This paper describes a new Fully-Implicit Black Oil Model, which is different from the conventional Black Oil Model in using a phase equilibrium concept. In this model four basic unknowns, pressure and three component masses per unit formation volume, are solved instead of pressure, water saturation and the third alternative variable (oil saturation or bubble point pressure) in the conventional Black Oil Model. Thus, we can avoid the complicated treatment of the third alternative variable and discontinuity at crossing bubble point in the solution of equations. Although the phase equilibrium calculation is included, the model still uses common PVT data of Black Oil Model. Solving equations is followed by a â??Flash Calculationâ?? for each time. The model includes both sequential and simultaneous schemes, so it can be used in various cases of reservoir simulation. The model has been tested with a three-phase coning study, which is from the Second Comparative Solution Project of Black Oil Model presented at the Sixth SPE Symposium on Reservoir Simulation, 1982.
The Black Oil Model is the most commonly used in reservoir simulation. The methods used in the Black Oil Model are also relatively perfect. In process of developing the model a variety of numerical methods, such as IMPES, Semi-Implicit, Fully-Implicit and Adaptive Implicit methods, have been developed, each of which has its own advantages in solution and can be used in certain cases. Although many numerical methods are still used up to now, we can say that the basic mathematical approach of Black Oil Model have not been changed yet. In all the methods mentioned above the equation system with three basic unknowns (alternative variable) is solved. The third variable is oil saturation at three-phase state (water, oil and gas phases presented) or bubble point pressure at two-phase state (water and oil phases presented). In softwares of some companies gas saturation and solution gas oil ratio are considered as the third alternative variable. May be, this kind of mathematical approaches is the most natural and easy to be accepted to solve Black Oil Model problems. In three mass-balance equations some rock and fluid properties, such as relative permeabilities, capillary pressures, oil formation volume factor, solution gas oil ratio and viscosities are functions of pressure, saturation and bubble point pressure as measured in laboratory. Therefore, when this kind of equations is solved, it is a simpler procedure to consider pressure, saturations and bubble point pressure as the unknowns.
In the development and application of Black Oil Model the mathematical approach has met some troubles in dealing with crossing bubble point. In IMPES and Semi-Implicit methods the equation coefficients can only be calculated by using old time step values of the unknowns. But the change of phase states might take place when one-time-step calculation is finished. As it is, the solutions might give some unreasonable results, such as negative gas saturation and the bubble point pressure above oil pressure. The problem cannot be avoided in conventional IMPES and Semi-Implicit methods. In order to eliminate unreasonable results in some softwares the material balance correction is applied to those grid blocks in which crossing bubble point has happened. It should be pointed out that the correction mentioned above is not a good treatment theoretically. Based on the single-grid-block-mass-balance principle, the pressure, saturations and bubble point pressure are corrected. But the flow variations between grid blocks cannot be considered in the treatment and the balance relationship of pressure is broken. So that for next time step calculation it can cause oscillation phenomenon or deconvergence of the solution.
