Reservoir Simulation of Variable Bubble-Point Problems
Authors: Thomas L.K. , Lumpkin W.B. , Reheis G.M. 
Published: December 2007

Abstract
This paper presents the development of a general beta reservoir simulator that will model conventional (fixed bubble-point) problems as well as problems involving a variable bubble point, such as gas injection projects above the original bubble point and water injection projects resulting in a collapsing gas saturation, Provisions are included for allowing the pressure to cross the bubble point with the same relative ease as in a conventional simulator.
Example problems are presented to demonstrate the utility of the model for gas and water injection problems. problems

Introduction
Many reservoir simulation problems involve treating a variable bubble point throughout the reservoir. For example, when gas is injected into an undersaturated reservoir, gas will go into solution, increasing the bubble point of the oil. As this oil moves away from the injector, the bubble point of surrounding areas also may increase point of surrounding areas also may increase because of mixing. During waterfloods of saturated reservoirs, the gas saturations in regions near the injectors frequently reduce to zero at pressures below the original bubble point. Thus, the bubble point will vary areally throughout the field. point will vary areally throughout the field.Recent publications have discussed certain aspects of the variable bubble-point problem. Most of these papers contain only a brief discussion of this problem. Ridings discusses the need for allowing saturation pressure to vary continuously throughout the reservoir as long as there is free gas associated with the oil. In the model presented by Cook et al., free gas saturation is monitored for saturated systems and the bubble point is set equal to the prevailing reservoir pressure when the gas saturation in a cell disappears. Bubble points for undersaturated cells are allowed to change because of the entrance of free gas and mixing. Spilletta et al. assume that a cell that is saturated or undersaturated at the beginning of a time step will remain so throughout the time step. They then solve their saturation equations for water and gas saturations. The bubble point of any undersaturated cell is adjusted to account for nonzero gas saturation, and the water saturation is modified to conserve oil.
Steffensen and Sheffield devote their paper to the reservoir simulation of a collapsing gas saturation during waterflooding. In their model, blocks that have a free gas saturation at the beginning of a time step and have zero or negative gas saturations at the end of a time step are detected, and the bubble points for these cells are set equal to the estimated pressure where Sg reduced to zero. Gas saturation for these blocks is set equal to zero and S is set to 1 - S . The oil saturation in adjacent saturated grid blocks is then adjusted so that oil material balance is maintained. Mixing caused by flow between undersaturated blocks is neglected.
This paper presents a comprehensive analysis of modeling variable bubble-point problems.* It treats the specific problems of simulating gas injection above the bubble point as well as waterflooding depleted reservoirs. It differs from previously reported work by accounting for the effect of bubble-point change on computed pressure change during a time step. Also, provisions are included that allow the pressure to cross the bubble point with the same relative ease as in a conventions! simulator In regard to waterflooding, mis paper differs from the work of Steffensen and Sheffield in mat it allows for bubble-point changes caused by mixing.

DEVELOPMENT OF FLOW EQUATIONS
To simulate the variable bubble-point problem, the expansion of the flow equations above the bubble point must include the effects of pressure and bubble point on fluid properties. Also, special consideration must be given to cells passing through the bubble point if both pressure and material-balance errors are to be eliminated.
SPEJ
P. 10