An Iterative Technique for Compositional Reservoir Models
Authors: Fussell L.T. , Fussell D.D. 
Published: December 2007

Original manuscript received in Society f Petroleum Engineers office Sept. 15, 1977. Paper accepted for publication July 11, 1978. Revised manuscript received March 26, 1979. Paper (SPE 6891) first presented at the SPE-AIME 52nd Annual Fall Technical Conference and Exhibition, held in Denver, Oct. 9-12, 1977.

Abstract
Compositional reservoir models are a class of phase equilibria models. The equations for these models, and especially those incorporating the Redlich-Kwong equation of state, are highly nonlinear and must be solved by an iterative method. The method of successive substitution commonly is used. This method, however, demonstrates linear convergence and almost always diverges or, at best, demonstrates poor convergence, for conditions where phase poor convergence, for conditions where phase equilibria calculations are required near the bubble point, dewpoint, or the critical point. Iterative point, dewpoint, or the critical point. Iterative methods that converge for these calculations are presented here. These iterative methods are called presented here. These iterative methods are called minimum-variable Newton-Raphson (MVNR) since they attempt to minimize the number of variables for which simultaneous iteration is required and use the Newton-Raphson method for the correction step. These methods demonstrate quadratic convergence near the solution. MVNR can be applied to one-, two-, and three-dimensional geometries. The hydrocarbon fluid is represented as an n component system. Flow equations for the water phase also are included; however, mass transfer between the hydrocarbon and waterphases is assumed negligible. The correction step uses a reordering technique that reduces the storage requirement and the computer cost per time step per grid block. An example problem is presented and discussed. problem is presented and discussed. Introduction
Compositional reservoir models are used to predict the performance of oil-recovery methods when interphase mass transfer depends on phase composition as well as pressure. These methods include depletion or cycling of volatile oil, high-shrinkage oil, gas condensate reservoirs, and enhanced oil recovery (e.g., CO2 and rich-gas displacements).
Most compositional models presented in the literature use table lookup K values (equilibrium ratios) or fits of the NGPA K values to describe the equilibrated distribution of components between phases. For complex oil-recovery methods, these phases. For complex oil-recovery methods, these techniques also require a convergence pressure correlation that accounts for the composition dependence on phase equilibria. One primary disadvantage of these techniques is that the predicted phase equilibria are not internally consistent - i.e. phase equilibria are not internally consistent - i.e. derivatives of thermodynamic quantities are not necessarily smooth or even continuous. This often leads to nonconvergence of the reservoir model. Using an equation of state removes this limitation and, furthermore, reduces the time and costs required for preparing data before reservoir simulation. Another significant advantage is that the phase-equilibria and flow equations can be used simultaneously to solve all variables. Table lookup models, presented in the literature, separate the flow equations from the phase-equilibria equations during the iteration toward the solution.
Fussell and Yanosik presented MVNR iterative methods for solving the equations for another phase-equilibria model - the single-stage separation phase-equilibria model - the single-stage separation unit. This study extends MVNR methods to the more complex, fully compositional models. Our methods demonstrate the same advantages of MVNR methods for the single-stage separation equations. The correction step of MVNR methods for compositional models includes a set of n + 1 equations for each grid block of the reservoir system, where n is the number of components in the fluid system.
SPEJ
P. 211