This paper gives a comprehensive analysis of algebras of Colombeau-type generalized functions in the range between the diffeomorphism-invariant quotient algebra G d = E M /N introduced in part I and Colombeau's original algebra G e . Three main results are established: First, a simple criterion describing membership in N (applicable to all types of Colombeau algebras) is given. Second, two counterexamples demonstrate that G d is not injectively included in G e . Finally, it is shown that in the range "between" G d and G e only one more construction leads to a diffeomorphism invariant algebra. In analyzing the latter, several classification results essential for obtaining an intrinsic description of G d on manifolds are derived.2000 Mathematics Subject Classification. Primary 46F30; Secondary 26E15, 46E50, 35D05.

A direct sequential method has been developed to simulate isothermal compositional systems. The solution technique is the same as that of the implicit pressure, explicit saturation (IMPES) method: one pressure is treated implicitly and (instead of the phase saturation) the component masses/moles are treated explicitly. A "volume balance" equation is used to obtain the pressure equation. A weighted sum of the conservation equations is used to eliminate the nonlinear saturation/concentration terms from the accumulation term of the pressure equation. The partial mass/mole volumes are used as "constants" to partial mass/mole volumes are used as "constants" to weight the mass/mole conservation equations. The method handles uniformly a range of cases from the simplified compositional (i.e., black-oil) models to the most complicated multiphase compositional models of incompressible and compressible fluid systems. The numerical solution is based on the integrated finite-difference method that allows one- (1D), two- (2D), and three-dimensional (3D) grids of regular or irregular volume elements to be handled with the same ease. The mathematical model makes it possible to develop modular versatile computer realizations; thus the model is highly suitable as a basis for general-purpose models. Introduction During the last three decades reservoir simulators have been well developed. The enormous progress in computer techniques has strongly contributed to the development of increasingly effective and sophisticated computer models. The key numerical techniques of modeling conventional displacement methods had been elaborated upon by the beginning of the 1970's, and it was possible to develop a single simulation model capable of addressing most reservoir problems encountered. Since the 1970's, however, because of the sharp rise in oil prices, the need for new enhanced recovery processes has forced reservoir-simulation experts to develop newer computer models that account for completely unknown effects of the new displacement mechanisms. The proliferation of recovery methods since the 1970's has resulted in a departure from the single-model concept because individual models tend to be developed to simulate each of the new recovery schemes. This proliferation of models, however, seems to be a less than ideal situation because of the expense involved in the development, maintenance, and applications training for the multiple new models. In addition, when different models are applied to simulate various enhanced recovery methods, no common basis exists to help survey, compare, and thus understand the different recovery mechanisms. The importance of a single, general simulator capable of modeling all or most recovery processes of interest was emphasized by Coats, who worked out a model as a step in this direction. Economic restrictions have also forced various companies to develop multiple-application reservoir models. The multiple-application reservoir simulator (MARS) program presented by Kendall et al. is one realization of the goal: a single program for multiple application. From a mathematical point of view, reservoir simulators consist of a set of partial differential equations and a set of algebraic equations, both with the appropriate initial and boundary conditions. In isothermal cases the partial differential equations, taking into account Darcy's law, describe the mass/mole/normal-volume conservation for each component of the reservoir fluid system. Phase and/or component transport caused by capillarity, gravity, and/or diffusion also can be taken into account. The algebraic equations describe the thermodynamic properties of the reservoir fluid/rock system. The existence of properties of the reservoir fluid/rock system. The existence of local and instant thermodynamic equilibria is a generally accepted assumption of reservoir simulation. This means that the number of mass/mole/normal-volume conservation equations is equal to the number of components used to describe the reservoir fluid/rock system. During the simulation the reservoir examined is divided into volume elements by a 1D, 2D, or 3D grid. Each of the volume elements is characterized by the appropriate reservoir properties and the displacement process is described by properties and the displacement process is described by a series of thermodynamic equilibria for each volume element. The difference between the simulators of conventional and enhanced recovery methods essentially arises from how many components are chosen as a means of appropriately describing the displacement process, and how the thermodynamic equilibria (thermodynamic properties) of the reservoir fluid/rock system are characterized. In cases of conventional technologies a simplified (black-oil) approach of the hydrocarbon system by a pseudogas and a pseudo-oil component generally is accepted, and the pseudo-oil component generally is accepted, and the thermodynamic properties of the given system depend only on the pressure. This approximation made it possible to develop the direct sequential IMPES solution technique, taking into account the advantage of black-oil models wherein the number of components is equal to the number of phases and thus the number of phases is equal to the number of conservation equations. SPEJ P. 543

This paper (SPE 52051) was revised for publication from paper SPE 37984, first presented at the 1997 SPE Reservoir Simulation Symposium held in Dallas, 8-11 June. Original manuscript received for review 31 July 1997. Revised manuscript revised 28 February 1998. Revised manuscript approved 25 June 1998. Summary The complexity and nonlinearity of reservoir simulation equations make it possible to apply a great number of linearization techniques. The SPE comparison projects and some other papers compare models for various problem classes. These papers generally compare the results and the speed of runs. The main object of this paper is to examine and compare the models on the basis of the numerical background of the linearization techniques. A general comparison method is proposed that can be applied to analyze these techniques. Recently, the proposed comparison method has been applied to examine implicit pressure, explicit saturations (IMPES) and implicit pressure, explicit masses (IMPEM) techniques. It has been revealed that the direct IMPES and IMPEM methods result in exactly the same pressure distribution, whereas the saturations and the masses are different. By applying the primary set transformation techniques, the saturation and mass distributions of the direct IMPES and IMPEM techniques can be transformed from one into the other. Now, the fully implicit methods are compared for a three-phase black oil system with one-dimensional (ID) radial grids. The relationships and convergence behavior of the implicit techniques are discussed. P. 316

In this paper an IMPES-type thermal volume balance technique is presented applying a direct solution of the set of mass and energy conservation equations. The effect of the convective energy transport can be considered in the isothermal pressure (volume balance) equation by a modification of the coefficients. The solution yields the pressure implicitly, component masses and energy explicitly (IMPEME technique) while keeping the mass, and energy balances. The explicit treatment demands timestep limitation. Volume balance errors are used to control timesteps. Compared with the direct isothermal volume balance approach, significant volume balance errors occur only at the front and at well regions. This is the portion of the reservoir in an adapti ve implicit method (AIM) where the fully implicit handling is used. The IMPEME method is proposed to be the explicit approximation of a thermal AIM. The method can also be incorporated in black oil or compositional isothermal volume balance models to include thermal effects.

The main object of this paper is to exarnine and compare models based on the numerical background of linearization techniques. A new transformation procedure is introduced that helps to separate the nonlinearity of the flow equations from the nonlinearity of fluid calculations. This transformation procedure can help to decide an optimal 'technique when developing a new model. In addition, it clears the real and the apparent differences among the mode Is, and it gives a possibility to arrive at a concise survey of the solution techniques. The detailed exarnination of the differences of IMPES and IMPEM techniques has revealed the origin of material balance error of the IMPES technique. The object-oriented C++ programming language was applied to the comparison studies. The basic C++ classes and member functions are presented in the Appendix at the end of the paper.