 from... 

The saturation and pressure equations of the simulation model are not amenable to analytical solution and must be solved numerically. We use the IMPES finite-difference method to solve this black-oil model. The IMPES method develops a set of algebraic equations that are implicit in pressure and explicit in the saturations. 

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