Continuous Compositional Volume-Balance Equations
Authors: Young Larry C. 
Published: December 2007

Abstract
This paper describes a continuous form of the governing equations for compositional reservoir simulation. A continuous overall balance or pressure equation is created from a weighted sum of the component balances. The overall balance is the continuous analog of the discrete pressure equation in the volume-balance method and other IMPES (IMplicit-Pressure Explicit Saturation) methods for reservoir simulation. The coupling between the overall balance and the component balances is analyzed, and some possible algorithmic improvements, which utilize this work, are suggested.

Introduction
In the early 1980's, three similar procedures for compositional simulation were independently developed.1,2,3 All three methods employ a weighted linear combination of the discrete component balance equations in order to obtain a discrete pressure equation. Discussions of these methods have shown that the basic methods are the same, although there are certainly some differences in the calculational details. 4,5,6 Furthermore, the weighting factors used in the linear combination are unique, and for black oil systems they are the familiar IMPES reduction factors which have been used for many years.6 Acs, et al.2 and Watts3 point out that the reduction factors are multiphase partial volumes. The pressure equation follows naturally when compositions are eliminated from an equation constraining the fluid volume or volume fractions (saturations), so all methods of this type may be viewed as volume-balance methods.
The methods, just described, all consider the discrete form of the conservation equations. All of the methods discretize the component balances before combining them to create the pressure equation. In the present work, we consider the continuous analog of this process, a subject that has received little attention previously. We are aware of only Martin's early work on black oil systems.7 This paper extends Martin's work to more complex fluid systems.
After deriving the continuous form of the overall balance or pressure equation, we consider the strength of the equations' dependence on composition and pressure, or the degree of coupling between the component balances and the overall balance. The literature contains mixed experiences regarding the strength of this coupling. The experience with fully implicit simulation (FIM) indicates that all equations must be solved simultaneously and that convergence is often slowed by strong nonlinear interactions between pressure and composition (or saturation).8 Recent works on streamline simulations claim that the equations are readily decoupled.9 In streamline simulations the overall balance is used to produce streamlines from the velocity field. The streamlines often remain fixed for many time steps in the component balances.
From the analysis of equation coupling, we attempt to reconcile the differing experiences reported in the litterature, and we offer several suggestions of algorithmic improvements and discuss their potential benefits.

Component Balance Equations
For isothermal flow, the component material balances are:Equation (1)
for i=1,…,N c. Where the phase velocities are defined by Darcy's law extended to multiphase flow:Equation (2)
These equations govern all isothermal reservoir fluid systems, which do not have major rock-fluid interactions. Simple 2-phase, oil/water or gas/water, or 3-phase black oil systems are special cases of Eq. (1). Only minor modifications would be needed to represent systems with rock-fluid interaction, since one of the arbitrary number of phases could represent an immobile solid (or rock) phase.