A Two-Dimensional, Two-Phase Compositional Model Which Uses A Moving Point Method
Authors: Ben-Omran Abdelhay M. , Green Don W. 
Published: December 2007

Abstract
A computer model has been developed to describe the isothermal flow of two multi-component compressible phases, oil and gas, in a two-dimensional porous medium. The composition in each phase and the mass transfer between phases are calculated. Capillary and gravity forces are included. The partial differential equation describing flow in each phase includes a source/sink term which is used to account for mass transfer between the phases. Convective transport is handled by introducing moving points. In this study the Benedict, Webb and Rubin (BWR) equation of state was applied as the "phase behavior package". The numerical solution proceeds in a leap-frog manner, by first solving the flow equations and then carrying out equilibrium calculations. Calculated output from the model includes pressure, saturation, and vapor-and liquid-phase compositions as functions of time and position. The simulator has behaved well for the position. The simulator has behaved well for the different conditions tested indicating the suitability of using a moving point method in a compositional simulator.

Introduction
Compositional models have been developed by a number of investigators. For these types of models, each hydrocarbon phase is assumed to consist of an N-component mixture. The fluid properties are functions of composition and pressure. Thus, composition should be specified for every mesh point at every time step. One approach to the development of compositional reservoir simulators has been to describe the behavior in terms of differential equations which represent a mass balance for each component in the system. This results in a large number of simultaneous equations which must be solved at each step in the solution. In another approach, a beta-type simulator has been modified to approximate compositional effects. In the latter approach, individual component compositions could not be calculated as a function of time and distance.
In this paper the hydrocarbon system is described basically by two partial differential equations, one describing flow in each phase. Mass transfer between phases is considered by including a sink/source term phases is considered by including a sink/source term in the two partial differential equations. The difference equations approximating the partial differential equations are formulated according to the implicit pressure-explicit saturation (IMPES) numerical method as presented by Stone and Garder. The difference equations, having pressures as the only unknowns, are solved directly using a D4 ordering scheme as proposed by Price and Coats. proposed by Price and Coats. To account for compositional changes resulting from convective transport of each phase, moving points are introduced. Each grid block is subdivided into cells of equal size and a moving point is defined to represent each phase in each cell, i.e., two moving points are used per cell. The points are moved in points are used per cell. The points are moved in each time step as dictated by the solution of the finite difference equations which describe the flow behavior.
Following the movement of the points for a single time step, equilibrium is re-established between phases by doing a flash equilibrium calculation. The phases by doing a flash equilibrium calculation. The mass-transfer rate which occurs is then used as the basis for the source/sink terms in the next tine step in the solution of the flow equations. The solution thus proceeds in a leap-frog manner, by first solving the flow equations and then carrying out equilibrium calculations.

THE MATHEMATICAL MODEL

The Flow Equations
In the flow equations of this model, the transfer of mass between phases is considered analogous to the production-injection (sink-source) process. The production-injection (sink-source) process. The withdrawal of a certain mass from one phase is balanced by an addition of an equal mass to the second phase. Based on this concept, the differential equations which conserve the mass of each phase may be written as follows:
For the oil phase,
(1)