No abstract

Given a set of observed data for a particular physical phenomenon, the problem of computing the "best fit" parameters for the mathematical model describing the phenomenon is a common problem in process or reaction mechanism identification. If the mathematical model comprises a set of non-linear ordinary differential equations, this leads to a non-linear boundary value problem. A very powerful way of attacking this class of problem uses an adaptation of the Newton-Raphson-Kantorovich procedure, called quasilinearization, which regards the non-linear problem as the limit of a sequence of linear problems. Starting from an initial trial solution, convergence if it does occur, occurs rapidly ; further, convergence is assured if the initial guess is "close enough" to the true solution. The diffieulty of making a good initial guess, a serious limitation of the method in the past, can in principle be overcome by the algorithm proposed. When a given vector may not be within the domain of convergence of the original problem, it must be within the domain of convergence of some other derived problem. The latter may then be perturbed towards the original problem in a finite number of steps. In the case of process identification, new data points are derived; these are subsequently adjusted until they coincide with the original data. The algorithm has been suceessfully applied to several examples from recent chemical engineering literature.

The alternating direction explicit procedure (ADEP) makes use of the boundary conditions to reduce multi-dimensional problems to a series of one-dimensional problems. The method, previously applied to reservoirs containing only an undersaturated oil, bas now been extended to cover the case of natural gas reservoirs. Although this involves solving a non-linear partial differential equation, application of the procedure is straight- forward and no calculational problems were encountered. Introduction There has been a growing interest in formulating mathematical models of petroleum and natural gas reservoirs - models which permit the engineer to examine and evaluate the physical and economic consequences of various alternative production policies. The tremendous reduction in the cost of solving such models in recent years has made possible their use as an almost routine management tool. This reduction is the result not only of improved computer hardware but also of the development of more efficient mathematical techniques. The present paper is concerned with the application of a recently proposed numerical method (ADEP) to two-dimensional gas reservoirs. STATEMENT OF THE PROBLEM Given a two-dimensional Region R, bounded by a closed Curve C (Fig. 1) such that the behavior on Curve is known, the differential mass balance for each fluid phase in R, neglecting gravitational effects and assuming Darcy's law for fluid transport, can be written as: A typical and common set of boundary conditions is given by (1) / = 0 on Curve C where r is the direction normal to Curve C; (2) p is known throughout Region R at some time t; and (3) wf is known for all x, y and t. Physically, this represents the case where the reservoir is bounded by impermeable media, where the initial pressure throughout the reservoir at the beginning of gas production is known and where the production rate at each well is specified at all times. For single-phase reservoirs, the problem is to determine the pressure throughout Region R at all times; for multi-phase reservoirs, in addition to the pressure, the value of the fluid saturation is also required. Since analytical solutions to this equation for the general case are not available, we must resort to numerical integrating techniques, using finite-difference approximations. SPEJ P. 137ˆ

The transient behavior of a packed bed of uniform spheres is considered, where heat (or mass) transfer inside the sphere is described by a linear parabolic second order partial differential equationthe standard diffusivity

A new experimental technique for measuring molecular diffusion through liquids, using microcell interferometry, is described. The main advantage is that an experiment can be performed in a few minutes. Aside from the saving in time, these short runs, which are inherently unsteady state, are more likely to be free of external effects. Concentration profiles are recorded on film at suitable time intervals. By analyzing the semidiscrete form of the diffusion equation and by a suitable linear transformation of the variables denoting the concentration profile, linear relationships are established between the logarithms of these transformed concentrations and time. This permits the determination of the diffusivity with maximum sensitivity. Conventional “least squares” curve fitting techniques are shown to be inadequate for analyzing the data from these experiments.