A three-parameter mathematical model for onedimensional flow in porous media is developed. The objective of the model is to predict accurately the longitudinal dispersion associated with the flow of either ga-'!!s or liquids over a wide range of Reynolds number.A qualitative analysis of the model parameters is given. Published experimental pulse-response curves are compared with results predicted by the model. Several general types of problems are given for which the model can be used.

A new single-well tracer method has been developed to measure residual oil saturations of watered-out formations within a precision of 2 to 3 PV percent. This in-situ method makes an average measurement over a large percent. This in-situ method makes an average measurement over a large reservoir volume by using trace chemicals dissolved in formation water. The technique is applicable in both sandstones and limestones for a wide range of conditions. Introduction Residual oil saturation is a basic item of data for many aspects of reservoir engineering. This number is required for normal material-balance calculations. Residual oil saturation is also extremely important in determining the economic attractiveness of a planned waterflood or a proposed tertiary recovery operation. Finally, in some areas proration is related to attainable residual oil saturation. Core analysis and well logging, the two most widely used methods for measuring residual oil saturations, are subject to a variety of well known limitations. One principal common fault is that both methods yield values that are averages over very small reservoir volumes. The chemical tracer method described in this paper samples a much larger volume of reservoir around a single well, The residual oil saturation measured represents an average over as much as several thousand barrels of pore space. Because this method makes an in-situ measurement, additional limitations of other methods are also avoided. In the single-well tracer technique, a primary tracer bank consisting of ethyl acetate tracer dissolved in formation water is injected into a formation that is at residual oil saturation. This bank is followed by a bank of tracer-free water. The well is then shut in to permit a portion of the ethyl acetate to hydrolyze to permit a portion of the ethyl acetate to hydrolyze to form ethanol, the secondary tracer. Finally, the well is produced and the concentration profiles of the two tracers are monitored. Ethyl acetate is soluble in both the water and oil phases, but ethanol is, for all practical purposes, phases, but ethanol is, for all practical purposes, soluble only in the water phase. As a result, the ethanol travels at a higher velocity and returns to the wellbore earlier than does the ethyl acetate. The difference in arrival times can be used to determine the residual oil saturation through the use of computer programs that simulate the tracer test (the greater the programs that simulate the tracer test (the greater the oil saturation, the greater the difference in arrival times). Field tests have demonstrated the reliability and applicability of this technique. This paper describes the tracer method, gives results of field experience, and presents a mathematical description of the process. One field application is described in detail, followed by a discussion of the scope and limitations of the technique. General Description of the Tracer Method Theoretical Basis The chemical tracer method depends on chromatographic retardation of a tracer chemical that is soluble both in formation water and in oil. Considering a system in which the oil is stationary (or at residual saturation) and the formation water is flowing at a-> velocity V w, the local velocity of a typical tracer molecule is-> -> JPT P. 211

Noteworthy experimental and theoretical progress has been made in recent years in elucidating various phenomena which occur in fixed-bed systems. The work of Wilhelm and coworkers (3, 7) on the nature of radial mixing in fixed beds is dehitive in this particular area. Kramers and Alberda ( 6 ) , McHenry and Wilhelm ( 8 ) , and others (1, 4, 5 ) have investigated the mechanism of axial mixing and proposed models for its description. Voluminous earlier work is available on the evaluation of heat transfer at the tube wall in packed beds and of heat and mass transfer between the packing and the flowing fluid in a bed. Furthermore the factors which influence the course of reactions within a single (catalyst) particle have been the subject of much effective research.Heretofore no practical mathematical model has been available for combining the environmental and reaction phenomena. The difficulties arising from coupling between material and heat sources in reactive systems were insurmountable, either for purely analytical reasons or because interactions with the environment could not be adequately expressed within the framework of the model.The purpose of the present work is the development of a mathematical model which is consistent with the available knowledge of mechanistic phenomena. The suitability of the model in terms of available high-speed digital computers for evaluation of solutions is also considered.The first part of this paper is concerned mainly with evaluating the model as a primary representation of the packed bed environ; nonreactive fluids are thus considered. The model is tested by treating it as if it were an actual physical system and subjecting it to analysis of the type performed previously (1, 3, 4, 5, 6, 7) . The numerical computations are carried out using an IBM-704 digital computer.Once the model is established as a predictive as well as a correlative tool it is extended to cover the behavior of reactive systems. The second part of the paper contains an analysis of the mathematical implications of this extension, along with a discussion of the numerical examples. By means of the latter the practical value of the model as a design tool is indicated.

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A one-dimensional column is considered i n which a number of chemical reactions with arbitrary kinetics may take place among an arbitrary number of components. initially, the column is i n complete chemical and physical equilibrium. A localized small p e r t u r b t i o n is introduced i n the column a t time t = 0. It i s shown that, i n general, this initial perturbation separates into a definite number of peaks which move with different velocities. Each peak broadens according t o an asymptotic relation, depending on a characteristic dispersion coefficient. I f n is the number of components, m the number of independent reactions, and u the number of equations of state t o be considered, there are n-m-u peaks. These peaks do not correspond t o single substances as in classical chromatography, but each peak has an eigencomposition. The velocities of the peaks are derived as functions of stoichiometry and equilibrium data. The dispersion coefficients depend, i n addition, on the kinetics of the chemical reactions and on the rate of mass transfer. Thus, perturbation chromatography offers a means of determining both equilibrium and rate data. The theory is illustrated by means of two examples.The term perturbation chromatography used in the title of this paper covers a broader class of phenomena than is normally associated with chromatography. A definition will now be given which, at the same time, delimits the scope of this work. Consider the steady flow of a multicomponent fluid over a uniformly distributed fixed adsorbent-catalyst phase. In the initial steady state condition, fluid composition is independent of position and time and determines the concentrations in the fixed phase through various physical and chemical equilibrium relations. At some time zero, the system is perturbed over a small portion of its length by a small change in composition of one of the phases. Perturbation chromatography is the behavior of the system after time zero. The description and use of the set of disturbances which propagates downstream is considered here.Earlier work in this area is summarized by Collins and Deans ( 2 ) , who discussed the number and velocity of peaks to be expected under ideal, local equilibrium conditions. In the present work, the equilibrium theory is generalized, and chemical and mass transport kinetics are taken into consideration. The behavior of the set of disturbances resulting from the initial perturbation will be related to the equilibrium composition of both phases, the stoichiometry of the reactions taking place, the chemical and physical equilibrium functions, and the interphase transfer and intraphase chemical reaction rates. These relations can be put to practical use in obtaining unknown equilibrium and rate information from observation of pulse behavior.The information to be gained is quite general. In particular, no limitation as to number of components or reactions is necessary; the restriction of infinite dilution Gemot Klauser is with the