An Efficient Multicomponent Numerical Simulator
Authors: Kazemi H. , Vestal C.R. , Shank Deane G. 
Published: December 2007

Original manuscript received in Society of Petroleum Engineers office Sept. 15, 1977. Paper accepted for publication April 21, 1978. Revised manuscript received July 27. 1978. Paper (SPE 6890) first presented at the SPE-AIME 52nd Annual Fall Technical Conference and Exhibition, held in Denver, Oct. 9-12, 1977.

Abstract
An efficient, three-dimensional three-phase, multicomponent, numerical reservoir simulator was developed to study petroleum reservoirs where interphase mass transfer is important. Flow equations have a volume balance on the water phase and a mole balance on the vapor-liquid hydrocarbon phases. Additional equations include the capillary phases. Additional equations include the capillary pressure, phase equilibrium, and saturation pressure, phase equilibrium, and saturation relations. Flow equations, in finite-difference form, are combined to obtain an implicit equation for the oil-phase pressure, an explicit equation for the over-all composition of each hydrocarbon component, an explicit water saturation equation, and explicit oil-gas saturation equations that satisfy thermo-dynamic equilibrium. Equations for oil pressure, water saturation, hydrocarbon compositions, and oil-gas saturations are sequentially solved in an iterative loop until convergence is achieved. The simplicity of the sequential solution algorithm presented here is believed to be a new contribution. Furthermore, if thermodynamic inconsistencies appear in the entry data, these can be detected readily from the pressure and other equations in the sequential pressure and other equations in the sequential algorithm.

Introduction
With deeper drilling, more reservoirs containing volatile crude oils and gas condensates have been found. To study the performance of such reservoirs and to assist in maximizing hydrocarbon recovery, compositional reservoir simulators are needed. These simulators account for multiphase flow and the interphase mass transfer of each component in the given hydrocarbon system. This simply means that at any given time the simulator tracks the motion of reservoir fluids and calculates the state of equilibrium at many strategic reservoir points (simulator nodes). Therefore, at each reservoir points the phase pressures, the phase saturations, the over-all composition, the mole fraction of each component in the liquid and in the vapor phase, and the liquid mole fraction are calculated with time.
Several tank material-balance methods provided the early computational approach to reservoir performance predictions. The experience gained performance predictions. The experience gained in the formulation and in the use of the tank models became the foundation of our knowledge for developing the multidimensional compositional simulator of this paper. Four papers, using finite-difference computational schemes, were instrumental in helping us make the transition from the tank model to the multidimensional case.
This paper provides formulation for a multicomponent numerical reservoir simulator. We have tried to provide the formulation with enough detail so that the reader can use it as an easily accessible starting point for his own research and development. This formulation is efficient in the sense that it is computationally less expensive than fully implicit schemes and can be used effectively for a wide range of practical problems.
Quantitatively, for an N-component, three-phase (water, oil, and gas) system, 3N + 7 variables (Pw, Po, Pg, Sw, Sg, and xi, yi, zi, i = 1,2,....., N, Po, Pg, Sw, Sg, and xi, yi, zi, i = 1,2,....., N, and L) are usually determined at each reservoir node at any given time. For comparison, in a black-oil simulator, six variables (Pw, Po, Pg, Sw, So, and Sg) are calculated at each node. This comparison points out that a compositional reservoir simulator requires more bookkeeping and more storage. After we have described the details of the computation in later sections, it will be recognized that the computation time can be several-fold, too. However, an efficient mathematical scheme, such as the one presented here, brings the computation time close to that of a black-oil simulator.
Flow equations have a volumetric balance on the water phase and a molar balance on the hydrocarbon phases. Hydrocarbon-phase equilibrium calculations phases. Hydrocarbon-phase equilibrium calculations use equilibrium ratios as a function of pressure and convergence pressure. Densities and viscosities are calculated in the most general case as functions of pressure and composition of the given phase. Flow equations are discretized in an implicit finite-difference form to obtain an implicit pressure equation, an explicit water-saturation equation, an explicit composition equation, and two explicit gas-oil saturation equations.
SPEJ
P. 355