Relationship of the Volume-Balance Method of Compositional Simulation to the Newton-Raphson Method

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Abstract
Summary: This paper provides an analytical exposition of the relationship between two fully compositional, isothermal, three-phase, numerical simulator types in the literature. The Newton-Raphson (NR) method is based on the standard NR solution of the mass- or mole-conservation equations and associated constraints. The volume-balance (VB) method is based on the principle that, in each simulator block, the PV equals the combined fluid volume of all the phases. These two types of models have been regarded as different approaches to the same problem. An analytical demonstration, however, proves rather surprisingly that, although the two model types were developed from fundamentally different viewpoints (i.e., mass conservation vs. volume conservation), they lead to the same matrix system of the finite-difference equations. This understanding yields many advantages, including (1) methods to improve efficiency in both model types, (2) fully implicit and iterative forms of the VB model, (3) the assignation of physical meaning to groupings of terms from the NR Jacobian matrix, and (4) an understanding of how to decouple the mass- or mole-conservation equations from the thermodynamic equilibrium constraints. Also, a unified approach is possible that offers the best features of both simulation model types.

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