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Pdf/1/05/111/2238662/Spe-949111-G.Pdf by Guest on 27 September 2021 T.P.2563 DISCUSSION OF THIS AND ALL FOLLOWING TECHNICAL PAPERS IS INVITED Discussion in writing (3 copies) may be sent to the Editor, Journal of Petroleum Technology, 601 Continental Building, Dallas 1, Texas, and will be considered for publication in the Transactions volume Petroleum Development and Technology. Discussion will close December 15, 1949. Any discussion offered thereafter should be in the form of a new paper. THEORETICAL GENERALIZATIONS LEADING TO THE EV ALlJATION OF RELATIVE PERMEABILITY WALTER ROSE, JUNIOR MEMBER AI ME GULF RESEARCH & DEVELOPMENT CO., PITTSBURGH, PENNSYLVANIA Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 ABSTRACT tain references can be cited"""-""·> where a qualitative aware­ ness of their existence and nature is indicated. This condition Theoretical expressions are presented to describe wetting is explainable in terms of the recognized complexity of the and non·wetting phase relative premeability relations. These problem, and in order to provide some basis for theoretical expressions have then been compared with existing published development, the general emphasis has been placed in the past data, the conformance noted being sufficiently good to satisfy on experimental procedures of evaluation, the validity of the requirements of some engineering use. As a consequence, which, in principle, could be confirmed by the analysis of well it may be supposed that relative permeability characteristics and reservoir performance. However, because of the experi­ of porous media now can be inferred from basic core analysis mental difficulties which have been encountered (notably, data, in a manner more convenient (although less direct) difficulties due to the so-called "end effects"), and because of than presently available methods of experimental evaluation. the general unreliability of field performance tests required to study the applicability of the data, relative permeability INTRODUCTION phenomena continue to be incompletely understood and This paper presents a new approach to the problem of described. t relative permeability evaluation. A classification of published In this paper is examined the possibility of predicting discussions of relative permeability concepts which have ap­ relative permeabilities entirely from fundamental considera­ peared in the petroleum literature will show that previously tions. In addition, attention is called to certain important emphasis has been placed on methods of experimental meas­ factors, previously unemphasized in the published literature, urement, on the interpretation of the data so obtained with which now can serve as a basis for the eventual experimental respect to the variable properties of the system, and on the­ confirmation of this and similar theoretical approaches, as oretical and practical considerations which relate relative well as the experimental solution of the problem in general. permeability to gross fluid behavior in petroleum reservoirs. Therefore, the analysis presented herein has the dual purpose Essentially, no detailed examination has appeared which both of orienting future experimental activity and also of pro­ treats the fundamental factors controlling the quantitative viding an immediate solution for the relative permeability features of the relative permeability relation*, although cer- porblem, sufficiently adequate for some engineering use. It will be seen that the conclusions reached are supported by theoreti­ • Intuitively, at least, we may suspect that the factors controlling the cal and intuitive considerations, and they are not contradicted quantitative features of relative permeability relations (where relative permeability of porous media to given fluid phases is plotted as function insofar as gross features are concerned by existing concepts of the saturation of these phases) will be describable in terms of basic rock texture and in terms of the character of the distribution of the and interpretations of published experimental data, except as immiscible fluids within the interstitial spaces. The term basic rock tex­ specifically noted. ture is used here to refer to the pore configurational ch~racters dependent on factors such as grain sizes and packing a& modified by cementation and other secondary processes. It will be realized that there are other factors Darcy's law, expressed simply as: of importance which in practical instances also will control the quantita­ tive features of the relative permeability relation, such as those related k dp to the various possibilities for variable interaction between the saturatinll q=-­ (1) fluids and the surfaces of the interstitial spaces. Thus, interstitial clays p. dx can be responsible for variable conditions of pore configuration obtaining in a given instance depending upon the nature of the saturating fluids. where q is linear rate of flow (assumed horizontal), k is the Variable conditions of preferential wettability also can obtain expressive of analogous possibilities in ways which will control heterogeneous fluid flow behavior. For the purposes of this paper, however, fixed pore con­ t The poor comparison usually obtained between laboratory and field figurational and wettability characters will be assumed. Moreover, no gas-oil ratios (c.f., Fig. 4 of paper by L. E. Elkins') is illustrative of attempt will be made to establish explicitly a dependence of relative conditions which discourage unqualified acceptance of the results of pre­ permeability on strictly fluid properties. vious studies except as presently useful approximations. In this connection, Manuscript received at Petroleum Branch office April 28, 1948. Pre­ Evinger and MuskatO have discussed in detail the discrepancy observed sented at Branch Fall Meeting, October 4-6, 1948. between field well produetivity values and values obtained from the References are given at end of paper. analysis of laboratory flow data. May, 1949 PETROLEUM TRANSACTIONS, AIME 111 T.P.2563 THEORETICAL GENERALIZATIONS LEADING TO THE EVALUATION OF RELATIVE PERMEABILITY permeability constant, f.' is fluid viscosity, and dp/ dx is the holds great promise, at least for engineering use, particularly pressure gradient, defines completely the viscous flow of if examination of newer data as they become available sub­ homogeneous fluids through porous media of uniform cro~s stantiates the general applicability of their correlations. section. The permeability constant, k, then is an expresi'ion of These attempts to explicitly relate permeability functions tu lhe properties of a particular porous body which collectively rock properties can be expressed by the following analytical require fluid flow to be controlled by fluid viscosity and pre;­ reasoning. Permeability can be defined functionally as: sure gradient in the manner indicated by Eq. (1). Now, in k = F(T) (2) media variously saturated with two or more immiscible con­ That is, there are a series of rock properties collectively rep­ tinuous fluid phases Darcy's law will still define the dynamic resented by the symbol T which control permeability accord­ character of flow for each phase separately, if effective perme­ ing to the prevailing physical requirements of the system. In ability constants are properly defined as functions of the general, the function, F (T), will be known in simple systems distributions of the various saturating fluids. This point of (e.g., capillary tube), whereas it will be only partially or iO view follows that accepted by Richards , who considered the imperfectly known in more complex systems (e.g., petroleum presence of one fluid saturating a porous body merely as a reservoir rock). In practice this is a serious limitation to the Downloaded from http://onepetro.org/JPT/article-pdf/1/05/111/2238662/spe-949111-g.pdf by guest on 27 September 2021 factor which controlled (in the same manner as the structure extent that the approximations which are discovered to apply of the porous body itself) the geometry of the paths available nnder one set of conditions become invalid under slightly for flow to another immiscible fluid simultaneously saturating different conditions such that empirical generalization is im­ the porous body. The analysis to follow requires assuming, as possible. By analogy the effective permeability, k" can also be a simplification, steady state flow with the fluid phase satura­ considered as formally defined in terms of other functions of tion distribution uniformly constant during the time interval rock properties, where at least one function of particular of measurement; and in the study of reservoir behavior this importance involves the fluid phase saturation distribution, or: usually can be done rigorously, at least in principle, by select­ k, = Fe(T"p), (3) ing sufficiently small elements for consideration wherein the where p is employed here to denote the saturation distribution macroscopic flow character is nonetheless retained. Under of the fluid phases. The relative permeability, k" then can be such conditions, the observed effective permeabilities become expressed as: significant constants functionally related to the saturation distribution in addition to the other porous body properties ke F,(Te,p) F ( ) (4) which, as noted above, themselves control completely the kr =T= F(T) IV r p dynamic aspects of single-phase (homogeneous fluid) flow. Eq. (4) calls attention to the simplification employed in the In developing the basis for the analysis to follow where present
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