The effectiveness of digital computing machines in making technicalcalculations depends on how well the work is arranged to utilize the capabilityof the machines. This note presents a particularly useful way of calculatinghydrocarbon vapor-liquid equilibrium in the flash vaporization (orcondensation) system. The method is well suited to sequence-controlledcomputing equipment. It is not limited to equilibrium calculations and may beused for solution of most implicit equations in one variable. Introduction There is increasing interest in the use of electronic digital computers inresearch and engineering calculations. This is a fortunate and inevitable trendin view of both the increasingly extensive numerical work which is becoming aroutine part of many daily production operations and growing demand for theoverwhelming amounts of calculations required by newly developed numericalmethods for solving heretofore unsolved problems. Machines are in many ways ideally suited to the task but of necessitypresent certain difficulties for a particular problem to be solved must oftenbe formulated quite differently from the way it would be arranged for manualsolution. This is done in order to take advantage of the inherent speed andprecision of electronic computers and at the same time to limit the need fornumber storage to the capacity of the machine. Therefore, this note issubmitted to present a general and quite powerful method of finding solutionsof the frequently encountered implicit equation: (Equation 1) where the root, x0, is to be found for a given set of y1… yn. Theprocedure is well suited for use with computing machines, for it usuallyrequires but little storage or programming beyond that necessary to evaluateF. Hydrocarbon Equilibrium The method is described in terms of the problem it was designed to solve, i.e., the calculation of hydrocarbon vapor-liquid equilibrium in flashvaporization. Given the composition of a hydrocarbon mixture and appropriatevalues of the equilibrium ratios K, to find the phase ratio and compositions ina closed system: let zi be the mol fraction of the i-th component in themixture. T.N. 136
The problem of unsteady-state gas flow through porous media leads to a second-order non-linear partial differential equation for which no analytical solution has been found. In this paper a stable numerical procedure is developed for solving the equation for production of gas at constant rate from linear and radial systems. An electronic digital computer is used to perform the numerical integration using an implicit form of an approximating difference equation. Solutions are presented in graphical form for various values of dimensionless parameters. The solutions are compared with the laboratory study of gas depletion in a linear system. Introduction Production of fluids from porous rock reservoirs is essentially a transient process. Transient gradients develop as soon as production begins, and further withdrawals continue to cause disturbances which propagate throughout the reservoir, each adding in some way to the prior ones. A correct mathematical analysis of this behavior is complicated by the fact that the transient or unsteady-state flow of compressible fluids must be described by difficult second-order partial differential equations. As a practical matter, three distinctly different cases arise:Flow of single-phase liquidFlow of gasesMultiphase flow The first of these has been found to give a linear second-order equation similar to the well-known heat flow equation. Solutions of Equation (1) for both linear and radial flow are available in several forms. Although a number of approximate solutions have, been proposed, each is limited in value by the associated simplifying assumptions. Inasmuch as the analysis of transient flow is limited to liquid systems, a solution of the second case is necessary if further progress is to be made in studying underground fluid movement. For this reason a solution of Equation (2) was undertaken by means of numerical integration of approximating difference equations.
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