THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING FLUID FLOW AND HEAT TRANSFER IN STRATIFIED SYSTEMS Mar.vi'n- L:'Katz A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan 1960 December', 1960 IP-484

ACKNOWLEDGEMENTS The author wishes to express his appreciation to the following individuals and organizations fQr their contributions to the research which was the basis of this dissertation: Professor Donald L. Katz for his assistance in formulating the problem, and for invaluable guidance throughout the author's academic career. Associate Professor M, Rasin Tek, chairman of the doctoral committee, for his sincere interes$t useful advice, and generous donation of time to discussion of the problems which arose in the course of the research. Assistant Professor Keith H. Coats, for his very generous donation of time for discussion of mrey of the mathematical details of the research. Assistant Professor Bearard A. Galler and Mrs. Shirley Callahan of General Motors for their aid in processing many of the preliminary calculations on the General Motors' ISM 704. The other members of the doctpral committee for their advice, constructive criticism, and encouragement. Professor R. C. F. Bartels of The University of Michigan Mathematics Department and Director of the Computing Center, for his valuable advice and for his aid in securing the use of the IBM 704 computer at the University. Messrs. Cleatis Bolen, Bill Hines, John Wurster, and Peter Severn of the Chemical and Metallurrgical Engineering Department for their help in construction of the experimental apparatus. E. I. DuPont de Nemours and Company for their fellowship grant for the year 1958-59. The American Gas Association for their fellowship, administered through UMRI Project N031, for the year 1959-60 and the fall of 1960. The Lukens Steel Company for donating the clad steel material from which the two-layer models were fabricated. The personnel in the Industry Program of the College of Engineering for their efficient and accurate rendering of the dissertation in its final form. ii

ABSTRACT The flow of fluids in stratified porous systems and the analogous problem of heat conduction in stratified solids have been investigated both from an experimental and a purely mathematical standpoint. The individual layers of the stratified systems studied were assumed to be both homogeneous and isotropic at all interior points. The models used allowed for complete communication between layers, so that interlayer fluid flow occurred in the unsteady state. A total of five heat transfer models were used to obtain experimental data, The primary purpose of obtaining the data was to verify the validity of the mathematical solutions obtained. The two two-layer models which were investigated were fabricated from a clad steel plate in which one quarter inch of type 316L stainless steel was uniformly bonded to one quarter inch of mild steel, The investigation of the problems from a mathematical standpoint was completely independent of the experimental investigation. Solutions were obtained directly from the partial differential equations governing flow in the stratified systems, using the technique of separation of variables. In the process of developing the mathematical theory of stratified systems, considerable work was done on single-layer homogeneous systems. Tables in the literature for dimensionless flux have been extended. Tables of dimensionless pressure (or temperature) distribution are also presented. iv

The mathematical solutions for multi-layer systems occur in the form of double infinite series. Tables of results for various values of the physical parameters have been computed and are presented in the Appendices. These tables permite the calculation of pressure (or temperature) distribution and various fluxes at the producing face in two-layer systems. The IBM 704 digital computer was used to calculate eigenvalues and sum the series solutions obtained. Tables of eigenvalues are presented in the Appendices. The majority of the mathematical work on stratified systems was concerned with two-layer systems, both linear and radial. The means of extending the two-layer solutions to the case of general multi-layer systems is presented. Tables of results were not computed for more than two layers The application of the solutions obtained to problems of fluid flow in petroleum reservoirs is discussed.Seven. example problems are presented to illustrate the use of the tables of results in such problemso Approximation methods are discussed for representing the behavior of rmuluti-; layer systems by a single homogeneous layer having mean physical propertieso V

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS.,..,..... o, o.............. o...o.,,.. ii. ABSTRACT..............................................o o. o o oo. iv LIST OF APPENDICES.....,,0..........,,.......,..... ix LIST OF TABLES.................... o.... e......o o.. 0.. LIST OF FIGURES........... o..e................ o o o.... xii NOMENCLATURE o............. 9 e e o o o o9 a XV I INTRODUCTION....o.....,.o.. o o oo o o.o. o..,... o II LITERATURE REVIEW..............,.,.... 5 III BASIC MATHEATICAL THEORY e.........o..o o... o...o. 8 A. Derivations of the Diffusivity Equation...O........... 8 B. The Necessity of an Unsteady-State Flow Analysis in Reservoir Systems.....,.,.....o..6 o...........oo 16 C. Methods of Solution of Partial Differential Equations............. eo................oo oo......O..a - 17 IV SINGLE-LAYER FLOW; MATHEMATICAL ANALYSIS.,.....oo....... 20 A. Single-Layer Linear FloW...ooooo.o o.o.o aoo..ooooo. 22 1. The Constant Terminal Pressure Case............. o o 22 2. The Constant Terminal Rate Case,................... 27 B. Single-Layer Radial Flow.....o.,...o.ooooo.o..o. 32 1, The Constant Terminal Pressure Case............... 34 a. Pressure Distribution............o........ 34 b. Dimensionless Cumulative Flux,.o.....,..... o o 37 2. The Constant Terminal Rate Case e...... o o o eo 42 C. Calculation of Eigenvalues from Characteristic Equations......... o... o.. o o o.........o oo..... 49 Do Example Calculations for Single-Layer Flow............ 53 1. Example Problem No. 1o..o...o...,.9.o9o..........o 53 2 Example Problem No. 2 o.. o o. o.o.. o.. o o.... 55 3. Example Problem No 3............ o o o....o..o o 58 4. Example Problem No 4......................... 65 5. Example Problem No o 5 o.....,............. 68 V MULTI-LAYER FLOW; MATHEMATICAL ANALYSIS.... o * o.o0, a 72 A Two-Layer Linear Flow; Constant Terminal Pressure..., 72 1. Pressure Distribution, P(xy,).. o..............o. 74 vi

TABLE OF CONTENTS (CONT D) Page 2. Dimensionless Flux at the Producing Face. o..pobo o 89 a. Individual Instantaneous Dimensionless Fluxoo. 89 bo Individual Cumulative Dimensionless Flux....... 93 co Total Cumulative Dimensionless Flux........... 93 Bo Two-Layer Radial Flow; Constant Terminal Pressure..o.oJ 97 1. Pressure Distribution, P(ryQ) o..o o............... 97 2, Dimensionless Flux at the Producing Face.......... oo06 C, Generalization of the Solutions for Two-Layer Systems to More Than Two Layers............o.... o 0 09 Do Convergence of the Analytical Solutions for Two-Layer Flow................oo...........o...... 120 E. Example Calculations......................... 0000000001 23 1. Example Problem 6oo......1...................... 124 2. Example Problem 7............... 12 VI EXPERIMENTAL WORK, MODEL STUDIES.......................00 133 A Design Considerations for Heat Transfer Models........ 136 B, Description of Heat Transfer Models Studied in This Investigation................... o o o o o o o o o o o o o o o. 142 1. Features Common to All Models..o o o o o.. o. o o..o. 143 2. Individual Features of the Models..oo............... 5 a Single-Layer Radial Model o......o o o o o o o 152 bo Single-Layer Radial "Fault.Plane" Model.. 0.... 154 c. Single-Layer Linear Model. O........o....... 155 d. Two-Layer Linear Model o...... o......... o o o o o. 1.60 e. Two-Layer Radial Model..................o o o o o o C Experimental Data... o.......... o...... o...... o..... VII DISCUSSION OF RESULTS............................0000000000177 A. Comparison of Experimental Data and Analytical Results............,,.o o,,,,,,.......... 177 B. Comparison of Two-Layer Flow with Single-Layer Flow.oo 187 1. Comparison of the Form of the Analytical Solutions. OQ o O O O O O O o o o o o o 187 2o Comparison Between Separated and IntercOmmunicating Two-Layer Systems,........... 0 0 0 0 0 O 89 3o Approximation of Multi-Layer Systems by a Single-Layer Homogeneous System Having Mean Physical Properties.......... o........... o.. 192 C, Steady-State and Unsteady-State Permeabilities of Stratified Systems.................. o............ 203 D, The Flow of Gas in Stratified Systems. o...o.ooo..... 204 vii

TABLE OF CONTENTS (CONT D) Page VIII RECOMMENDATIONS FOR FUTURE WORK....o...oo.oo. o o o,... 208 BIBLIOGRAPHY..................o...........o.oooo. 211 APPENDICES,..................o......,,...,................. 215- 3o4 viii

LIST OF APPENDICES APPENDIX Page A. Derivation of the Orthogonality Conditions for the "Y" Equations in Two-Layer Flow..... o.... o... o o..... o. o o 215 B. Eigenvalues, b, for Single-Layer and Multi-Layer Radial Flow, Constant Terminal Pressure.................., 221 C. Eigenvalues, a, for Single-Layer Radial Flow, Constant Terminal Rate.......,. o o o.. o...... o.. o. o *... o. 239 D. Eigenvalues, a, for Two-Layer Linear Flow, Constant Terminal Pressure...0, o,... o O O. O O O o o o. o o. o o.. o. o o 0. e o 250 E, Eigenvalues, a, for Two-Layer Radial Flow, Constant Terminal Pressure................. o o.......................... o........ e o 266 F. Dimensionless Pressure Distribution, P(x,Q), for Single-Layer Linear Flow, Constant Terminal Pressure....... o.....,... 268 G. Dimensionless Pressure Distribution, P(r,Q), for Single-Layer Radial Flow, Constant Terminal Pressure............. 270 H. Dimensionless Pressure Distribution, P(r,O), for Single-Layer Radial Flow, Constant Terminal Rate...................... 275 I. Dimensionless Cumulative Flux, Q(t), for Single-Layer Radial Flow, Constant Terminal Pressure....o..,,.,... o,,...,.... 280 J. Dimensionless Pressure Distribution, P(xy,O), for Two-Layer Linear Flow, Constant Terminal Pressure........O..O.ooOO. 286 K. Dimensionless Pressure Distribution, P(r,yO), for Two-Layer Radial Flow, Constant Terminal Pressure................... 306 L. Individual Instantaneous Dimensionless Fluxes, ql, and q2) for Two-Layer Linear Flow, Constant Terminal Pressure 5...o 3108 M. Individual Instantaneous Dimensionless Fluxes, ql, and q2, for Two-Layer Radial Flow, Constant Terminal Pressure, e.,. o, o 329 N. Total Cumulative Dimensionless Flux, Q(t), for Both Layers, for Two-Layer Linear Flow, Constant Terminal Pressure....oo 338 0. Total Cumulative Dimensionless Flux, Q(t), for Both Layers, for Two-Layer Radial Flow, Constant Terminal Rate o..... o.. 344 ix

LIST OF TABLES Table Page IV-1 Temperature Distribution for Example Problem......... 55 IV-2 Pressure Distribution for Example Problem 3......... 62 IV-3 Change in Reservoir Pore Volume for Example Problem 3 6. 64 IV-4 Dimensionless Pressure Distribution, P(rQ), for Example Problem 4...................................... 66 IV-5 Actual Pressure Distribution, P(rata)), for Example Problem 4..o................................ 68 IV-6 Pressure Distribution for Example Problem 5 <........... 71 VI-1 Basic Experimental Data for the Single-Layer Radial Model............................. o o o o o o... o... o o e o o 166 VI-2 Dimensionless Temperatures for the Single-Layer Radial Model...................... o o o o o o o....... 167 VI-3 Thermocouple Locations for the Single-Layer Radial Fault-Plane Model............. o,........ o O... o,,. o 16 8 VI-4 Basic Experimental Data for the Single-Layer Radial Fault Plane Model........o o....................,o.oD 169 VI-5 Dimensionless Temperatures for the Single-Layer Radial Fault Plane Model..o.... o......oo....o,... 170 VI-6 Basic Experimental Data for the Single-Layer Linear Model... o..o...ooo.,,.o. o o...... o o. o. oooo o. 171 VI-7 Dimensionless Temperatures for the Single-Layer Linear Model..e....,.o o o........o o o......o o o o.. o... 172 VI-8 Basic Experimental Data for the Two-Layer Linear Model..o................oa..........0..a...........o ooooo.73 VI-9 Dimensionless Temperature Distribution for the Two-Layer Linear Model.........ooooo.....o.............oe 174 VI-10 Basic Experimental Data for the Two-Layer Radial Model.. O..o.......................................... o 175 VI-11 Dimensionless Temperatures for the Two-Layer Radial Model................ D.....,0 o., 176 x

LIST OF TABLES (CONT'D) Table Page VII-1 Dimensionless Temperature Distribution for the TwoLayer Linear Model as Calculated from the Analytical Solution.,1.O, o o..o o o.o,0. o o o..o 179 VII-2 Dimensionless Cumulative Fluxes for Separated and. Intercommunicating Two-Layer Systems............. o o o. 19 VII-3 Dimensionless Cumulative Fluxes for the Approximation of a Two-Layer Radial System by a Homogeneous SingleLayer System,................., o...........o oo oo oo 199 VII-4 Dimensionless Pressure Distribution for the Approximation of a Two-Layer Radial System by a Homogeneous Single-Layer Systemo............... O. O O. 0 0 0 0, 0 0 0 o 200 xi

LIST OF FIGURES Figure P I-1 Crossflow in a Two-Layer System................ 2 III-1 Heat Conduction Through a Differential Volume Element.. 10 III-2 Volume Element for Derivation of the Continuity Equation for Fluid Flow....................... 0 10 IV-1 Mathematical Model for Single-Layer Linear Flow....... 23 IV-2 Dimensionless Pressure Distribution for Single-Layer Linear Flow, Constant Terminal Pressure............... 28 IV-3 Mathematical Model for Single-Layer Radial Flow,,3.... 33 IV-4 Functional Properties of a Characteristic Equation.... 51 IV-5 Hypothetical Reservoir for Example Problem 2.......... 56 IV-6 Dimensionless Pressure Distribution, P(r,9)for R =100, for Example Problem 3...............e o o e........e.. 61 IV-7 Aquifer Pressure Profile for Example Problem 3...... 63 IV-8 Dimensionless Pressure Distribution for Example Problem 4............................,......o. 67 V-l Mathematical Model for Two-Layer Linear Flow........ oo 73 V-2 Mathematical Model for Two-Layer Radial Flow.......... 98 V-3 Mathematical Model for Three-Layer Linear Flow.......e 110 V-4 Heat Transfer Model for Example Problem 6........... 125 V-5 Temperature Distribution for Example Problem 6 o..oe..e 125 VI-1 Schematic Drawing of a Heterogeneous Steady-State Flow Model................. 0 0................... 135 VI-2 Pressure Distribution for Heterogeneous SteadyState Flow Model e,.,...........e.g...e.e e e...eo.. 135 xii

LIST OF FIGURES (CONT D.) Fiure Page VI-3 Single-Layer Linear Experimental Model..............0. 144 VI-4 Single-Layer Radial Experimental Model..........eo.,e 1.44 VI-5 Single Layer Radial "Fault-Plane" Experimental Model... 145 VI-6 Two-Layer Linear Experimental Model o.o.......... 146 VI-7 Two-Layer Radial Experimental Model O.OO o....o..O..... 14"7 VI-8 Steam Fitting for Linear Experimental Models....... 148 VI-9 Photographs of the Single-Layer Radial Fault-Plane Experimental Model............. a................. 149 VI-10 Line Diagram of Steam Apparatus for Single-Layer Radial and "Fault-Plane" Experimental Models.. o..,.. 153 VI-11 Sheet Metal "Channel" for Insulating Linear Experimental Models oo...,....a................. 157 VI-12 Line Diagram of Steam Apparatus for Both Linear and the Two-Layer Radial Experimental Models.o....oo. o.... 159 VI-13 Thermocouple Installatio n in the Two-Layer Radial Experimental Model..............o.................... 161 VII-1 Comparison Between Experimental Data and the Analytical Solution for the Single-Layer Radial Model.eo. e...... 180 VII-2 Comparison Between Experimental Data and the Analytical Solution for the Single-Layer Linear Model - No Insulation on Model..........o...V..........0 o* o, 183 VII-3 Comparison Between Experimental Data and the Analytical Solution for the Single-Layer Linear Model - Asbestos, Aluninum Foil, and Glass Wool Insulation, Without Sheet Steel Channel.e o0 e,. 0....0 0 0 0, 0 o e...0.. 0 e.. o 184 xiii

LIST OF FIGURES (CONT'D.) Figure Page VII-4 Comparison Between Experimental Data and the Analytical Solu-tiOn for the Single-Layer Linear Model - Complete Insulation, Including Sheet Steel Channel....................... o..... *.. 185 VII-5 Comparison Between Experimental Data and the Analytical Solution for the Two-Layer Linear Model.1...................................... 186 xiv

NOMENCLATURE Cp Heat capacity, BTU/#~F c Fractional parameter denoting location of the interface in twolayer models. Also used to denote fluid plus formation compressibility in dimensionless time terms. cw Water plus formation compressibility, vol/vol atm or vol/vol psi a Eigenvalues for the separated Y equation in multi-layer flow b Eigenvalues for the separated X equation in both single-layer and multi-layer flow f,g Fractional parameters denoting interface location in three-layer models h Thickness of a layer, centimeters or feet H Total thickness of a multi-layer system, centimeters or feet JoJl Bessel functions, first kind, zeroth and first order respectively La Real length, centimeters or feet L Dimensionless length, La/H k Thermal conductivity, Btu/hr ft2(~F/ft) for heat conduction Permeability, darcys, for fluid flow K Dimensionless ratio of permeabilities and porosities, or therm:.al properties for multi-layer systems N Number of layers in a multi-layer system P Dimensionless pressure or temperature. Also real pressure, psia p Functional parameter for the finite Fourier cosine transformation q Instantaneous flux, vol/sec or Btu/sec Q Cumulative flux, volume or Btu qli g2 Dimensionless instantaneous individual fluxes for two-layer systems Q(t) Dimensionless cumulative total flux for both single-layer and twolayer systems xv

NOMENCLATURE (CONT D ) ra Real radius, centimeters or feet r Dimensionless radius re Exterior radius, centimeters or feet rb Interior radius, centimeters or feet R Dimensionless radius ratio, re/rb s Functional parameter for the LaPlace transformation Ta Real temperature, ~F T Dimensionless temperature ta Real time, as minutes or hours Uvw Velocity components for derivation of the diffusivity equation We Cumulative flux, Btu or cubic feet Xa Ya Real cartesian space coordinates, centimeters or feet x7y Dimensionless cartesian space coordinates YoQYL Bessel functions, second kind, zeroth and first order, respectively Greek Letters da Separation constant; also eigenvalues for the constant termina.l rate case A function of K2 plus the eigenvalues a and b A function of the eigenvalues a and b A dimensionless ratio used in defining Q(t) X Separation constant Porosity, fractional,ui F.lFuid viscosity, centipoise p Density, #/ft3 O Dimensionless time xvi

NOMENCLATURE (CONT D) Subscripts i Denotes the layer of a multi-layer system rmn Running interfers for double series evaluation a Denotes an actual, rather than dimensionless value 1,2 Denote quantities for layers 1 or 2, respectively xvii

I INTRODUCTION The primary purpose of the research which is the basis of the dissertation has been to develop a more quantitative understanding of the flow properties of fluids in underground formations, particularly in aquifers connected with oil and gas reservoirs. Such formations frequently exhibit changes in the physical properties of the porous rock matrix as a function of depth. These inherent inhomogeneities can often be approximated by a "layer-cake" type of model, or other stratified system, Since the flow of heat in solids is governed by the same equation as the flow of slightly compressible liquids (such as water) in porous imedia, the- two problems may be investigated concurrently. Considerable use has been made of this fact, since the experimental models which were studied in this investigation were heat transfer analogies of the fluid flow problems. It should perhaps be explicitly stated that the research reported herein in no instance involves simultaneous heat transfer and fluid flow. The results obtained with respect to heat transfer are simply corollaries of the basic fluid flow problem, resulting from, the identity of the governing flow equations. The primary factor distinguishing the study of stratified systems from that of homogeneous, "single-layer" systems is the phenomenon of flow of fluid (or heat) between adjacent layers of the stratified system due to driving forces (pressure or temperature differences) induced in unsteadystate flow. The phenomenon herein referred to as "inter-layer flow"' or "crossflow" is indicated schematically in Figure I-1. Thus, an extra dimension is necessary in solving for the flow properties of the system. -1

-2{4-, / ^i ) q) 0 CH 0 k / oV H 0 I 0 rq hh Z * w 0

-3For example, the temperature in a solid, homogeneous steel bar which is being heated at one end can be expressed in terms of only one length dimension plus time. However, if the bar is composed of two parallel pieces of metal, intimately bonded at the interface, the solution for temperature distribution must be a function of position with regard to the axis perpendicular to this interface, as well as distance from the end being heated, and time. Problems of flow in stratified systems have been investigated both experimentally and. analytically in this researcho A total of five heat transfer models were used. to obtain experimental data on temperature distribution in the unsteady state. These data were used to verify analytical solutions which were independently obtained from the governing partial differential. equations. The majority of the work which has been done is concerned with two-layer systems. The analytical solutions obtained for the two-layer case have been extended to the more general case of several layers. However, it will be shown that the complexity of the solutions mounts so rapidly with increasing number of layers that solutions of the type obtained are of only theoretical interest for more than three or four layers, since obtaining numerical answers then becomes extremely difficult. All of the results presented in this dissertation are for bounded systems, i.e. for systems in which the length or radius is finiteo It is possible, however, to make use of the solutions presented for solving problems associated with infinite systemso This can be done by considering the time range of the solution for which the exterior bounary of the ursystem does not as yet have a signifi.cant effect on the results.

-4Considerable use has been made of the IBM 704 digital, computer in this investigation. All of the analytical solutions which have been obtained are in the form of infinite Fourier and Fourier-Bessel series. The calculation of numerical results from these solutions by hand is, in nearly all cases, sufficiently tedious as to make this impractical. Using the IBM 704, however, it has proven possible to compute extensive tables for dimensionless pressure distribution (or analogously, temperature distribution) and flux for several important flow models.

II LITERATURE REVIEW To the best of the author's knowledge, there does not now exist in the literature any material directly relating to the analytical treatment of flow in stratified systems where the effect of interlayer flow is considered. There does exist, however, a considerable volume of material relating to the solution of the diffusivity equation in homogeneous systems with various boundary conditions, as well as on experimental model studies relating to both homogeneous and heterogeneous flow systems. The diffusivity equation, which governs both the flow of slightly compressible fluids in porous media and heat conduction in solids, is discussed in some detail in many mathematics textbooks, such as those by Churchill,(89) Wylie, (44) Snedden (35) and Mickley, Sherwood, and Reed.(30) A very comprehensive treatment of solutions of this equation as applied to heat conduction has been given in a book by Carslaw and Jaeger. (5) The paper by Van Everdingen and Hurst(40) in 1949 presented the first comprehensive treatment of the application of solutions of the diffusivity equation to unsteady state radial flow in reservoirs. (302 33) Other authors, such as Muskat(3233) had previously presented analytical solutions which could be applied to certain reservoir problems, but the paper by Van Everdingen and Hurst was certainly a milestone in the field. Since that time, noteworthy papers have also been published by Chatas,(7) Mortada,(31) Coats, (10,11,12) and others. The two basic types of boundary conditions considered in these papers are the "constant terminal pressure" (CTP) case, in which a step function change in pressure is -5

-6applied at some boundary, and the "constant terminal rate" (CTR) case in which a step function change in flux is applied. Dimensionless cumulative flux, Q(t), for the CTP case and dimensionless pressure change at the boundary, P(t), for the CTR case have been tabulated both by Van Everdingen and Hurst and by Chatas. Mortada includes graphs showing pressure distribution for the infinite CTR case in his paper. Coats has shown the application of these results to actual reservoir problems, using field data. (11,12) The work in the above papers is all principally concerned with single-phase isothermal horizontal radial flow of slightly compressible liquids in homogeneous and isotropic porous media. Considerable work.has also been done in recent years on developing analytical solutions for other, more complex flow systems in reservoirs. Solutions are now available for a thick-sand model and a hemispherical flow model.(26) Some solutions to the diffusivity equation in elliptic coordinates have been obtained.(10) Stevens and Thodos have presented some point source solutions.(37) Interference effects between wells or between storage fields situated on a common aquifer have been studied.(31,37,12) Material which has heretofore been published on heterogeneous or anisotropic flow systems in reservoirs has been to a considerable degree based on an approximate, statistical, or semi-empirical approach to the problems. Such is true of the works by Muskat(33) and Law(28). Cardwell and Parsons have published some results for simple cases of heterogeneity,(4) but this author believes that their conclusions with regard to unsteady state behavior are incorrect. Some recent papers(l9'2l)

-7have made use of "influence" - or "resistance-functions" to utilize known field data to predict future reservoir performance without the necessity of postulating a model. A recent paper by Lefkovits, et al, has been directly concerned with the analytical treatment of stratified systemso( -)his paper is a very significant contribution to the knowledge of the behavior of such systemso However, it was assumed in the paper that the various layers of the porous matrix were connected only at the well bore (or analogously at the reservoir-aqu.ifer interface)o Thus, there was no consideration of cross-flow between adjacent layers of the system. This was also true in the earlier paper by Standing, et al. 36 Much of the current knowledge of the behavior of stratified systems has been obtained from model studieso Several informative papers, (1.6) (6) such as those by Ferrell, et aL, (1 Caudle and Witte, Craig, et al.,(13) Gaucher and Lindley(l7) and Van Meurs(41) have recently been published dealing with the use of fluid flow modelso In most cases the models used involved two phase flow. In addition to fluid flow models, some work has recently been done on analog models. Potentiometric models, such as those described by Fatt(l5) are of considerable use in studying steady-state flow, but are not usually applicable to unsteady-state investigationso A much more useful model for transient phenomena is the thermal or heat-transfer model, which has been used to some extent by Crawford and Landrum, et alo (l, 27 Without wishing to deprecate these latter articles in any way, this author must take issue with the idea expressed therein that the use of a heat transfer model. is something basically "new," since the identity of the controlling partial differential equnations for fluid flow and heat transfer has been known for many yearso

III. BASIC MAHEMATEMATICAL TEORY This section of the dissertation is devoted to: 1# Derivation of the diffusivity equation, whi.ch governs both heat conduction in solids and flow of slightly compressible fluids in porous media. 2o Discussion of the necessity of an unsteady-state floow analysis in reservoir systems, 3 A brief discussion of some of the methods available for the solution of partial differential equations such as the diffusivity equation~ Ao Derivations of the Diffusivity Equation The diffusivity equation for heat conduction in solids has been derived in several references, The following derivation is based largely on the work of Jakob(23) and that of Carslaw and Jaeger,(5 but uses the nomenclature of this dissertation where applicable. The basic law of heat conduction is q =k A (III-) It originates from Biot (1804, 1816) but is generally called Fourier's equation, because Fourier (1822) used it as a fundamental equation in his analytic theory of heat(5 The basic equation of heat storage is* q = pCpV (II2).ta. Due to the frequent use of dimensionless variables in the derivations contained in this dissertation, the subscript "a" has been applied to real spatial and time variables, Thus, for example, xa denotes an actual spatial distance, as feet, inches, centimeters, etc.-"-whereas x denotes the corresponding dimensionless variable -8

-9where q is the heat energy stored in unit time in the volume V of a medium of density p and specific heat Cp, when the temperature increases by AT in a time interval Atao Equation (III-1) may also be written in the differential form d.Q kA dt (111-) a where dQ is the heat conducted in the direction xa during the time interval dtao Consider the differential parallelepipedon shown in Figure III-1. Initially assume the conducting solid to be homogeneous and isotropic tiroughout this differential element* A heat balance will be set up which is valid in the differential of time dtat The heat entering from the left will be dQ1 x= -k(dyadZa) xT dta (III-4) and the heat leaving at the right side will be dVx = -k(dyadza) (T + - dxa) dta.a axaa (III-5),3T ~2T x dta - k(YaZa) - + a x) ata Similar equations exist for the heat quantities d.Qy; dQ2,y dQlz and dQ. which are conducted in the directions Ya and zao The heat energy stored in the differential body is dQs = pCp(dxadyadza) T dta (III-6) Tta The total heat energy entering the differential element in time dta is dQl = d.Qlx + dQly + dQ1,z (III7)

-10/z oI g dzo Za yo i -- Xo Figure III-1. Heat Conduction Through a Differential Volume Element. Yo / / / / / / Za zo Figure III-2. Volume Element for Derivation of the Continuity Equation for Fluid Flow.

-11and that leaving is dQ2 - dQ2 + dQ2 + ddQ2, ( III8) Assuming no heat sources or sinks within the differential element, the law of conservation of energy may be written dQ1 - dQ2 + dQs (III9) Substituting for dQ1, dQ2, and dQs and simplifying, PC_ 6T 62T 62T 6.2 = + + (2 (III-lO) k ta a ya a This is the form in which the diffusivity equation for heat.conduction is most usually written. For the steady-state case, where temperature is not a function of time, this reduces to LaPlace's equation 2T a2T a2T a2T -T =, -- + +. = (II-ll) aXa2 aya2 Za22 Now consider the case where thermal conductivity, k, is a function of position~ Then Equation (III5) must be written dQ2 = -(dada)dz k T + L (k dxa dta (II- 12) xa 6xa ax Proceeding as before, the diffusivity equation in this case takes the form pC T k + k T + P ta dXa aXa ya Ya a \- OZa This form of the equation will be used in this dissertation in all cases involving flow in heterogeneous systems, The assumptions made in the derivation of Equation (III-13) were:

-121, The conducting medium is isotropic at all interior points~ 2. There are no heat sources or sinks within the conducting medium. 3. The density, p, and the heat capacity, Cp, Of the conducting medium are independent of both position and temperature. The diffusivity equation for flow of slightly compressible fluids has also been derived by many authors. The following derivation is largely excepted from the Handbook of Natural Gas Engineering. Consider an element of volume such as that shown in Figure II-2, where the fluid density at the center of the element is p. The superficial fluid velocities u, v, and w in the Xa, Ya, and Za directions respectively, at the center of the element will be considered. These superficial velocities are based on the entire cross-sectional flow area without regard to the fractional porosity of the porous medium, The mass rate of flow through any face of this volume element is (velocity) (density) (area). Thus the mass rate of flow in the xa direction into the left face is [ ( ) a- ( Alxa ) j vaAz (irI-14) and the mass rate of flow in the xa direction out of the right face is [u+ (xa. ) ( 2 P) ][a + ( 2)a]a (III-15) The net rate of flow into the volume element in the xa direction is then - AxayaAza [(pu)] (III-16)

-13Similarly, the net mass flow rates into the volume element in the ya and Za directions, respectively, are A't z La v)] -Z-x A ] (III-17) and aa The rate of accumulation of fluid within the volume element, due to a change in density of the fluid within the element, is katyah a (a ) (III-l9) etti th t t a tothaof contin Setting the net input equal to the accumulation gives the equation of continuity, (pu) + (pv) (pw). (III —2o) axa aYa aZa. ta Note that no assumptions with regard to homogeneity and/or isotropy are necessary in the derivation of this equation of continuity, Darcy's law in its original form is written q _ k ^ d (III-21) A p dL In terms of the velocity components, this is equivalent to the three equations k aP k vP k aP u - i i.i; v =- w = (III-22) b xa B aya 1 za The remaining equation necessary in the derivation is an equation of state relating fluid density to pressure. It is at this point that the concept of what constitutes a "slight~ly compressible" liquid becomes important0 In this

-14dissertation, a slightly compressible liquid is defined as any liquid which has a pressure versus density relationship expressible as p = p [1 + c(P - Po)] (III-23) over the pressure range of interest in a given problem. This definition designates water and most crude oils as slightly compressible liquids in nearly all problems associated with flow in uxiderground reservoirs' Combining Equations (III-20), (III-22), and (III-23) gives the diffusivity equation for the flow of slightly compressible fluids in porous media, a / k aP \.a k P a {k dP aP l — - + - ( ) + 1- w ( 1124) a Xa ay Yaa Ya a Za a ta For permeability and fluid viscosity independent of position, this simplifies to 82 + 8 +.+.: Xc., (111-25) axa2 2ya2 aza k ata The identity of the governing equations for heat condauction and flow of slightly compressible fluids in porous media is thus apparent, with of course the exception of different factors appearing in the coefficient of the time termso The assumptions made in deriving the diffusivity equation for fluid flow were: 1. Negligible capillary or gravity effects. 2 Single phase flow of isothermal slightly compressible liquid. 3* Isotropic porous matrix at any interior points 46 Darcy's law applies,

-15The diffusion equation may be written in terms of cylindrical coordinates by a simple transformation of variables of the fozm x. = r cos a a a za r sin o' (III-26) a a 2p a2 2 2 ds = dra+ ra d + dya For radial flow with no gradient in the circumferential direction (i,e.,, aP/a6 = 0), and with homogeneity in the radial direction (ioe,, ak/ara = 0), the diffusion equation thus becomes (, /c^IP 1 P \ sP \ ~k T + ) c (II-27) a a a oya a, a for. fluid flow, and k - _ + _ (k III-28) ( r2 p at.6ra 0ra ra "a a ta for heat conduction. The unsteady state flow equations for gas flow and flow of highly compressible liquiads are derived in the Handxbook of Natural Gas Engineering and will be shown here for comparison in the case of linear flowo Gas Flow: a (III-29) 3Xa2 kP 8ta Flow of highly compressible liquids~ ^, +e f~ f -- C- (:II-30) x 2 + c. k -t a a a whereas for slightly compressible liquid's axa2 k ta

B. The Necessity of An UnsteadyState Flow Analysis in Reservoir Systems Chatas has published a detailed discussion of criteria for the necessity of'using an unsteady-state flow model for reservoir systems,'' A few of the pertinent points from. his paper will be discussed, hereO There are two ratios which serve to indicaethethe degree to which a system will behave in an unsteady-state mranner upon the application of a. pressure disturbance at its boundaries, The first ratio is that of the speed of propagation of disturbances in a reservoir to the maximimn speed of fluid movemenet through the porous matrix Since ethe fonrmer of these is the speed of sound in the fluid, this ratio may be expressed as irb(n C(I) rB k(Pe Pb) where B is the bulk modulus of the reservoir fluid, A second informative ratio is that of the rate of change in the fluid-mass content of a resezvaoir caused by pressure variations at its boundaries to the stleady-state mass-flow ability of the reservoirs This ratio may be approximated by re.2c inj (rT di - ~ ~ ~ ~~ —---— 2 —-~,1 i."III-32 2 (Q) (Pe-Pb) pk e 0 where PB is the pressure at a boundary~, This expression serves as a measure of the time required for the readjustmente of the inte.rnnal pressure distribution in a reservoir to a steady-state distribution when pressure'variations occur at the boundaries, For completely incompressible fluids, contained in a completely incompressible porous matrix, the ratio (ITI531) becomes infinitely great,

-17whereas that of (III-52) vanishes, In this case, steady-state flow always prevails, with the applicable flow equation for radial flow being a2P 1 aP 2 + - = O (III-33) ra a ra r This is true because in incompressible fluids the transmission of pressure variations at a boundary and the readjustment of internal pressure distributions occur instantaneously and without attenuation, For actual reservoir fluids such as water or crude oil, the magnitude of the ratio (III-53) is very large That is, boundary disturbances are transmitted many thousands of times faster than fluid can flow in the system. However, the large extent of these underground reservoirs usually causes the ratio (III-32) also to be quite large4 Thus, steady-state flow no longer obtains, and an unsteady-state flow analysis is necessary. The effect of various parameters on the ratios of (III-31) and (III-32) is of interest. Increasing the extent of the system (re), the porosity (0) or the fluid compressibility (c) all tend to increase the time required for the system to reach steady-state, Increasing the permeability k, on the other hand, tends to make the system approach steady-state more quickly. C. Methods of Solution of Partial Differential Equations The author certainly does not plan to present here a comprehensive treatise on means of solving partial differential equations. However, some justification of the mathematical methods of attack used on the problems in this research seems to be in order.

-18There are two basic mathematical means of attack on partial differential equations The first involves setting up a grid or network for the problem and using finite-dlifference techniques to obtain a solution.. This method has been facilitated by the recent availability of comparatively high speed computing machinery such as the IBM 704. The justification for not using such methods in this investigation is largely based on the number of parameters involved in the problems0 It is entirely possible that for a given set of physical parameters a finite-difference solution would be more advantageous, However, the general solution to the problem, quantitatively showing the functional relationship of the many different physical parametric groups, would require the Solution of many different grid net-works, It was therefore decided at the beginning of the research that an analytic solution to the problems, if obtainable, would be preferred. The second basic mathematical method of attack involves conversion of the partial differential equation into ordinary differential equation(s), This can be most easily accomplished using either some transformation technique (LaPlace transformations, Fourier transfornations, etc.) or the classical method of separation of variables, As previously noted, Van Everdingen and Hurst(40) and other authors have principally used transm formation methods of solution, in particular the LaPlace transformation~ The question of whether or not problems exist which can be solved by separation of variables but cannot be solved. using transformation techniques, and vice versa, is of considerable interest. The author cannot cite any examples of such problems. It is certainly true, however, that many problems are more amenable to attack by one means Lthan the others

-19In the case of single layer flow in homogeneous, isotropic, bounded systems, there appears to be little basis to choose between the method of separation of variables and that using the LaPlace transformation, The method of separation of variables is somewhat more straightforward, but the LaPlace transformation solution has the advantage of presenting a clearer picture of the behavior of the system at vezy large and very small values of dimensionless time0 In the case of flow in stratified systems, this author believes the method of separation of variables to be superior. A solution of the problems was attempted. using a LaPlace transformation with respect to time followed by finite Fourier trigonometric transformations with respect to the space variables Sufficient difficulties were encountered, particularly with regard to introduction of the interface conditions, to cause the abandonment of this method at an early stage4, The author does, however, believe that a solution based on some type of transformation methhod should be possible, The method of separation of variables has been used in nearly all cases in this derivation (the single:exception being the solution for singlelayer linear flow of liquids in the constant terminal rate mode). For a detailed description of the method of separation of variables the reader is referred to the books by Churchill(8'9) and by Mickley, Sherwood, and Reed.(30) For a comparison of methods on the same problem, the reader may compare the solutions in this dissertation for radial flow in pingle-layer systems to those preseted by Van Erein and Hst(4o) systems to those presented by Van Everdingen and HT:.rst4

IV SINGLE-LAYER FLOW; MATHEMATICAL ANALYSIS In the course of the investigation of flow in stratified systems, a considerable amount of work has been done on single layer homogeneous systems. The reasons for this work are three-fold. The primary reason was the necessity of having a complete set of tables for singlelayer flow in order to adequately compare the results with those obtained. in stratified systems. Although most of the problems presented in this section have been previously investigated by other authors, notably Van Everdingen and Hurst,(4) the tables which are available in the current literature are not sufficiently complete, especially with regard to pressure (or temperature) distribution, to enable an accurate comparison to be made. The second reason for the work was to check the results obtained from the single-layer experimental models studied in this investigation. Again, the tables in the literature are not sufficiently complete for an accurate check. A third reason concerns the means by which mathematical solutions may be obtained. The solutions obtained by Van Everdingen and Hurst(4) which were amplified and extended by Chatas, (7) Mortada, (31) and others, were all based on the use of the La Place Transformation. It was decided to use the classical method of separation of variables in attacking problems of flow in stratified systems in the research leading to this dissertation. Thus the work reported herein on single layer flow, which was also based largely on the technique of separation of variables, served as a mathematical foundation for the more complex work on stratified systems. -20

-21The work reported herein is in two categorieso First the mathematical models used will be described and the solutions presented. Then the means of using some of the results will be explained using example calculations. The tables of results obtained are presented in appendices.

-22 — A. Single-layer Linear Flow Figure IV-I shows schematically the mathematical model used for both the constant terminal pressure and constant terminal rate caseso It is postulated in the following derivations that the model shown in Figure IV-1 is homogeneous and isotropic throughout. Considered as a liquid-flow system, the additional specifications are the assumptions used in deriving the diffusivity equation, namely: 1. Negligible capillary or gravity effects 2. Single phase flow of an isothermal slightly compressible liquid 3. Darcy's law applies. Considered as a heat transfer model, it is specified that the thermal diffusivity of the conducting solid be independent of temperature. The nomenclature used in the derivations will be that for liquid flow. The solutions will then also be written in terms of heat transfer nomenclature where any significant difference occurs. 1. The Constant Terminal Pressure Case The equation governing the flow in this case is the diffusivity equation for k independent of position, as expressed in cartesian coordinates. This equation was derived in Section III of this dissertation and appears there as equation III-25. 2P 2P a6 P Arc aP (IV 1) CX' a2 y, + 6ya2 + 2 k ta Since in this case there will be no net flow in the Ya or za directions, the equation reduces to: 82P = [4c'P i C k A+. (IV - 2)

-2-U r4 0 I - H a. NaN

-24Defining dimensionless variables as follows: x a La (IV- 7) k t _ ta (IV- 4)'~cLa the equation may be written 2p 2 ap (IV - 5) v-2 2 9 -The applicable boundary and initial conditions for the constant terminal pressure case are Initial Condition: P (x,o) = 1 (IV - 6) Boundary Conditions: P (o,G) =0 (IV - 7) 6P x (,G)= o (IV - 8) Equations IV-6 and IV-7 serve to introduce the step function in pressure at the x=O (i.e. x:a = O) end of the mathematical model. Equation IV-8 expresses the condition hat the oher end of the model have no flow across it. It is also known that lim P9) e- co (iv - 9) The method of separation of variables will be used as follows. It is first desired to find all solutions of the type P = X(x) T(Q) (IV - 10)

-25which satisfy the equation (IV-5), the boundary conditions (IV-7) and (IV-8), and the limiting condition of (IV-9)o Substituting (IV-10) into (IV-5), X" T = X T' (IV - 11) Therefore X "= - a (IV -12) X T The reader unfamiliar with the basis of the technique of separation of variables is again referred to the references cited(8 9 30) for an explanation of this equation. The resulting ordinary differential equations are thus X" + a2X = 0 (IV - 13) T' + a2T = 0 (IV - 14) For a = 0: X = C1 + C2 x (IV - 15) T = C3 (IV - 16) But from (IV-9, C3 = Oo The case of a = o thus contributes no terms to the solution. For a f o: X = CC cos ax + C5 sin ax T C6 e2 From (IV-7), X (o) = O, so that C4 = 0 From (IV-8), X' (1) =0 = Cg cos a (IV- 17 ) But C5 4 0 or a trivial solution results. The case of a = 0 has already been considered,

-26Therefore cos a = 0 (IV-18) This equation is of the type to be herein referred to as a characteristic equation, and defines eigenvalues as a = (- ) t; m = 1, 2, 3....... (IV-19) Therefore P -X(x) T(e) - C sin ax e-a2 (IV-20) There is no single value of the constant C which will satisfy the initial condition (IV-6) for all values of xo However, since the original differential equation (IV-5) was linear, any sum of solutions of the form of (IV-20) is also a solution. The initial condition may thus be written 00 1 = ) Cm sin ax (IV-21) m=l Multiplying both sides of this equation by sin ax, integrating both sides with respect to x over the range x = 0 to x = 1, and applying the applicable orthogonality condition that o sin au x sin a x dx = 0 for u v (I-22) 0 0 The equation: fr C is found to be m sin ax d 2 cos a -1 em r1i2A r^ <n (IV-23) sin ax dx sin a cosa -a uo~~~ _]

-27or since from (.IV-18), cos a = 0, Cm =2 a The solution is thus 00 sin ax e a9(IV-24) m=l a where. a = ( m ) i; m = 1, 2, 3,...... (IV-19) The orthogonality condition for well-known cases such as this will be stated as such without proof. In some of the more complicated cases to be encountered in multi-layer flow, the derivation of the applicable orthogonality condition will be given. In terms of heat transfer, the dimensionless time, 9, is defined as kta (9 - pCpLa (IV-25) and the solution is again 00 T I=) 2 sin ax e-a2 (IV-26) nS: a A table of the dimensionless pressure (or analogously the dimensionless temperature) for this case, with parameters of x and 9, is presented in Appendix F. These results are also shown graphically in Figure IV-2. 2. The Constant Terminal Rate Case The governing equation for this case is again 32P v u0_ aP (IV-2) a k. a

-280. 8 \ X \ _ \ \ \' \ \ 2 _ _ _ \I _ a 1 g \ | \ \ \ \\ \.8' ~r'-8'V. \__ m XinXm.rX\ e,6' 10' 10 Lo Figure IV-2. Dimensionless Pressure Distribution for Single-Layer Linear Flow, Constant Terminal Pressure. (b~ Linear Flow, Constant Terminal Pressure.

-29This case will be used to illustrate a transformation method of solution. A finite Fourier cosine transformation will be employed on the spatial dimension in conjunction with a LaPlace transformation with respect to timeo Defining dimensionless variables as: kt (IV-27) l T cLa x=t I-) (IV-28) equation (IV-2) may be written as 82p = _P ax2 a9 (IV-5) The applicable boundary and initial conditions for the constant terminal rate case are P (x,o) = (IV-29) a (o,e) - -1 (Iv-30) 6P (At,) = (IV-31) Ox Equations (IV-29) and (IV-30) introduce the step function in flux at x = 0, and equation (IV-31) expresses the condition of no flow across the other end of the modelo The operational property of the finite Fourier cosine transformation, as given by Churchill, (8) is Cn {F"(x)} = -n2fc(n) -F'(o) + (-1)n F'(ic) (IV-32) for n=0, 1, 2,..

-30Let P represent the finite Fourier cosine transform of P (x,G) with respect to the variable x. Then dt+ n2 P 1 = ~ (IV-33) Note that the transformation has reduced the partial differential equation (IV-5) to a total differential equation, and has at the same time introduced the boundary conditions (IV-30) and (IV-31) This ordinary differential equation will be solved using the LaPlace trans: formation for purposes of illustration. Let P represent the LaPlace transform of P with respect to dimensionless time G. Then sP + n2P - 1 = 0 (IV-34) s Solving algebraically 1 P = s(s+n2) (Iv-35) This is of the form 1 P = (s-a)(s-b) (I-36) where a = 0 b =-n2 From a table of inverse transformations, (8) the solution for P is thus 1 2G 1 e-2 P n2 (l-e-n ) =n2 - n2 (IV-37) The necessary property for inverting the transform P given by (8) Churchill as 1 2 ~ F(x) = i fc (o0) + t ) fc(n) cos nx n=l (IV-38) for 0 < x < 7t

-31At n = 0 P= L{no} = L{ 2}1 = (IV-39) Therefore P (n,G) = - e G (IV-37) i7 n" n2 P (o,o) = 0 (IV-40) The inverse transformation then yields 00 lt-x)2 n a 2~e~"2 CI oo - n P(x,~) J= (t -x.. cos nx (IV-41) 27i 6 t n= l n2 For the heat transfer case, dimensionless variables are defined as k tat pC L pa = t Xa (IV-28) La and the solution is again (-x) 2 2 n2 T(x,G) = 2nT 6 it T(x,~) = 2i 6- A;;^ 2 cos mex (IV-43) It is of interest to note that this form of the solution is somewhat easier to work with than the solution given by Carslaw and Jaeger, case III on p. 310 of their book. It has been verified that this solution, IV-435 converges rapidly for values of ~ greater than 1.0o Tables of results have not, however, been computed, since many values of interest would unfortunately occur in the range of ~ < lo0.

-32B. Single-Layer Radial Flow Figure IV-3 shows schematically the mathematical model used for both the constant terminal pressure and constant terminal rate cases. The same assumptions with regard to homogeneity, isotropy, etc. will be made in the radial case as in the case of single-layer linear flow. The equation governing the flow in the case of single-layer radial flow is the diffusivity equation for k independent of position, as expressed in radial coordinates. The equation can be obtained directly from equation III-27 by assuming k constant. 62p 1 aP 82p,4c aP 2+ - - + iv-44 ara ra ra aa k ta Since in this case there will be no gradient in the ya direction, the equation reduces to 82p 1 3P _ tc aP;2+ - - = - Et(Iv-45) 672 rar Va ha a a a Defining dimensionless variables as ra ~~~~~~r = av^~ ~(IV-46) R=r (IV-47) rb = ta for fluid flow (IV-48) Scrb kt for heat conduction 9 = ~r 2 PCprb

- 335 I I If Y =I I I I~ I I e A Iz I I Figure IV-3. Mathematical Model for Single-Layer Radial Flow.

-34equation (IV-45) may be written a2p 1 p P P-4 T^- - v-}49) r r r d~ 1. The Constant Terminal Pressure Case The solution for P(r,G) will first be obtained using the technique of separation of variables. The cumulative dimensionless flux across the producing face will then be obtained by appropriate differentiation and integration of this solution. a. Pressure Distribution The boundary and initial conditions for this constant terminal pressure case are Initial Condition: P(r,O) = 1 (IV-50) Boundary Conditions: P(l,Q) = 0 (IV-51) Q(R,O) = 0 (IV-52) It is also known that lim P = 0 9 -* co (IV-53) It is first desired to find all solutions of the type P = X(r) Y(@) (IV-54) which satisfy equations (IV-49), (IV-51), and (IV-52). y" X" 1 X 2 Y = X + = -b2 (IV-55) Y X r X X' + b2 Y = 0 (IV-56) X" + l X' + b2 = 0 (IV-57) The equation (IV-57) is one form of Bessel's equation. For b = 0: Y = C4 (IV-58) X = C5 ln r + C6 (IV-59)

-35But from (IV-53), C4 = 0 Therefore b 0. For b 0: Y = C e.b (IV-60) X = C1 Jo(br) + C2 Yo(br) (IV-61) From (IV-51): x (1) = o = C1 Jo(b) + C2 Yo(b) (IV-62) X' (R)= 0 = -b[C1Jl(bR) + C2Yl(bR)] (IV-63) Since b / 0 C1Jl(bR) + C2Y1(bR) = 0 (IV-64) In order that there be a non-trivial solution, the determinant of the coefficients in (IV-62) and (IV-64) must equal zero. This gives the characteristic equation Jo(b)Y(bR) Yo(b)Jl(bR) - ( (IV-65) which defines an infinite number of real and positive eigenvalues b for each value of R. The means by which eigenvalues were determined from characteristic equations of the type of (IV-65) in this research are discussed in section (IV-C) of this dissertation. From (IV-64) Jl(bR) C = -c Y(bR) (iv-66) Then from (IV-54) 2 P = Ce-b [Jo (br)Y (bR) -J (bR)Y (br)] (IV-67) No single value of the constant C will satisfy the initial condition (IV-50). However, due to the linearity of (IV-49), any sum of solutions

-36of the form of (IV-67) will also be a solution. The equation for the initial condition may thus be written 00 1 =- Cm [Jo(br)Y1(bR)-J1(bR)Yo(br)] (IV-68) m=l Define U(br) = JO(br)Yl(bR) - Yo(br)Jl(bR) (IV-69) Then multiplying both sides of equation (IV-68) by rU(br), integrating over the range r = 1 to r = R, and applying the applicable orthogonality condition that 1R rUm(br (br(br) dr = 0 for m n n (IV-70) we obtain fR rU (br) dr - =_.. (IV-71) flRrU (br) dr From Muskat,(32) page 631: l rU (br) dr = - [r 1 (IV-72) I1 rU2(br) dr 1 U Rl rU2(br) dr = 1r u2' + r I }I (IV-73) < -2 > U (bR)= rbR (IV-74) Also, by direct differentiation -b[ Jl(br)Y (bR)-Yl(br)Jl(bR)] (IV-75) 3u U (bR) 0 (IV-76) From (IV-65) U (b) = 0 (Iv-77)

-37Define V(br) Jl(br)Yl(bR) - Yl(br)Jl(bR) (IV-78) Then aU(br) -bV(br) (IV-79) ror and V(bR) _ 0 (IV-80) Substituting into (IV-71) -1 Cm -72 [bV(b)] The solution is thus bG /(IV-81) 2bV(b) 00 P(r,) = Cm U(br)eb (IV-82) m=l where Cm is given by (IV-81) and (IV-78), and U(br) is defined by (IV-69). Equation (IV-82) was used to compute the tables of 1(r,G) appearing in Appendix G. b. Dimensionless Cumulative Flux A dimensionless cumulative flux across the producing face (at ra=rb, r=l) may be defined as Qh) s t (dap di (IV-83) This is the definition used by Van Everdingen and Hurst. (4)

-38( -)r=l {Z Cme y [ia trij Lm=l J r=l VX00b ~ ~ eb@~ ~(IV-84) = _ 2~-b2G = b Cm V(b)e m=l m=l Q(t) = E b C ) m=l 00 X=- Cm V(b) 1-e (IV-85) m=l Equation (IV-85) may be conveniently separated into two series, the first independent of time the second time idependento 00 - V(b) m=l 00 + C -b2e (IV-86) m=l b V(b)e This solution is perfectly valid} and was actually used to compute several values of Q(t). Unfortunately~ however, the time independent series converges extremely slowly, requiring of the order of 500 terms to attain four-place accuracy. In this particular instance, the technique of the LaPlace Transformation proves to be superior~ Van Everdingen and Hurst have used this method to obtain the solution

-39 Q(t) - ) e-bJ202(bR) (IV-87) V ~,2 b2[Jo2(b) - J12(bR) It can easily be shown that the Equations (IV-85), (IV-86) and (IV-87) are mathematically identical. In both solutions the eigenvalues, b, are obtained from the characteristic equation Jo(b)Yl(bR) - Yo(b)Jl(bR) = 0 (Iv-65) Rearranging Yl(bR) = Jl(bR)Yo(b) (IV-88) Jo(b) V(b) = J(b)Yl(bR) - Jl(bR)Y1(b) _ J(bR) [Jl(b)Yo(b) - Yl(b)Jo(b)] Jo(b) J1(bR) 2 1 (bR) (2 (IV-89) Jo(b) \ b Substituting into (IV-85) (t) -2 bD -e - bV2(b) zz1 SC2 Z b [le-b2Vb - (b) / [ 2(bR) 5,b LJ2(b) 4 = I. J^(b) 2 - JL(bs1JJ 2 [l-e-b2Q ] J2 (bR)'2 2. 2 2 2 t b2[J12(bR) - J2(b)] m-l e 2 z J12(bR) W2/, b[J 2(b) - JT1(bR)] hi f b2 [Jo2 ( ) _ 2 (bR) -b2 [J0(bbR) (Iv-9o) z b22)- J12(bR)] kfal |

-40The solutions are thus equivalent if R2-1 2 j 12 (bR) (IV-91) T =2 7.(iv-91) 2 L, b2[Jo2(b) J12 (bR) ] From (IV-68) 1 = CmU (br) - b 2 2V2(b) L U(br) (IV-92) b 2v2(b) - (4T) Substituting for V(b) from (IV-89) 2b J1 (bR) 2 b - --- WS 1 2b -— Job- ~ U(br) b2 J12 (bR) 4 4 J 2(b) I2b2 rt2 Jo (b) el (bR) = JU(b)Jl(bR (b) (IV-93' [J12(bR) Jo2(b) (br) L^ = / Multiplying both sides by r and integrating with respect to r between 1 and R: rdr Cm j rU(br)dr -R l7'0[l~b7- - ar ]i I (Iv-94) 2 = ~ m rJ~r J f-^ b|;vvlYMrl

-4100 =r SX Cm [- L V(b)] m=l 00 Jo(b)Jl(bR)V(b) m=l b [J (b) - J(bR)] (Iv-95) 00 R2 _, = J2(bR) J1(bR) 2 2 ) m=1 b 2r2 (b) J1 (bR)] Jo(b) b -96) (IV-96) 2m b[J2(b) J1(bR) This is the same as equation (IV-91)o The two solutions are thus mathematically equivalent. It is evident that the solution obtained using separation of variables utilizes a very slowly convergent series to represent the -2l\ simple algebraic quantity k -2-. The LaPlace Transformation solution represented by equation (IV-87) was therefore used to compute the table of results presented in Appendix I, using the IBM 704 computer. These tables are an extension of the tables presented by Van Everdingen and Hurst, who presented values of Q(t) up to R = 10. Appendix I presents tables of Q(t) from R = 10 through R = 1000. Comparison with Van Everdingen and Hurst's tables is possible at R = 10. The maximum difference between tables is 0.2%. This slight error is due to two factors. Firstly, the first eigenvalue, bl, used by Van Everdingen and Hurst is slightly in error (0.1104 should be 0.110269). Secondly, and more important, Van Everdingen and Hurst used only two terms in the series for all values of dimensionless time, whereas at low values of 0 (where the discrepancy between tables occurs)

-42it was actually found necessary to use up to six terms to obtain fourplace accuracy. 2, The Constant Terminal Rate Case The governing flow equation is again. 2P 1 P P (Iv-49) ar The applicable boundary and initial conditions for the case of constant terminal rate are: Initial condition: P(r,o) = O (IV-97) 6P Boundary conditions: r(1,G) = -1 (IV-98) aP(R, ) = 0 (IV-99) or A variation of the usual technique of separation of variables will be used for this case. Both a product-type and a sum-type of separated solution will be postulated. First, consider all solutions of the form P = X(r)Y(G) (Iwhich satisfy (IV-49), (IV-98) and (IV-99). Then YI X" 1 X' = X + x = - 2- (IV-101) The resultant ordinary differential equations are y + c2y 0 (IV-102) X" + + X' + X = 0 (IV-103) r For Q0O: 2 Y = C3e - G (IV-104) X = C1JJo(cr) + C2Y0o(r) (IV-lo5) p = ea Clj(o) + C2YO(or)] (IV-106)

-43For = 0: Y = c4 (Iv-o07) X = Q5 Qn r + C6 (IV-108) P C3 C n r + C4 (IV-109) Now, consider all solutions of the form P X(r) + Y(G) which satisfy (IV-49), (IV-98), and V[V-99), For this case, Y' X=:" + - = (IV-llo) r 7 Solving Y = C7G (IV-11l) 2 x= C7 r (IV-112) P = C ( + ) = C(r2 + 4'9) (IV-113) Since a sum of these solutions will also be a solution, let 2 P = e [C1Jo((cr) + C2Y,(car)] (IV-114) + C3in r +. C4 + C (r2+4q) Differentiating, aP = -+o[C1J1(oCr) + C2Y1(CTr) Je: (IV-115) + C3 + 2C5r r From (IV-98) ()l = -1 = -a[ClJl(a) + C2Yl(c) ]e-2 (IV-116) + C3 + 2C5

-44From (IV-99) -a2e (r)r=R = -[ClJl(o ) + C2Yl(R) ]e(IV-117) C+ R + 2C5R Since equations (IV-116) and (IV-117) must be valid for all values of G9 ClJl () + C2Yl(a) = 0 (IV-.18) CiJ1(ciR) + C2Yl(CR) =0 (IV-119) In order that there be a non-trivial solution, the determinant of the coefficients must equal zero. This defines the characteristic equation J1 (a)Y1(aR) - Y1(a) J1 (aR) = 0 (IV-120) from which an infinite number of eigenvalues a, all real and positive, may be determined for a given value of R. The criteria of (IV-116) and (IV-117) being valid for all values of.9 also necessitates C3 + 2C = -1 (IV-l2.1) 3 + 2C5R = O (IV-12) R - Therefore -R2 C -- 3 - R21 (IV-123) C = 5 2(R2-1) (IV-124) Since the initial pressure was set arbitrarily, the additive constant, C4, may be set equal to zero. From (IV-118) Jl(ca) c~ = -% v-Trs (IV-125)

-45Therefore [CiJo(ar) + C2Y0(oxr)] [ClJO((cr) -C1 J(C) Yo(o I)] Yl ( ) (IV-126) C[JCo(m)Yl(a) - Jl(a)Yo(ar)] Define U(oW) = Joo()Yl(a) - Ji(O)y o(1) (IV-127) Substituting into (IV-114), P =CU(C) e-a0 R9 P' = a~in r'[ 2 R2 1 ) (IV-128) There is no single value of the constant C which will satisfy the initial condition (IV-97) for all values of r. However, the initial condition can be represented by 00 P(r,o) = 0 = C3 in r + C5r2 + CnU(cr) (IV-129) n=l Multiplying both sides of equation (IV-129) by rU(oar), integrating with respect to r from 1 to R, and applying the orthogonality condition (IV-70), rR 2 pR C rU (Oc)r = -C3 rU(cx) inr dr R (IV-130) -C< r3u(or)dr It is evident that (au) = (IV-131) kr/ r=4

-46From (IV-120) (asu = ~0 (IV-132) \rr= R From Muskat(32), p. 631: U (a) -2 ((IV-133) Itc Evaluiating the integrals: r2U(or)dr = ~ r2U + =2 r - RB2U2(ceR) - 2 (IV-134) r 2 |r 2a rin r U(o=r)dr =- 2 U-r r nr = U ) + 3 (I-135) r1 6u d r - a2 3 r3U(car)dr = - 2 + + (IV-136),-l /'... O? cir O2' a4 7 | 2R2U(R) + 4 a2 jC3 uR2 U(CR) 2 - - 1 2R2u(dlR) 41-R2.O a. 2(R2-1) 2 U3 Cn [ R22a2U2(a ) 4 1 2t+2a2... 4Tc o[{R2n2c 2U2(cR) - (IV-137) From (IV-120), rearranging, Jl(a) Y1 (ceR) Yl(a) - (Iv-138) J -,(ceR

-47Substituting into (IV-127) U(oR) = Jo(OR) Y1 (a) - Ji (ca) Y (cR) =Jl(aR [ o((R) Yl(oR) - Yo(aR)Jl(aR)] Jw ) (] (IV-139) Substituting (IV-139) into (IV-137), 4rc n a AtB2a2 4kJl2(Qa) 4 EJ IC) (M) L, ~~~2a2 V^R:Pj24x (aR) -I (iv-140o) itJl2 oiR) a[Jl2(a) - J2(aR)] Then it should be true that 2 2 -a2r ~r,)- in (r + (Gv-141) P(r,() CnU(r (IV-141) n=l But this equation does not satisfy the initial condition (IV-97). It can be shown that letting P(rO) = B. carrying out the previous analysis, then setting B = 0, results in the same solution, since R B 1 rU(or)dr = 0 (IV-142) J1 In order to fit the initial condition, a material balance will be employed. -k ap ft3 q = 2hrb () rb se 23thk ft3 sec (IV-143)

-48Q.' q dGe = crb dG k o = 2.hcrb2G ft3 (IV-144) W = 2ithcrb2pG lbs. (IV-145) At any time G it must also be true that R W =1 P 2ith(crb prdr. R = 2Srhocrb2p 1 rPdr (iv-146) Consider large G, where e 0 G = 0. Assume that one additional tern, which must be independent of G, is necessary in the solution (IV-141), and let this term be represented by "b". Then W = 2jh(ocrb p LR[ 3 2r R2rlnr + brl dr = 2sthcrb2pG ('1J7-147) rR | r3 2 t 2br + ~2-` )- ln 1 dr +21jhGcrb p 1 2(B-l)) Then, from (IV-145), J br +- -2 ) (R2-1) dr= 0 (IV-148) b (R -) + 1 (R -1) - R2 lR 1 (R21)1 2 (R_-1) R -1 2 or b = 4R41nR - 3R4 + R2 + 1 (IV149) 4 (R2-1)

-49The final solution is then P r2 + 4 R21n r 2(R -1) R -J + (1 + 2R2 +:4RnR - 3R4) r(R2_1)2 i+ t eC (oR) |J(a)Y(ar) - Y (a)J(oar) n=i a [J2 (R) - -J(a)] (IV-150) This is the same solution which Van Everdingen and Hurst derived using LaPlace Transforms. It should be noted that in their paper, (4) in equations (VII-17) and (VII-18), a misprint occurs in the numerator of the Fourier-Bessel series (the term Jl(PnR) ). Using equation (IV-150), tables of dimensionless P(r, G) for the constant terminal rate case have been computed on an IBM 704. These tables appear in Appendix H. Provision was made in the computer program to use up to 100 terms in the series. It was found that, in nearly all cases, less than 10 terms were actually required to obtain the desired accuracy. Appendix H presents P(r,G) for the constant terminal rate case for values of R from 1.5 to 100. It should be noted that P(l, G) in Appendix H corresponds to the dimensionless P t)defined by Van Everdingen and Hurst. Comparison with their tables indicates excellent agreement between P(1,G) and P(t) in all cases where comparison is possible. C, Calculation of Eigenvalues from Characteristic Equations The simplest types of characteristic equations, such as cos a = 0 (IV-18)

-50can be solved by inspection to obtain the eigenvalues a - ); m = 1,2,3,... (IV-19) In the more complicated cases, such as Jo(b) Y1 (bR) - Yo(b) Jl(bR) (IV-65) the interrelation between terms of the equation is such as to make a solution by inspection impossible. It is this second type of characteristic equation with which this discussion is concerned. The eigenvalues which are desired occur as roots of the characteristic equation, i.e. when the value of the function does indeed equal zero. This is shown graphically in Figure IV-4, in which the value of the function F = Jo(b) Y1 (bR) - Yo(b) J1 (bR) (IV-l51) is plotted versus the variable b, for R = 10o'The method used to obtain eigenvalues from all of the characteristic equations in this dissertation (except of course (IV-18) wh.ch can be solved by inspection) was the "half-interval" method. This is an interative type of method, in which successively better approximations to the eigenvalue are made until the desired accuracy is obtained. An IBM 704 digital computer was used to obtain final values in all cases. Some preliminary work was done on a Bendix G-15 computer. In the half-interval method, the search for roots of the characteristic equation is initiated by specifying some initial point for the variable (b=O in the case of (IV-65)) and some initial search increment. The increment is successively added to the current value of the variable until the function changes in sign. At this point, the interval

-51F =J(b)Y,(bR)-Yo(b)J,(bR) I I I I I I t + + + + + ii+ N 0-i in ED.N N 0 N aN Oi aD Ii Figure IV-4. Functional Properties o a Characteristic Equation. a m bo - - - - - - - - - - - ezz^^:-=-~~~~~~~~~~~~~~~~~E ______ ~~~~~~~~~~~~~~~~~~i_~__ i * to -- --- -- --- -- --- -- -^ -- - g - ----- -- / I ^ -- ----------- J - -- -- -- -- - Fiue V4.Fncinl rpeteso Cact risi qain

-52in which the change in sign occurs is successively "halved" or divided by two, always retaining that half in which the change in sign occurs. When the size of the interval has decreased to the limits on accuracy desired, the eigenvalue is taken as the midpoint of this interval. The procedure of adding the search increment is then repeated until the next change in sign occurs, etc. The search increment is frequently modified during the course of the search, once a few eigenvalues have been obtained. In all cases, the eigenvalues obtained in this investigation were tested to at least six significant figures. These eigenvalues are presented in Appendices B. C, D, and E. Since the half-interval method is a relatively slow method for determining the eigenvalues, consideration was given to using a more rapid method, such as Newton's method. It was decided, however, that the relative simplicity of the half-interval method, which allowed the use of simple computer programs, minimized the possibility of error sufficiently to compensate for its deficiency in speed. In particular, the half interval method, properly used, makes it virtually impossible to "skip" an eigenvalue. This feature was of particular importance in the case of two-layer linear flow, in which the distance between eigenvalues was not monotone decreasing, but varied in a somewhat random manner.

-53D. Example Calculations for Single-Layer Flow In this section, the use of the tables of results presented in the appendices to this dissertation'will be illustrated, for the case of single layer flow. The tables to be used are those in Appendices F, G, H, and I. A total of five example problems will be used here for illustration. The use of Appendix F will be illustrated by both a heat transfer problem and a reservoir flow problem in order to emphasie.,e thel, d.ua.:l usefulness of such tableso The use of Appendices G and I will. be illustrated by a reservoir flow problem involving the constant terminal pressure case. An aquifer storage reservoir problem will illustrate the use of Appendix H. A final problem will then illustrate how the tables, in particular those of Appendix G, may be used to obtain solutions for problems involving "infinite" aquifers. The example problems in this dissertation have in all cases been kept as simple as possible. The problems are intended solely to illustrate the use of the tables in the Appendiceso For this reason, problems involving the use of the superposition principle have not been included. The use of the superposition principle is necessary for example, when a varying pressure at thegas-water interface in a gas storage reservoir-aquifer system is known as a function of time. The reader is referred to works by Coats(' ( 2 ) and others(24) for a more complete treatment of reservoir flow problemso 1. Example.Problem No, 1 Consider a steel bar which is initially at a uniform temperature of 70 Fo The pertinent dimensions and physical properties of the

-54bar are known to be Length, L = 20 inches Width, wa = 1 inch Height, ha = 1 inch Thermal conductivity, k = 26 Btu/hr ft 2(~F/ft) Density, = 490 #/ft3 Heat Capacity, Cp = 0.12 Btu/ F At time zero, one end of this bar (xa 0) is rapidly heated to 270uF, and is maintained at this temperature thereaftero It is desired to calculate the temperature distribution in the bar after one hour, assuming that there are no heat losses from the bare This problem is a typical example for which the "constant terminal pressure" case tables may be used for heat transfero Since heat flow will be in only one direction, the width, wa and height, ha, are irrelevant, and the proper table is that in Appendix Fo The results in Appendix F are shown graphically in Figure IV-2o Since Appendix F involves only dimensionless quantities, it is necessary to first dedimensionalize the known physical parame~terso The applicable equations for this dedimensionalization are given on the first page of Appendix F as: xa Xa x = vL (IV-152) kta 9= a PCpLa (IV-153) At a time of one hour, in this problem (26 ) (1) Btu ft _hr ft3 oF = T490)(0ol2)2 hr t2 oF Btu

-550.1592 (dimensionless) (IV-154) The length conversion for this problem is x, = 20x inches (IV-155) The conversion from dimensionless temperature to actual temperature in this problem is given by Ta = 270 - (270-70)T a = 270 - 200T ~F (IV-156) Reading values from Figure IV-2 at G = 0,1592, or from Appendix F (interpolating with respect to G) the values in Table IV-1 may be obtained. This table shows the desired temperature distribution after one hour. Table IV - 1 Temperature Distribution for Example Problem 1 x Xa T Ta (inches) (OF).05 1.069 256.2.1 2.137 242.6.15 3.203 3 29.4.2 4.268 216.4.3 6.391 191.8.4 8.520 166.0.5 10.617 146.6.6 12.700 130.0.7 14.765 117.0.8 16.815 107.0 9 18.836 102.8 1.0 20.849 100.2 2. Example Problem 2 Consider a gas reservoir which is bounded on three sides by faulting planes, such as that shown in Figure IV-5- Such a reservoiraquifer system would exhibit essentially linear flow in the aquifer. The pertinent dimensions and physical properties of the aquifer are known to be

-56L \ k \i 0 \ a ~ I <U W QL 4\ ~ ujCI~~ \ 2 4o r) U)!<~ W \ 3 0 \r

-57Linear Extent of the aquifer, L = 30 miles a. Permeability, k 2000 millidarcies Porosity, = 20% Water viscosity, = 1.0 centipoise Water plus formation compressibility, c = 7 X 10 vol/vol psi If the pressure throughout the gas reservoir is suddenly dropped by 100 psi, and maintained at the lower pressure, it is desired to find the time at which an observation well six miles from the gas-water interface (i.e. xa = 6 miles) will indicate a pressure decline of 10 psi in the aquifer. Use will again be made of Figure IV-2 and/or Appendix F, for single-layer linear flow. The applicable equations for dedimensionalization of the physical parameters are in this case. Xa x La (IV-157) kt G = a 40cLa (IV-158) A compatible set of units to make ~ dimensionless are: k = 2 darcys 4 = 1 centipoise = 0.2 (dimensionless) c = 7 x 106 x 147 vol/vol atm La = 30 x 5280 x 30.48 cm ta = time, seconds The dimensionless distance of the observation well is x _ = 6 miles =0.2 (IV-159) La 30 miles

-58To find the time desired, the dimensionless time at which P(0.2,Q) = / 10\ 1- 0100 = 0.9 is simply converted back to real time using equation IV-158. From Figure IV-2 or Appendix F, this value of G is 0.0074. The time in question is thus ta cLa0 k = (l)(O.2)(7xlO 6x14.7)(30x5280x30.48)2 (.0074) (2) = 1.772 x 106 seconds 1.772x106 20.5 days (Iv-6o) (3600) (24) - 205 days It should be noted that this analysis implicitly assumes a stationary gas-water interface, whereas in actuality the interface would shift somewhat due to water encroachment into the reservoir. 3. Example Problem 3 Consider a circular gas reservoir surrounded on all sides by a bounded uniform aquifer, with dimensions and physical properties as follow h = 100 feet, thickness of aquifer formation rb = 2000 feet, radius at gas-water interface = 0.8 centipoise, water viscosity = 0.1 porosity -6 c = 7 x 10 vol/vol psi water plus formation compressibility k = 1000 millidarcy permeability R = 100 so that the exterior boundary radius, re, of the aquifer is 200,000 feet, or about 38 miles

-59If this gas reservoir is maintained at 100 psi above its initial pressure, it is desired to find the time at which the effect of the exterior aquifer Qbundary becomes significant, and to determine the pressure profile in the aquifer at and after this time. It is also desired to find the increase in the pore volume of the reservoir at and after the effect of the exterior aquifer boundary has become significant, The system in this case should be approximated by the singlelayer radial flow mathematical model, The pressure profile will therefore be obtained using the tables in Appendix G. It is necessary to assune a stationary gas-water interface in determining the pressure profile. The increase in the reservoir pore volume can be found using the tables in Appendix I. The relation between actual and dimensionless time must first be calculated kta t a kta2 (IV-48) k 1 darcy -6 2 Srcrb (0.8cp)(0.1)(7xlO x 14.7 vol/vol atm)(2000x30,48cm) = 3.26 x 10-5 sec1 (-6) Now 1 year = (365)(24)(3600) = 3.155x107 sec Therefore k r = (3.26xl05) (3.155x107) 1030 year"1 Seb Then 9 = 1030ta for ta in years (IV-162) From the tables in Appendix G, at R = 100, the dimensionless pressure begins to deviate significantly from 1.0 at about 9 = 103.

-60Thus, the time at which the effect of the exterior boundary becomes significant is approximately one year. Also, using the table at R =100, the values of P(r, G) may be determined for succeeding values of time. These values are plotted in Figure IV-6. The pressure profile in question is shown in Table IV-2o As an example from this table, the dimensionless pressure P after two years at r= 20 is P = 0.69. Thus, at a point in the aquifer (20)(2000)= 40,000 feet (7.58 miles) from the center of the gas reservoir, the pressure at this time is (1-0o69)(Q10) = 31 psi above the initial pressure. A plot of the aquifer pressure profile for this problem is shown in Figure IV-7. In order to calculate the increase in the pore volume of the reservoir, the total cumulative water efflux will be calculated, using the tables in Appendix I. A convenient formula for calculating the water efflux in field units is2 We = 6 283 Icrb h(Pb-Pf)Q() (IV-163) where: We = cumulative water influx or efflux, cuofto = fractional porosity c = formation plus water compressibility, vol/vol psi h = formation thickness, ft. rb = inner boundary radius fto Pb = inner boundary pressure, psia Pf = initial aquifer pressure, psia

-610 cf ~ E~ I% eH / " III0 C) -P O 9) c; m I I In I Cs ~ O, (%) I, /D ) ~ ____:I T I l' o / 0 CJ I) 0 0 ajcnssaj: 0 0 0 0 ~~'a~n~seJ~~ SSGTU0TSU~~tJ2T/3[

-62o0 O co oC - o P-, *r e * * o c I m) 0 O\ L L-\ 0 P- P 0 At \ H H C k c5 CQl) H 4UmCD rc\ L( _ C O ON rd 0 L —- (1\ O H C C o O OL P LCN LC\ \N H Cd a) *) - (j m — cu.. O LC N Ln 0 O V2 PH H O C0 L Hr P P- L - Ln 0 H a) crj E 0 EH Q CU CD rl O P-l *4 L \ 0 -P H CD H C 00 02 LN 0- u 0 a) C a 0. in O IO Ln CU n 1 D -P * OrH ti A L C C ) m Xr I U] H +> ) 0 M <+-i 0 O^ 0 0 0 0 n (l O O O O O *H (D ) CH Cu I0 C N H C O O a)^~ ~i H Cd 1> C^ L0 t O a H 0000

OD I,_____. I__,S1_._ H,/r --- --- -- --- --- HrH 0 111 - M / U) CH!sd wW~l-d\J!5d_ __ _ _ ff C' - __O~~ __ O~~ NO N K!sd —--

-64The relation between actual and dimensionless time is the same as that used in calculating the pressure profile. The water efflux, W, in this case denotes the increase in pore volume of the reservoir, in cubic feet. This change in volume is shown in Table IV-3, at times of ones two and three years. The relation between Q(t) and We used in this case was We = 6o283(Ool)(7xlO6)(2000)2(l00 )(100)Q(t) = 176 x 105 Q(t) cubic feet (IV-164) Table IV-3 Change In Reservoir Pore Volume for Example Problem 3 G ta Q(t) We V dimensionless years dimensionless cubic feet 1030 1 300.1 5 28x107 2060 2 546o0 9262x107 3090 3 776o0 13o66xlo7 It is of interest to note that the original reservoir pore volume was V =- frb2h = r(0.1)(2000) (100) = 12o6 x 107 cubic feet (IV -16) After three years, the reservoir would have more than doubled in volume. The reservoir radius after three years would be \ V \ O12' +Oi 3;66) x o0 rb = t () h = I t(0l) (100) 2890 feet (IV-165) This much change in the reservoir radius (from 2000 to 2890 feet) is somewhat incompatible with the assumption of a constant radius. The results obtained should, however, be reasonable approximations. The exact solution to the problem, considering the effect of the moving boundary, is considerably more complicated and is beyond the scope of this example problem.

4. Example Problem 4 Consider an aquifer storage reservoir with the following dimensions and physical properties Po = 700 psia h 80 feet rb = 1000 feet k = 400 millidarcys c = 7 x 10-6 vol/vl psi I0 = 0.17 =1 centipoise re = 100,000 feet It is desired to grow this storage reservoir at a constant rate of 1.5 MMcf pore volume per month. The required reservoir pressure and the aquifer pressure profile are to be calculated for 1, 2, 35 4, and 6 months after initiation of the growth. The tables in Appendix H will be used for these calculations. The applicable relation between real and dimensionless time for these tables is kta 0 = k — (IV-166) O crb2 For this case, at one month (0,4) (30) (24) (3600) (1)(0.17)(7 x 10-6 x 14.7)(1000 x 30.48)2 = 64 (IV-167) From Appendix H, the dimensionless time, @, at which the effect of the exterior boundary at R = 100 first becomes significant is approximately

-66Q = 500. Moreover, values of P(r,~) are not listed in the table for R 100 for values of @ less than 100o This difficulty may be alleviated by using the value of P(r,G) from the table for R = 50 at the time ~ = 64. Since at this time the effect of the exterior boundary at R -= 50 has no significant effect either, the result should be exact. This technique will be covered in more detail in Example Problem 5. To facilitate the interpolation with respect to ~ from the tables in Appendix H, it is convenient to plot P(r,~) for the Q range of interest This plot is shown in Figure IV-8o The pertinent values of P(r,~) are then shown in Table IV-4. Table IV-4 Dimensionless Pressure Distribution, P(r,@), for Example Problem 4 ~~/ta Dimensionless Pressure, P(r,0) months dimensionless r=l.r= r —6 r-20 r-=50 r-100 1 64 2.51 1.41 0,77 0.05 0.00 0,00 2 128 2.85 1.75 1o07 0o16 0.00 0.00 3 192 3.05 1.95 1o26 0.27 0.00 0.00 4 256 3519 2.09 1.40 0.37 0.01 0.00 6 384 3.39 2.28 1.60 0.51 0o04 0o00 By equating the rate of growth of the pore volume to the rate of efflux of water into the aquifer 1. x 106 ft3/month 50000 ft3/day (IV-168) 30 days/month One form of the equation for transforming the dimensionless P(r,~) to actual pressures is(26) [units as for Equation (IV-1635] p = p + 25.2 q ~ P(r,@) (IV-169) kh

-67P(r, ) (o uA b, i. \ \ -- --- ----- -- - - - - ----.\ \I __I _I I __ aU) ____ ___ — 7 ~.. FiueI-8-0 - esol sPesr OnfrEapePo em4 Figure IV-8. Dimensionless Pressure Distribution for Example Problem 4.

-68In this problem (252)(50,000)(1) P = 700 + P(rG) (40oo)(80) = 700 + 39.4 P(r,Q) (IV-170) Applying Equation (IV-170) to the dimensionless P(r,Q) in Table IV-4, the actual pressure distribution, P(ra, ta) is that shown in Table IV-5. The pressure at ra 1000 feet is the required reservoir pressure. Table IV-5 Actual Pressure Distribution, P(ra, ta), for Example Problem 4 Pressure, P(ra, ta), psia ta ra =rb r r ra = ra months 1,000 ft 3,000 ft 6,000 ft 20,000 ft 50,000 ft 100,000 ft 1 798.9 755.5 730.3 702.0 700.0 700o0 2 812.1 768 9 742.1 706.3 700.0 700.0 3 820.1 776.8 749.6 710.6 700.0 700.0 4 825.9 782.4 755.1 714.6 700.4 700.0 6 833.6 789.8 763.0 720.1 701.6 700.0 It should be noted that this analysis again assumed the gas-water interface to be essentially stationary throughout the time period of interest. 5. Example Problem 5 This problem is intended to illustrate the means by which the tables, such as those in Appendix G, for finite or bounded systems may be used in solving problems where the extent of an aquifer is assumed to be infinite. The technique to be illustrated is also applicable to determination of factors such as P(r,Q) and Q(t) for those low values of time

-69at which the known exterior boundary has no significant effect, and for which values do not appear in the table with the particular R value desired. An "infinite" aquifer is usually defined as one in which no significant effect of an exterior boundary can be observed. The behavior of any bounded system at time values sufficiently low that the effect of its exterior boundary is negligible thus is exactly the same as that for the "infinite" system. Thus, for example, all values in Appendix G at times, Q, for which P(R,G) is approximately 1.0 can be used for the calculation of the behavior of infinite systems. Consider a circular reservoir which is surrounded on all sides by an aquifer of essentially infinite extent. The known dimensions and: physical parameters are: PO 1100 psia rb 1000 feet, radius of reservoir = 1 centipoise water viscosity 0 0.1 porosity c 7 x 106 vol/vol psi water plus formation compressibility k = 1000 millidarcy permeability If this reservoir is maintained at a pressure of 1000 psia, it is desired to find the pressure profile in the aquifer as a function of time. kta Q >,1crbE (l)ta (1)(0.1)(7 x 14.7)(1000 x 30.48)2 1.046 x 10-4 ta for ta in seconds = 1.046 x 10-4 x 3600 x 24 ta -9.03ta for ta in days (IV-171)

-70Values of P(r,g) will be obtained from the tables in Appendix Go The conversion from dimensionless P(r,Q) to real P(ra, ta) is then P(ra, ta) = 1000 + (1100 - 1000) P(r,@) = 1000 + 100 P(r,G) psia Xa = 1000 x, feet (IV-172) The values of P(r,Q) and corresponding P(ra,, ta) are shown in Table IV-6. The column "R Table" in Table IV-6 indicates the value of R in Appendix G from which the values of P(r,G) were obtained. at that value of 0 or ta. The only limit on this method is the range of R values for which the factors desired are available. In this case, the maximum value of 0 which may be used is 0 = 105.

-73Lo U) CJ CU H H' O00 F O Lt \L0O O O O O O0.'.O cu t- " — 0 0000o 4-P b- r- C H\ r HHHHUL HH 0(-d NOO O CO OO l- O H CO 0 Co 0CC) 0 - O -H Lr\ 0\0 CH co *spi 0 O \ t- \O Lr\Lr\n -:-t \ 0 0 0 HOOOOOOOO O 110 \O H"- \O CU Ot0 p CQ 00000 0\cOO0- K.\O Q OJ 0O, H 0 HHHHHHH CH (D HcdU Lf\ C) G\O ) O 0\\ H' ~ O r-"' O O O O 1 oN L \ Lr\ Lr\ __:-' H D LrH 0 0- H tLr\ CH Lr\ HO H-N anf 11 CI -a o co o r — t I) H L oO\~ OO 4' H H H H H o II O5......CtO0 O m OOHCSOt^ -o 0 o C DOo L I I I PF }-0Rt 000000 4P <) H- - COU Pc CU - H., - c; oc;d oI:,, H 11 0 0C.......... CA O O OO O O\ 0CO l- O \O -:g -C0 r-H rO -I0 ^" P H HHH! O O O O O O 90 Ho \ HHH \HOHH- 0H v cO. -. tn. =.-!- I, L l _RI CO H -0rX i- I I.H -~, O\ M' L a\ UC FC. LA\ r4 O n rA CU r p 0o H II ^. r\ O I -~ > - II I........ 0 C-\OPK 0 r111111111 r 4 OI\ r\ \ a P P 4-> 4-', ~Qo O \ — f- VO CUJ - Hr 0 ~r 11 It 0.......... p K\ CD i O H O O C\ X Lr\- t L o oO O 3 1 I I I 0 I I r I I Pi r 0 H H O O O O O O O b- H! r-q O1H J 1 r-H r-I H,-'r-! C~~~~~~~~~~~~~Q~~~~C C C CN \O C\ oO O P c CM 0\ O C CD - 0 kLr\ L r- \ O \00 S,, K,\ o 0.,M o,.. "~-'" \""o co 8~,: -H C\ H 0 a) a t m d-I 0 0rl 0 1- -r-0 H 0 oH "H r- H O r-!

Vi MULTI-LAYER FLOW; MATHEMATICAL ANALYSIS The principal purpose of the research which was the basis of this dissertation was the investigation of flow in stratified systems, ie., multi-layer flow, This section presents the mathematical analysis which was done on this problem. All of the mathematical analysis in this section is based on the case of constant terminal pressure, The majority of the section is devoted to presenting analytical solutions for the pressure distribution and the fluxes across the producing face in two-layer systems, both linear and radial, Tables of results for these quantities, in dimensionless form, are presented in Appendices J through 0. Example problems demonstrating the use of these tables are included in this section. Part of the section will also cover the extension of the solutions derived for two-layer systems to three, four, and more layers. A. Two-Layer Linear Flow; Constant Terminal Pressure The mathematical model used for two-layer linear flow is shown in Figure V-1. It is postulated in the following derivations that each of the two layers of this model is in itself homogeneous and isotropic, A perfect bond, i.e, no resistance to flow, is postulated at the interface between layers. Considered as a liquid flow system, the additional require, ments are the assumptions used in deriving the applicable diffusivity equation, namely: 1. Negligible capillary or gravity effects. 2. Single phase flow of an isothermal slightly compressible liquid. 3* Darcy's law applies..-72

-753-:L Y:C' —-- We X=O Figure V-1. Mathematical Model for Two-Layer Linear Flow.

-74Considered as a heat transfer model, it is postulated that the thermal diffusivity of the conducting solid is independent of temperature* 1. Pressure Distribution, P(x,y,@) The equation governing the flow in this case is the diffusivity equation as expressed in cartesian coordinates, as derived with k/. being a function of position within the flow model.. This equation was derived in Chapter III of this dissertation, and appears there as Equation (III-24), a /k aP /k 6P /k P p,,.y+ t_)+ - -— = -c t - - (V,-1) X,' a1 Xa Ya p. ay,' $- \ aZay ata 6xa Ya 6x' ta 6z ^ya a Za a. Referring to Figure V-lj there will be no flow in the za direction. Equation (V-l) thus reduces to f(k aP ) + a. ap) = c ap (V-2) ax, \ axa/ yx 6;a 5ya ata Define dimensionless variables x = Xa (V-5) H -Y - (V-4) H L La (v-5) H Equation (V-2) may then be written 6 /k aP\ +a (k \ 3P _ -O2 _P - ax a a~x/ y y 6y/.ta or k\ a2P aP a \, k 2P 6P a /k\ t.'.~x- + -T ~x'+')7 + y t (V-7) x x J y \H ata

-75 Define k = k O<y<c -1 (v-8) = k2 O < y Then k= 0 O < x < L (-9) dx and = O < y < c and c < y < 1. (V10) ay Since the model is postulated to have the same fluid in both layers, viscosity kp and compressibility c are the same in both layers. Equation (V-7) may then be written k o2P k a2P aP pcH2 x2 0cH2 - y2 ota for all x, 0 < y < c and c < y < 1. (V-ll) Define a= 1 0 O<y<c (V-12) =02 c <y<l K = 1 0<y<c k2 2 * _ k — (V-14) K 2 k i02 e= k_ a (v-15) p1cH'

-76Q is thus the dimensionless time in this case, The governing flow equation now reduces to K a P A K L a (v-l6) ax2 ay2 a~ The applicable initial and boundary conditions for the case of constant terminal pressure in this model are Initial Condition: P(x,y,O) 1 (V-i7) Boundary Conditions _ P (x,0o,) 0 (V-18) Jy (x,l,o) =. (V-19) by P(O,y,S) = 0 (V 20) (L,y, @).0 (v421) ax Equations (V-17) and (V-20) introduce the step function pressure change at the face x O0o Equations (V-18) and (Vl-9) characterize the fact of no flow across the y = 0 and. y = 1 faces, and Equation (V-21) does likewise for the end face x =Lo Two equations apply at the interface y c, denoting the continuity of both pressure and flux at this interface, P(c-) = P(c+) (V-22) 6P (c-) > P (c+) by 6y A solution will be obtained using the technique of separation of variables. It is first desired to find all solutions of the form P X(x) Y(y) T(~) (V-24)

-77which satisfy Equation (V-16) and the boundary and interface conditions (V-.18) through (V-23 ) X" Y" T' i K +K-" -. (V=25) X Y T',.... ~ X ~ ~ X. (v-26) X Y K It is of interest to note that in this case the order of separation of variables is important. It is impossible, for example, to separate variables by initially setting a function of.y equal to a function of X and;0, due to the fact that K is a function of yo The ordinary differential equations resulting from this separation of variables, together with their associated boundary conditions are I -.aT = 0 (V-27) -oJ 0 (V-28) X" + AX =0 (V-29) X(o) X(L) = 0 (V-30) KY" - [a + XK]Y = 0 (V-31) Y'(o) Y (l) = 0 (V-32) If a =: T = C1 (constant) But from (V-28), this is impossible. Therefore, a i 0. If X = 0: X = C2 + C3x From (VT30) X(o) = 0 Therefore C2 = 0 X'(L) = 0 Therefore C3 _ 0 Therefore, X i 0.

-78For a 0: a~ T = Cle In order to satisfy (V-28), a must be negative. Therefore, define a- -a2 (V-33) If X is negative: Let X = -g2 Then (V-29) becomes X" - g2X 0 and X = C1 sinh gx + C2 cosh gx From (V-30) X(o) = 0 Therefore C2 = 0 X"(L) = 0 = lg cosh gL But g ~ 0, or else X = 0. If C1 = 0, the solution is trivial. Therefore it is required that cosh gL = 0 But cosh gL is positive for all real g,L. There are therefore no real roots g to this equation. Therefore X must be positive. Define X = +b2 (V-34) Equations (IV-27) through (IV-32) then become T' + a2T 0 (V-35) lim T = (V-36) Q-> 00 X" + b2X 0 (V-37) X(o) X (L) = 0 (V-38) KY" + [a2 - b2K] Y = 0 (V-59) Yl(o) = Y(l) = 0 (V-40 )

-79By inspection 2 T = Ce Q (V-41a) and X = C6 sin bx + C7 cos bx (V-41b) From (V-38) X (o) = 0o Therefore C7 = 0. X'(L) = O - C6b cos bL Since neither C6 nor b can be zero cos bL = 0 (V-42) This characteristic equation, (V-42), defines an infinite number of real and positive eigenvalues b for each value of Lo b = 1 (V-43) b; m= 1,2,53,.o.. The solution to Equation (V-39) presents the most difficulty. Rewriting for convenience KY" + [a2 - b2K] Y O (IV-39) K = 1 <y<c (Iv-14) K c < y < 1 Since K is a constant in each region, it is possible to divide both sides of Equation (IV-39) by K, obtaining Y" + [a2, b2] Y = 0 < y< c (v-44) Y" + [ b] Y= o c < y < 1 (V-45) Define 7 _= a2 - b2 (v-46) f = 7f -b2 (V-47) K2

-80Equations (1f-44) and (IV-45) are then zY" + 7 = 0 < < c (v 48) Y" +'Y' 0 oc < < 1 (V49) Solving by inspection Y = C1 sin yy ~ C2 os'y y < c (V-50) Y = C3 sin y + C4 cos Py c < y 1 (V-51) From (V-40o) Y'(o) = C1 cos 7Yy C2^ sin yy (V-52) y = 0 Therefore C1 0 and Y = C2 cos Xy 0 y< c (V-53) Also from (V-40): Yg () = 0 = C3D cos P - C4p sin or sin C3 = C4 cs (54) Substituting (V-54) into (V-51) Y C4 sn sin py + cos Py * -|_cos p J = 4L sin P sin Py + cos cos y (v-55) 4 cos (1-y) c < y < cos p The interface conditions (V-22) and (V-23) may be written Y(c-) -:(o+) 5(v'56) Y(c-) - KcpY~(c+) (v-57) Substituting C2 cos 7c = - 4 cos B (l-c) (V-58) P L ecos C

-81-C27 sin yc = L[1o ] K2 sin (1-c) (V-59) In order that there be a non-trivial solution, the determinant of the coefficients must equal zero, ie. PK2sin p(l-c) cos yc + y sin yc cos p(l-c) = 0 (V-60) This characteristic equation implicitly defines an infinite number of eigenvalues, a, for each value of b, This can perhaps be more clearly seen by substituting the definitions for 7 and P from (V-46) and (V-47) into (V-60), K2 bK sinL(l-c) K2 b] cos Lc 2 - b2] -K2 2 (V-61) + 4a2 - b2 sin c fa- b2 cos [(l-c) b2 = 0 From (V-58) C4 cos yc cos P C2 cos P(l-c) Then the solution for Y may be written Y = C2cos yy 0 < y < c (V-53) = os YC cos P(l-y) c < y < 1 (V-62) 2 cos P(l-c) Substituting (V-41a), (V-4lb) (V-53), and (V-62) into (V-24), and combining the values of the arbitrary constants C2, C5, and C6, -a2 P =C sin:bx e cos ry o < y < c (v-63) -a y 1 ye = C sin.bxe cos os P(l-y) c < y < 1 LTcos st (l-c)j a V6 t o This solution, (V-63), satisfies the flow Equation (V-16), the boundary conditions (V-18) through (V-21), and the interface conditions (V-22) and

-82(V-23)o There is) however) no single value of the constant C which will satisfy the initial condition P(xy, 0. 1 (v-1 7 for all values of x and y, Due to the linearity of (V-16), any sum of terms of the +ype (V-63) will also be a solution, Since two parameters, x and y, are involved, this leads to the representation of the initial condition by a double infinite series, analogous to the single infinite series obtained for single-layer flow<, The necessary orthogonality conditions for this case are L X(bm, xm X(bmx)m=r dx = 0 fO or r Tv64) 0 and. /Y(m an, ) y)n, (amnY)n- dy 3 - 0 0 for S.~ r, any nm (V-65) The ort:hogonality condition, (V7-64), is a'well known case, and will thus be stated as such without proof, The condition ('vi65), however, is not self evident, and is not a well known case, The derivation of (V-65) has therefore been included in this dissertation, as Appendix Aa The reader who is interested in somne general theory of differential equations with discontinuous coefficients, together with associated orthogonality properties, is referred to the work of Sangren (3 Proceding analogously to the previously shown single-layer case: 00 00 1 - )7 - CY sinbx (V-66) m-1 r^

-83L 00 J f sin bx dx -[cos bL - 1] Ly -------- -4- _____ (V-67) sin bx dx - [bL-cos bL sin bL] 0 From (V-42), cos bL = O Thus 00 21=1~2 CY b (v-68) Since (2/bL) is independ.ent of n, Equation (V-66) may now be written as 00 00 1 i= [ i C yJ 2( ) sin bx (V-69) m=l n=l m where: 00 Z Crm = 1 (V-70) n=l Then 1 bC f Ydy cos yy dy +c COS 7C cos (1l-y) dy = - 2 21 cos,-lec) (vT71) fm day y cos yy dy + f cos (-y) dy 0o c Cos (1- c) Evaluating the integrals =os yy dy - sin yc (V-72) Y7 0 1 1 PCos Ye /0- ^ - cs ye J cos P(l1-) cosy ( c sin P (ly C cos yc sin (l-c) (V-75) p cos p(l-c)

-841 cos2 yy dy = c [2yec sin 2yc] (v-74) 0 1 s2C —- cos21'ly) dy cos2 (1c-c!(i75) - os"- 2 - [2P(1-c) +. sin 2p(1 c)] 4p cos p,1-le) Substituting into'(V-71) sn 7- reos ~-ZrE si~n:(l ~. L 2 p cos p(l-).1) Fin.' ~o s sin 27 p(li-[c) + sin 2]lL 47v - 4p cosS ( l-e J Simplifying 4 cos P(1-) [p sin 7~e cos p(l-c) + y cos /c s in (l-c)].mn L[2y +c s in 27Cy] cos(l-c) Y- osC ( -' 2; ] (Wv-77) Equation (V,-77) for CQn is perfectly -valid, but is very highly sensitive to slight errors in the ei.genvalues a and b, Working with a Bendix G-15 digital computer, using double precision arithmetic which amounted'to working with about 12 decimal digits,, the author foznd tthat in one case an error of one 7 part in 10 in the eigenvalue a caused the value for Cmn to be incorrect by a factor of more than tw'o A much. more stable equivalent equation for C^ can be obtained for the case of K K (ie 1 - )} Consider the term in (V-'77) F 3p sin 7'c cos p(l-c) + y cos ec sin p(l-c) (V-78)

-85From the characteristic equation, (V-60) or (V-61), defining the eigenvalues a: 0 = y sin 7y cos p(l-c) + PK2 cos 7c sin P(l-c) (V-79) Rearranging F = _i sin 7c cos p(l-c) + cos yc sin (1-.c) (V-80) 7 7 0 = -- sin yc cos p(l-c) + cos ye sin B(l-c) (V-81) PK2 Subtracting (V-81) from, (V-80) F _= ~ - 2] sin yec cos P(l-c) (V-82) 7 LY PKKS Now using the definitions for y and p, from (V-46) and (V-47), letting K2 K0 L e - -y] _ 2 -a2 b K\[a2T2] b(l-K) (v-85) _ 7 73K^2 YpK2 2K K2 F = - sin yc cos p(1-c) (V-84) Subbstituting ('V-4) into (V-77) F 4(l-Kb2) 1 2) 2 LK2 j -) sin ye cos 3(l-c) m cos2p(l-c)[2yc + sin 2yc] + y cos2yc[2p(1-c) + sin 2e(1-c)] (V-85) Equation (V-85) is much less sensitive to slight errors in the eigenvalues a and b than is (V'-77).

-86Summarizing, the solution is thus: 00 0 P(x,y,~) e [a y B L') (i1 86,inbx m=l n=l M where: Cmn is defined by(V-85) n s y, coS < 7y < e cs ye (v87)'Co $I-lc cos 1y) < y 1 ib n- M1),2, (-43 ) 2 L 7 sin re cos P(l-c) + PK2 sin lc) yco c = 0 (V-6o) y = a- ba (2T 46 P = 2 I (V-47) One very important part of the derivation yet remains0 It has been shown that all the eigenvalues b and. a are real and positive, Moreover, by inspection it can be seen that the sign chosen for the squa-re root in (V-46) and (v4'7) defining y and P is immaterial, since the cosine fuxection is an even fmctlion. However, it has not yet bee n shown whethe.r y and P are real or imaginary I't can be seen that the characte:rist ic equation', (V60o) or (V61l) has an in~finite ntmiber -of roo4ts, a, for each value of b when both bo and. P are reala The question thus becomes wht;heo any eigenvalues are obtained as roots of (v-60) when y, P, or both are imaginary, If both y and. P are imaginary: Define 7 = I:1 (V-88 = 1 cL (v-89)

-87The applicable relations between real and imaginary functions are: sin ( x) = i sinh x (V-90) cos (ix) = cosh x (V-91) Substituting into (V-6o) elsinh e1c cosh c2(l-c) + E2K2cosh c c sinh c2(l-c) = 0 (V-92) Now the terms cosh clC, cosh c2(1-c), c2sinh e2(1-c), and elsinh Eic are always positive for all real values of -1' 2' and cg Moreover, K2 is always positive, There are thus no real roots el, ~2 of Equation (V-92). It therefore follows that the case of both y and P being imaginary contributes no eigenvalues a, and thus no terms to the series solution for P(x,y,@~) Now, for convenience, let the layers always be numbered such that * k2p1 K2 - > 1 (V-93) k102 Then in the range b < a < b 1K2 (V-94) y is real, but P is imaginary, Define P 03Ia) (V-95) Substituting into (V-6o), using (V-9G) and. (V-91), y sin yc cosh. o(l-c) - K2 sinh A(l-c) cos ye = 0 (V"96) This characteristic equation, (V-r96) does yield a finite number of roots, a, for each value of the eigenvalue b. The number of roots of (V-96) (which of course applies only over the range specified in (V-94)) is usually small for low values of m, and increases as m increases, due largely to the fact that

-88the range specified'by (V-94) increases as m increases for a given value of L~, For example, it has been found that for the values K2 ~ K2 = 10 L = 10 (V 97) c 05 the number of eigenvalues which appear as roots of (V-96), in the range (V-94), is as follows for the first forty values of bm. Number of eigenalues m._ in'the range (V-94) 1 7 1 8 1 2 14 20 3 21 27 4 28 33 5 3 40 6 It is, of course, necessary to make use of the relations (V-90) and (V-91) between imaginary and, real functions when computing the values of Cn and Ymn from the eigenvalues found in the range (V-94)o Equation (V-85) for C then becomes r (K2-I) b2 sin' c 2 i K-.2 ~L- sin ye cosh:( l- e ) L K2 _k} C mn cosh.i(l-c) [2yc + sin 2.cl + y cos2yc[2(l-c) + sinh 2~il-e] (v-98) This may be written as [ 4(K 1 ( ) sin 7'c., L ^ j ^,4 /_________ (V99) mn [2 sn2c; sy r,c + tan 1r.e[2y'e + sin 2'c] +4 2'y cos7c tanhL>e cQS o s lc

-89The form (V-99) is convenient for further simplification at high values of cu(l-c), at which point cosh2Lc(l-c) becomes very large, and tanh((l-c) approaches oneo The equations defining Ymn in the range (V-94) become'Ymn = cos y < cos 7c cosh (l-y) c < y < 1 (V-1001 cosh o(l-c) = Values of dimensionless P(x,y,Q) are presented in Appendix J. The use of these tables will be illustrated by an example problem later in this sectiono 2. Dimensionless Flux at the Producing Face There are three types of dimensionless quantities useful in calculating the flux across the producing face (x xa = 0). Two of these, the individ.ual instantaneous dimensionless fluxes, cl and q2, and the individual cumulative dimensionless fluxes Q1 and Q2, may be derived directly from the solution for P(x,y,9) by appropriate differentiation and integrationo The third, which is the total cumulative dimensionless flux from both layers taken as a whole, Q(t), is analogous to the dimensionless flux Q(t) for the case of single-layer flow. The expression for Q(t) may be most easily found by combining the solution for P(x,y,Q) with a mass balance (or analogously, a heat balance) on the model. a. Individual Instantaneous Dimensionless Flux The dimensionless ql and q2 will be derived considering the model to be a heat transfer modelo

-90For either layer: Btu = kA T _ _ kA [(T) 1 T (V-101) hr LKAxa/ exax J ea mean H x x mean ( T ] ( /fT dy 0 < y < c ax xmean c 3x X-=O l1 i C ( E ) x dO.y c < y < 1 (V-102) Let the layer 0 < y < c be referred to as "layer 1", and the layer c < y < 1 be referred to as "layer 2". From the derivation of P(x,y,Q), Equation (V-93), it is required. that layer 2 have the higher thermal diffusivity Substituting (V-102) into (V-lOl) (-) - k — ax) ykwAT 7 dy (V-o03) Bt - kl-c)HwAT dy klw AT T ) dy (V-104) hr J (l-c)H o -x x O /Btu k2 (1-)HwT ) dy ( -i04) hr (1-C)H Txx -= o The solution for T(x,y,Q) is given by (V-86). T(x,y,g) = [ Cne-a2 Y (b) sin bx (V-86) The only changes in the liquid flow solution to convert it to a heat transfer solution are that for heat transfer -: kJlt (V-105) P]Cpl1

-91and * k2p1Cp (v-106) klP2Cp2 In both layers C e-a2mn (2 b cos blx=O J 71 7 Cm e-a2 m (V-107) L *v-= h-J Define ql=/ (aT) dy (V-108) __i?, (v-.lo8) q1 x \ax^=0 6/ gq2^.= (/a dy (V-109) Then /Btu h = waTklq1 (V-110) and ( Btu = wATk2q2 (V-lll) hr /2 Now, evaluating integrals c rC YdY - cos rYdy = sin yc (V-1l2) d cosy os n (c) d. (..... / ^B cos 7 (CcY sin, Ll-c) Y-d = - % o dc (V-113)

-92The solutions are thus 1a 2 I e Cm ea29 sin Yc (V-114) q<l 2 - Cae2 [ cos 7c sin P(l-c) v-5) 12 - T Cmne a (V-115) ZL Z_ L 0 cos 8P(1-c) It is, of course9 necessary to use both real and imaginary values of ^ as in computing T(xy9Q)O An analogous derivation of qL and q2 for the flow of slightly compressible fluids leads to the equations (ft) = o026355( )wAPPqc (V-10a) hr 1 - (hrf t o 0.02635(? A )TPqc2 (V-llOb) where k19 k2' darcys 1 = centipoises w -- feet of width AP original pressure difference driving force9 psia The numerical constant in these equations results from the conversion of units to those designated aboveo Tables of ql and q2 are presented in Appendix Lo An example calculation illustrating the use of these tables appears later in this section.

-93bo Individual Cumulative Dimensionless Flux The individual cumulative dimensionless fluxes, Q1 and Q2, are obtained directly by an integration of q and q with respect to time. e a2Q (V-116) a2 Therefore oo oo W-a2- -a 1 =L Cmn 7 r a2- (V-117) Mz I,,_ I Q2 2 Cmn cos Yc sin P(l-c) l-eV-118) L P cos P(l-c) JI a2 J hnv= I I - Both Equations (V-117) for Q1 and (V-ll8) for Q2 may be separated into two series, one independent of time, the other time dependent. Unfortunately, the time independent series is, for both Q1 and Q2, very slow to converge. Unlike the case of Q(t) for single layer radial flow, where a simple algebraic quantity was found to be the mathematical equivalent of the time independent series, there is no great simplification possible in this ease. Since the preparation of tables of Q1 and Q2 would require the calculation of many more values of the eigenvalues, a, than are necessary for calculating T(x,y,Q) and Q(t), it was not thought to be worthwhile to calculate such tables for this dissertation. It is believed that the availability of tables of ql, q2, and Q(t) should be sufficient for the great majority of practical problems. The expressions for Q1 and Q2 are included in this dissertation only for those readers who might wish to calculate these quantities for some particular values of the controlling parameters. Co Total Cumulative Dimensionless Flux The total cumulative dimensionless flux from both layers considered as a whole may be derived by combining a heat ( or mass) balance

-94with the expression previously derived for T(x,y,Q). This derivation will again treat the model as a heat transfer model, The initial "heat-in-place" for the model, assuming unit width, is given by Qi = VpCpTi Btu (V-119) V1 = HLa cH2L (V-120) V2 = H(l-c)La - (1-c)H2L (V-121) Therefore, since a ATi of loO was used for this model Qi = H2L[cp Cp1 + (l-c)P2Cp2] (V-122) At any time, the heat-in-place, Qhip) must be given by a volumetric integration of the temperature, i.e. Qhip H L{ lCpl [ATmean]1 + (l-c)2Cp2p [ATmean]2} H2 P1C T dy dx (l-c)p L [ T dy dx ( *-H 2L [Tdydx+ T dyx (V-125) Define Pl.Cp cT dy dx + P2a2 T dy I x L To L c cplCp + (l-c)P2Cp2 dy dx + P2C2 T dy dx L- Pl...C+.( pl-c) C.. (v-124) L [c + (1-c) P2C ]

-95Then E represents the fraction of the heat energy still present at any time, and. (l-e) is the fraction of the total heat energy initially present which has been lost. Then, per unit width Q - H2L [cpiCp + (l-c)p2Cp2] (1-E)AT (V-125) is the Btu of heat loss. Define Q(,) = L(1-E) (V-126) Then Q(Btu) = H2wAT [cplCpl + (l-c)P2Cp2] Q(t) (V-127) where w is the width of the model, and AT is the original overall temperature difference driving force. Note that 0 < Q(t) < L. Rewriting the solution for T(x,y,Q) for convenience T = sin bx (V T = Z- L e CYmne-aYm ( sin bx (V-86) where Ymn cos yy 0 y< c cos yc co cs (l-y) c < y < 1 (V-87) cos P(1-c) Evaluating integrals rL 1L 1 L o sin bx dx = cos bx (V-128) *JQ~~o bo b C C Ymndy= cos yy dy sin = (V-129) Ydy = cos Yc cos p(l-y)dy _ cos 7c sin B(l-c) (V-130)'< n... R os 1(-c yn p cos ( l-c)

-96Then O0 0 1i C T dy dx = Cme'a2g sin Yc ( (v-131) - Il h-t and P2.dy dT -a20 ( 2 cos y-c sin P(l-c) LpCP21 T dy dx P2Cp2 _ (b2L). ( LPlCpl PlCpl b2L 8 cos P(1-c) (V-132) Therefore Z? -a2Q ( 2 i fsin yc P2C2 cos5 sin P(l-c) e = m~l n-l Lbm Y[ PC cos P(l-c) (v-153) [c + (l-c) P2CP2] rl~!I-~, PlCpl and, as defined. Q(t) = L(1-e) (V-126) The corresponding derivation using a mass balance instead of a heat balance is similar in all respects and yields ~ -a2 / 2 s [sin yc 02 cos 7c sin P(l-c) 1 mE l nl ZbC ( ) L Cos 1 P-c) J ( ~E=: ---------------— j: ---------—,-, —---------------------- (V -134 ) [c + (1-c) 1.1 and Q(t) = L(l-E) (V-126) The use of the quantity Q(t) for computing the actual flux for the flow of slightly compressible fluids is given by Q(ft3) - H2wcwaP[c1 + (l-c) 2] Q(t) (V-127a) where cw is the formation plus water compressibility, and AP is the original overall pressure difference driving force.

-97Tables of Q(t) are presented in Appendix N. All of these tables are based on the criterion that 01 = 02 or analogously plCpl = P2Cp2. The use of these tables will be illustrated by an example calculation later in this dissertation~ B. Two-Layer Radial Flow; Constant Terminal Pressure The mathematical model used for two-layer radial flow is shown in Figure V-2o The same specifications with regard to homogeneity, isotropy, etco, apply to this model as applied to the two-layer linear model. Since much of thej mathematical analysis of this case is similar to that of either single layer radial flow or two-layer linear flow, reference will be made to these solutions rather than repeating the corresponding part of the derivations whenever possible. 1o Pressure Distribution, P(ryQ) The applicable flow equation in this case is k Pap +1, p + a- k 6pc _0 (IIIr Lr a ra aYa Ya.a ta Define dimensionless variables ra r - (v-135) rb R re (V-136) rb y Ya (V-137) H A _ rb (V-138) H K2 k l- V K2 -- k201; K2 = (V-139) klta', crb2 (v-140)

-984e,. k2 k, YI LAYER 2 k Y> of >'Y=:O i M M for Two-Laer Radial Flow. Figure V-2. Mathematical Model for Two-Layer Radial Flow.

-99The governing flow equation may then be written K [aP + 1 + A2K a p (V-141) ar2 r 6r Jy2 as where: K 1 0 < y < c - K c < y 1 (V-14) The applicable initial, boundary, and interface conditions for the case of constant terminal pressure in this model are Initial Condition: P(r^y,O) - 1 (V-142) Boundary Conditions: aP(r OQ) 0 (V-143) ay 7(r.l,~) = 0 (V-144) P(l,y,Q) - 0 (V-145) ap(R,yG) - (V-146) Tr Interface Conditions: P(c-) P(c+) (V-147) a(_(c) K2 a (c+) (v-148) ay ay The technique of separation of variables will be used to obtain a solutiono It is first desired to find all solutions of the form P = X(r) Y(y) T(o) (V-149) which satisfy Equation (V-141), and the boundary and interface conditions (V-143) through (V-148) K [_ l- + A2K YK - T a ((V-150) [' x 1 x. 2 (v-151)

-100The resultant ordinary differential equations, together with associated. boundary conditions are thus TF - a T = 0 (V-152) lim T - 0 (0Y155) 9 -> oo X'1+- X + XX 0 (V-154) r X(1) = X (R) - 0 (V-155) A2KT - [a + XK]Y = 0 (V-156) Yc (O) - Y (1) - ~O T1.57) As in the linear case, OC must be negative Define a -a2 (V-33) If X 0: X" + - XS - 0 r X C1 + C2 in r? C2 r x9 (R) C2 = 0 R Therefore, C2 = 0 X = C1 X(l) = C1 = 0 Therefore X # 0 If X is negative~ Let X -g2 X" + X - g2 X = 0 r2X" + r X - g2r2X = 0

-101This is one form of Bessels equation~ X - CJIo(gr) + C2Ko(gr) ax Clgl(gr) - C2gKl(gr) 6r X(1) 0 = ClIo(g) + C2Ko(g) (V-158) X'(R) 0 = g[ClIl(gR) - C2K1(gR)] (V-159) Since X 0, g 0. Therefore, for a non-trivial solution Io(g)Kl(gR) + Ko(g)Il(gR) = 0 (V-160) But for all real g, R, the values of all terms in this characteristic equation are always positive. There are thus no real roots, g, to Equation (V-160). Consequently, X must be positive. Define X - +b2 (V-161) Substituting (V-161) and. (V-33) into (V-152) through (V-157), the ordinary differential equations become T' + a2T = 0 (V-162) lim T - 0 (V-163) ~ -- 00 x" + 1 X| + b2X 0 (V-164) r X(l) - X (R) - 0 (V-165) A2KY" + [a2 - b2K]Y 0 (V-166) Y (o) = Y (1) = 0 (v-67) By inspection T Clea (V-168)

-102Equation (V-164) is one form of Bessels equationo It is identical to Equation (IV-57) for single layer radial flow, and has the same boundary conditions. From Equations (IV-57) through (IV-67), X - C4[Jo(br)Y (bR) - Yo(br)J1(bR)] (V-169) and J (b)Yl(bR) - Y((b)J1(bR) - (V-170) The characteristic Equation (V-170) is identical to Equation (IV-65). The eigenvalues, b, are thus the same for both single-layer and two-layer radial flow. Since K is constant in both layer 1 and layer 2 individually, Equation (V-166) may be written + 1 _ - b2 Y = 0 < y < c, c < y<l (Y-171) Define 1 7 _ /a2 _ b2 (V-12) p V 7J ^. b1 <(V-175) A K2 Then Y + 72Y 0 0 < y < c (V-174) y,, + p2Y 0 c < y < 1 (V-175) Now Equations (V-174) and (V-175) are identical to Equations (V-48) and (V-49) for two-layer linear flow, and have the same boundary and interface conditions. Thus, from (V-48) through (V-62) Y = C5 cos 7y 0 < y < c Y =C5 cos cYy < = os -- cos p(l-y) c < y < 1 (V-176) cos Kl-c)

-103and PK2 sin P(1-c) cos 7c + 7 sin 7c cos P(l-c) = 0 (V-177) Equations (V-177) and (V-6o) are identical, yielding an identical set of eigenvalues, a, for a given b. However, since the values of b are not the same for two-layer linear and two-layer radial flow, the eigenvalues, a, are also different. Now, as in the case of single-layer radial flow, define U(br)= Jo(br)Y1(bR) - Y0(br)J (bR) (V-178) Then, substituting into (V-149), P = Ce-a2@U(br) cos yy 0 < y < c = Ce-a2u(br) cos c cos P(ly)] c < y < 1 (V-179) Equation (V-179) satisfies Equations (V-141) and (V-143) through (V-148). Again, however, there is no single- value of the constant C which satisfies the initial condition (V-142). As for two-layer linear flow, this initial condition may be represented by a double infinite series. The necessary orthogonality conditions have been given previously as fY(m,amny)n-= Y(mami)nr dy 0= for Q.r, any m (V-65) rR rUm(br)Un(br)dr 0. for m / n (V-70) It should again be noted that Equation (V-65) is derived in Appendix A of this dissertation.

-104Proceding as before, 1 Z 2_ ) CY U(br) (V-18o) R o rU(br)dr 1 CY --- (V-181) f rU2(br)dr 1 From (IV-71) through (IV-81) ZCY 2bV(b)- (V-182) i, b2V2(b) - where: V(br) = Jl(br)Yl(bR) - Yl(br)Jl(bR) (IV-78) Define 2by(b) F(b) = CY 2 ) (V-185) n= _ _ b2V2(b) - 2 Then 1 f Ydy C = Cn = (V-184) F(b) n 1 f Y2dy 0 From Equations (V-71) through (V-77) C = C 4 cos P(1-c)[g sin 7c cos P(l-c) + 7 cos 7c sin P(l-c)] F(b) mn [2yc + sin 2yc]cos2p(1-c) + y cos2yc[2p(1-c) + sin 2P(1-c)] (v-185) As in the case of (V-77), this equation for Cn is very highly sensitive to slight errors in the eigenvalues a and b. A modification is possible, analogous to that of Equations (V-78) through (V-85) for the case of singlelayer linear flow. The only difference in these simplifications occurs due to the slightly different definitions of y and pc

-105Equation (V-83) now becomes (for 01 = 02 K2 = K2) _ Jy _. 2K2 - y2 y7 K2 7 K2 1 a2 [ b K2 2 - b2] 7 P K2 b2(1 - K2) (v-186) A y 7 P K2 The modified equation for Cn then becomes [4(1-K2) (b2)sin yc cos2p(l-c) Cmn = -A2K2 1 (V-187) P cos2p(1-c)[27c + sin 2yc] + y cos2yc[2P(1-c) + sin 2(1l-c)] Equation (V-187) is much less sensitive to slight errors in the eigenvalues a and b than is Equation (V-185), although both should give identical numerical'. answers for absolutely correct values of the eigenvalues. The solution may then be written P(r,y,@) = C mne a Ymn m F(b)U(br) (V-188) where: Cmn is given by (V-187) mn -- cos 7y 0 < y < c cos 7c oCOS (C ) cos P(l-y) c< y < 1 (V-176) cos P(l-c) y = _,/a2 _ b2 (V-172) 7 b2 (V-172) A K

-106F(b) = bV(b) (V-185) b2v2(b) - (4 V(br) Jl(br)Yl(bR) - Yl(br)Jl(bR) (IV-78) U(br) = Jo(br)Yl(bR) - Yo(br)Jl(bR) (V-178) and the characteristic equations are U(b) = 0 (IV-77) and y sin yc cos P(1-c) + PK2 sin P(l-c) cos yc = 0 (V-177) As in the case of two-layer linear flow, both real and imaginary values of P must be considered in order to obtain valid numerical results. Values of dimensionless P(ry,@) are presented in Appendix K. The use of these tables will be illustrated by an example problem later in this sectiono 2. Dimensionless Flux at the Producing Face The three types of dimensionless flux for the radial case are analogous to those for the linear case, namely 1. Individual instantaneous flux, ql and q2, 2. Individual cumulative flux, Q1 and Q2, and 3. Total cumulative flux, Q(t)~ The derivations of these quantities are sufficiently similar to the derivations for two-layer linear flow that the author does not feel that including these derivations would be worthwhileo The results will therefore be stated without proof.

-107The individual inbtantaneous fluxes for the radial case are, 7l = Cmne a4 ( 2b2V2 (b) ] sin 7. (V-189) tZ -I b V2(b) J —'I" I h-I - Vc e-a2 2b2v(b) 1 cos Yc sin P(1-c)1 -19 12 I_ L b2V2(b) - cos B(l-c) The individual cumulative fluxes for the radial case are: [mn a2 J v( — 191 X > [ 1 - e)a b2V2(b) ]-'7 X [i-e-a2e.. b2V2(b) F2 cos 7c sin p(l-c) L -.i[a2,b2V2(b) - cos P(l-c) (V-192) The total cumulative flux from both layers for the radial caBe is Q(t>) -=( (1 - ()) (V-1-95) where: /(R 2 )\' co [6, sin re P2Cp2 cos 7c sin $(!-t) |[ 2V2(b). (y ~si~~~ ~l~-~LC e Iftajg' R m —_ nnl Y p C Cne" cos P(l-c) CJ'jl - b' L.. [ c. C plCpl ] (PV^1) for heat transfer, or the same with replaced by for liquid flow.

-108The equations by which these dimensionless quantities may be used for heat transfer computations are given by Btu ) 2nHklqljT (V-195) hr I (Btu. 2ntHk2q2AT (V-196) hr 2 (Btu)l 2 tHklQ1AT (V-197) (Btu)2 2 2tHk228T (V-198) Total Btu 2crb2H[cplCpl + (l-c)P2Cp2]Q(t)AT (V-199) Corresponding equations for fluid flow are (ft3 - 0.1654( k )HAPq1 (V-195a) hr 2 2 where: kl, k2 - darcys H = feet = centipoises AP = original overall pressure difference driving force, psia Total Q(ft3) 2rirb2cwHAP[ cl + (l-c)12]Q(t) (V-199a) where: cw = water plus formation compressibility Tables of ql and q2 are presented in Appendix M of this dissertationo Tables of Q(t) for the case a, - a2

-109or analogously PCpl = P2Cp2 appear in Appendix Q. Tables of Q1 and Q2 were not computed, due to the convergence difficulties discussed in the corresponding section for twolayer linear flow. The means of using the tables is illustrated by an example calculation in a later part of this dissertation. C. Generalization of the Solutions for Two-Layer Systems to More Than Two Layers Derivations have now been presented for two-layer flow in both linear and radial systems. This section will show how these solutions may be generalized to apply to a model of any number of layers. Since the only essential difference in the derivation of the solutions for more than two layers occurs in the separated Y equation, which is the same for both linear and radial flow, it unnecessary to consider linear and radial flow separately. The derivations will be presented for linear flow models. The corresponding radial flow solutions will then be presented without a complete rederivation. Consider the three-layer mathematical model shown schematically in Figure V-3. The specifications with regard to homogeneity, isotropy, etc. will be taken as the same used for two-layer linear flow. Defining dimensionless variables x = (V-200) y = (V-201) L - (V-202) H

-110- ys| cf —-------- r /'^' bS~ / xs^ Y=l y=g -- 1 o X=O Figure V-3. Mathematical Model for Three-Layer Linear Flow.

- 11the governing flow equation may be written k+K a ( ) a - 0cH2 a (V-203) dx'- dy 5y / 6ta For the constant terminal pressure case, the applicable initial and boundary conditions are Initial Condition: P(x,y,o) = 1 (V-204) Boundary Conditions: P(O,y,ta) = 0 (V-205) bP (L,y,ta) = 0 (V-206) ax P (x,O,ta) = 0 (V-207) -P (x,l,ta) - 0 (V-208) by Separating variables in the usual manner, for P = X(x) Y(y) T(ta) (V-209) the ordinary differential equations from the separation are X" + b2X: 0 (V-210) a2 T' + - T - 0 (V-211) ky + (k Y1 + a2 - b2 )] Y = 0 (V-212) The solutions to (V-210) and (V-211) are, by inspection X = C sin bx (V-213) -a2_ T = C e7-c#H (V-214)

-112where: b -=( _ ) -; m-1,2,33... (V-215) The question of whether a and b are real or imaginary, etc., is the same as for two-layer linear flow. The difference from the two-layer flow derivation occurs in the solution of the Y equation, (V-212). Let the permeability, k, and porosity, 0, of the layers be that indicated in Figure V-3. The layers are assumed to be homogeneous and isotropic at all interior points. The applicable boundary and interface conditions become Y'(o) = Y (1) = 0 (V-216) Y(f-) = Y(f+) (V-217) Y(g-) = Y(g+) (V-218) k2 Y(f+) Y'(f-) Y(+) (V-219) kl Y (g-) = ~ Y, g+) (v-220o) k2. Define kL [a2 ( b2 1 )] (V-221) kl I 2. Fa2 - b2 k2 X2 (V-222) k2 L 2 k3 [ ( 03 (-2 The equations and corresponding solutions are then Y11 + a2y = 0 < y < f (V-224) Y = C1 sin Qy + C2 cos Cry

-113Y" + X2Y = 0 f <y< g (V-225) Y = C3 sin Xy + C4 cos Xy Y"1 + c2Y = 0 g < y < 1 (V-226) Y - C5 sin ey + C6 cos Ey From (V-216) Y' (o) = 0 = [OC1 cos Oy - OC2 sin ay]yO = QC1 Therefore C1 0 Y'(1) = 0 = [ C5 cos E - C C6 sin c] Therefore C5 = C6 tan c The solutions may thus be written Y = C2 cos Oy 0 < y < f = C3 sin ky + C4 cos Xy f < y < g C7 cos e(l-y) g < y < 1 (V-227) The interface conditions (V-217) through (V-220) may next be applied. From (V-217) C2 cos af = C3 sin Xf + C4 cos Xf (V-228) From (V-218) C3 sin Xg + C4 cos Xg = C7 cos c(1-g) (V-229) From (V-219) k2: -C2 sin af = kl2 [C3 X cos Xf - C4 Xinin f] (V-230) From (V-220) X[C- cos Xg - C4 sin Xg] = C7 sin (-g) (V-231) 2 ~~~~k2 C7 sin e(i-g)(v2)

-114This set of four equations [(V-228) through (V-231)] can be considered as four equations in the four unknowns C2, C3, C4, and C7. In order that there be a non-trivial solution (i.e., C2 C3 4 C4 f C7 / 0) the fourth order determinant of the coefficients must be equal to zero. Define K k2 (V-232) k2 K L3- (V-233) Rewriting the equations C2[cos caf] + C3[-sin Xf] + C4[-cos Xf] = 0 (V-234) C3[sin Xg] + C4[cos Xg] + C7[-cos c(l-g)] = 0 (V-235) C2[-a sin af] + C3[-K2X cos Xf] + C4[K2X sin Xf] = 0 (V-236) C3[x cos Xg] + C4[-X sin Xg] + C7[-K3e sin e(l-g)] = 0 (V-237) Define: F1 = cos af F7 = -a sin af F2 = -sin Xf F8 = -K2X cos Xf F3 = -cos Xf F9 = K2X sin Xf F4 = sin Xg F10- X cos kg F = cos Xg Fl -X sin Xg F6 = -cos E(l-g) F12 = -K3C sin e(l-g) (V-238) The characteristic equation defining the eigenvalues, a, may then be written in the form of the determinant F1 F2 F3 0 0 F4 F5 F6 D F5 F6 0 (V-239) F7 F8 F9 0 0 F10 F11 F12

-115Expanding by minors, the equation becomes F1F4F9F12 - F1F5F8F12 + F1F6F8F11 - F1F6gFlo + F2F5F7F12 - F2F6F7F11 - FF4F7F12 + F3F6F7F1 = 0 (V-24o) Due to the zeros present, the 24 terms of the fourth order determinant simplify to only eight. The characteristic equation for the three-layer case thus contains eight terms, each of which is composed of four elements, Fi. Now consider a four-layer linear case. The solutions to the separated Y equation would in this case contain eight constants, C1 - C8. The boundary conditions Y'(o) = Y'(1) = o (V-216) would eliminate two of these. The six interface conditions would then provide the necessary six equations for determining these six constants. The characteristic equation is thus of the form of a sixth order determinant equaling zeroo Such a determinant, expanded by minors, would normally involve 6! = 720 terms, each of which would have six elements. It can be shown that the zeros present in the determinant reduce the number of nonzero factors in the characteristic equation to thirty-two terms, each of which involves six elements. The extension to more than four layers is apparent. Let the number of layers in the model be N. The number of constants appearing in the solutions to the separated equation will be 2N. Of these, two may always be eliminated using the boundary conditions (V-216). Now for N layers, there will be (N-l) interfaces between layers. Since at each interface, both the pressure and flux must be equal in the adjoining

-116layers, a total of 2(N-l) = 2N-2 equations will result from the interface conditions The characteristic equation for an N-layer system is thus in the form of a determinant of order (2N-2), set equal to zero. Expansion by minors of this determinant will yield 2(2N-3) terms, each of which will contain (2N-2) elements. For example, a five-layer system would have a characteristic equation of the form of an eighth order determinant equal to zero. Expansion by minors would yield 128 terms, each of which would contain eight elements. Let us now return to consideration of the three-layer system. The characteristic Equation (V-239) or (V-240) defines an infinite number of the desired eigenvalues, a, for each value of the parameter b. From inspection of (V-221) through (V-223), it is apparent that the factors a, X, and c might be either real or imaginary, depending on the relationship between a, b, and the ratio (k/0). As in the two-layer flow case, it is necessary to consider the complete spectrum of real and positive values of the eigenvalue, a, for each value of b The elements, F, appearing in the characteristic Equation (V-240) can always be written in terms of real quantities, even though a, X, and/or c may be imaginary by using the equations sin ix = i sinh x (V-241) cos ix = cosh x (V-242) (i)2 = -1 (V-243) relating real and imaginary numbers. Having determined the eigenvalues a and b, the only remaining difficulty in obtaining the solution for three-layer flow is the establishmelmnt.

-117of the applicable orthogonality conditionso The orthogonality conditions for two separate and distinct layers, and for a continuous variation of (k/0) with y are derived in Appendix A of this dissertation. In both cases it was found that YmYndY =0 for m n (V-244) The applicable condition for three or more separate and distinct layers may be easily derived using the same basic procedure as for two layers. The author has done so for a three-layer system, but does not believe the inclusion of the derivation in this dissertation is warranted, due to the similarity to the two-layer flow derivation. It was found that the condition (V-244) also applies to three-layer systems. Careful examination of the two- and three-layer derivations shows, moreover, that Equation (V-244) should be valid for any number of separate and distinct layers, as long as the criteria of continuity of pressure and flux obtain at all interfaces. The three-layer solution continues by determining the relation between C3 and C4 in Equation (V-227), in the range f < y < g. This can be most easily done using (V-234) and (V-236), eliminating C2. Substituting from (V-238) into these equations. C2F1 + C3F2 + C4F3 = O (v-245) C2F7 + C3F8 + C4Fg - (V-246) Eliminating C2, it is found that C4 i F2F - FlF8 ) 719 FF

-118Define F2F7 - FF8 (-248) FlF9 - F3F7 The expansion to fit the initial condition (V-204) then proceeds as follows 9 1 -[ i, CY sin bx (V-249) l'l = M I J L f sin bx dx CY = ~L -_ b (V-250) L bL h-l I sin2 bx dx 0 Then 1 Y l = Croy(bL ) sin bx (V-251) where 1 f Y dy Cmn - (V-252) f Y2dy o and Y = cos Oy 0 < y < f = sin Xy + E cos Xy f < y < g - cos E(l-y) g < y < g (V-253) Expanding (V-252), substituting from (V-253) f g 1 J cos cy dy + f [sin Xy + E cos Xy]dy + J cos E(l-y)dy p o,.;. __......................... Cmn f g 1 f cos2ay dy + f [sin Xy + E cos Xy]2dy I cos2e(l-y)dy o f g I1 + I2 + I3 (v-254) 14 + I5 + I6

-119Evaluating the integrals 1 = cos cy dy in (V-255) C) 12 = [sin Xy + E cos Xy]dy 1 E = j[cos Xf - cos Xg] +:[sin Xg - sin Xf] (V-256) I3 = cos e(l-y)dy = sn (l-g) (V-257) I = cos Cy dy = [sin. Of cos af + af] (V-258) I4 = 2e 15 [sin Xy + E cos Xy]2 dy = 2- [gX - sin gX cos gX - fX + sin fX cos fX] E + - [cos 2Xf - cos 2Xg] E2 + [gX + sin gX cos gX - fX - sin fX cos fX] (V-259) 2X I6' cos2E(1-y)dy - L [2e(l-g) + sin 2E(1-g)] (V-260) The solution for three-layer linear flow in the constant terminal pressure case is then c00 -a2t P(x,y,ta) = CmnY( ) sin bx e (V-261) where Y is given by (V-248) and (V-253) Cmn is given by (V-252) through (V-26o) b is defined by (V-215) a is defined by (V-238) through (V-240)

-120The corresponding solution for three-layer radial flow, constant terminal pressure is PZ^yta = ~ -a2ta P(r,y,ta) - CYej c2- F(b)U(br) (V-262) where Cmn and Y are as for linear flow F(b) is given by (V-183) U(br) is given by (V-178) b is defined by (IV-77), and a is defined by (V-238) through (V-240) with the change that 2 k2 [1a2 b2 ( ) (v-264) A2k2( )] ( E2_ [a2 b2( )j] (v-265) rather than by (V-221) through (V-223). D. Convergence of the Analytical Solutions For Two-Layer Flow All of the solutions obtained for two-layer flow, and also for multi-layer flow, have been found in the form of double Fourier or FourierBessel series. In the case of linear flow, the eigenvalues b are found by inspection, but the eigenvalues a are defined by an implicit expression. In the case of two-layer radial flow, the eigenvalues b were defined by the same characteristic equation as for single-layer radial flow, and the eigenvalues a are again defined by an implicit equation. The eigenvalues for the two-layer case were calculated on the IBM 704 using the

-121half-interval method, as explained in Section IV-C of this dissertation. Since the calculation of the eigenvalues for the series involved a considerable amount of time on the IBM 704 (a total of approximately twelve hours for the eigenvalues in this dissertation) the rate of convergence of the analytical solutions was important. The number of eigenvalues necessary to obtain the desired accuracy in the numerical tables was determined by trial and error. In the linear case, a twelve by eight matrix of eigenvalues was used for all values of L. That is, a total of eight eigenvalues, a, were computed for each of twelve values of the eigenvalue b for each value of L. This proved to be adequate for the computation of P(r,y,G), cl and q2, and Q(t). As noted in Section V-A-2-b, however, this proved to be entirely insufficient for calculation of the individual cumulative dimensionless fluxes'Q1 and Q2 [Equations (V-117) and (V-118)]. A forty by eight matrix of eigenvalues was computed for L = 10 and used in an attempt to compute Q1 and Q2. This, too, was found to be insufficient. The calculation of tables of Q1 and Q2 was therefore abandoned. In the radial case, unlike the linear case, it was found that the number of eigenvalues required was a function of a parameter in the problem, the ratio (rb/H). The number of eigenvalues b required was essentially constant. The number of eigenvalues a required per b was strongly dependent on the value of (rb/H). It was originally intended that the solution for two-layer radial flow should have two principal applications. First, the solutions should apply to reservoir-aquifer systems (where rb denotes the radius of the reservoir) and second, they should apply to the flow within an oil reservoir

-122to a given well bore (where rb denotes the radius of the well bore). In the former case, the value of (rb/H) is normally quite large, of the order of ten to a hundred.or moreo In the latter case, however, the ratio is normally quite small, of the order of 0.01o The number of eigenvalues a per eigenvalues b required for an accurate numerical result was found to be only one for values of (rb/H) greater than 10.0. This may be contrasted with the linear case, where 3 to 8 such eigenvalues were usually necessary. It was found, moreover, that the solution for P(r,yQ) was essentially independent of the parameter (rb/H) for values of (rb/H) = 10 or more. Consequently, values of ql, q2, and Q(t) were also independent of (rb/H) in this range. For values of (rb/H) less than about 1.0, on the other hand, it proved to be nearly impossible to obtain valid numerical results. The number of eigenvalues a per eigenvalue b required was found to be several hundred at (rb/H) = 0.05. This would require a prohibitive amount of computer time for computation of eigenvalueso Moreover, the problem of round-off error with such a large number of terms'in the series might very likely invalidate any numerical results thus obtained. For the two-layer radial heat transfer model in this investigation, (rb/H) =- o6. A ten by forty matrix of eigenvalues (ioe. forty a's per each of ten bgs) proved insufficient to give even an approximate check of the experimental data. It thus proved to be impossible to compute values of P(r,y,@), etc., for values of (rb/H) which would be useful for flow to a well in an oil reservoiro It seems probable that a point source solution to the problem would alleviate this difficulty.

-123It is of interest to consider the physical reason behind the failure of the series to converge for low values of (rb/H). The author believes that this failure is largely due to the nature of the pressure gradient in the y direction in such cases. For high H, and thus low (rb/H), points far removed from the interface between layers are largely unaffected by the presence of the other layer. It is only near the interface that a comparatively sharp change in pressure with respect to y at a given x occurs. The approximation of such a gradient byaSerIies solution is tantamount to fitting a step-function curve. Thus, a very large number of series terms are required for an accurate approximation. All values in the Appendices of this dissertation for two-layer radial flow were computed using a five by one matrix of eigenvalues. This proved to be sufficient for accurate calculation of all factors desired. E. Example Calculations This section is intended to illustrate the use of the tables in Appendices J. L, and N for two-layer linear flow, and Appendices K, M, and 0 for two-layer radial flow. Two example problems will be presented. In order to once again illustrate the dual usefuleness of the tables, the problem for two-layer linear flow will be considered to be a heat transfer problem, while that for two-layer radial flow will be concerned with a reservoir-aquifer system. Both problems to be considered here will be for bounded systems. The use of the tables in Appendices J through 0 for infinite systems is analogous to that indicated in Example Problem 5 for single-layer flow.

-1241. Example Problem 6 Consider a two-layer bar such as that shown schematically in Figure V-4, with the following dimensions and physical properties. La = 24 inches - 2.0 feet H = 0.48 inches = 0.04 feet w =1.0 inches (width) c = 0.5 (i.e. interface between layers is at ya = 0.24 inches) kL = 16 Btu/hr ft2(OF/ft) k2 = 160 Btu/hr ft2(OF/ft) = P2 = 500 lb/ft3 Cpl Cp2 = 0.1 Btu/lb ~F The bar is initially at 70~F throughout. At time zero (i.e. ta = 0) one end of the bar (at x = xa O0) is rapidly heated to 270~F and maintained at that temperature thereafter. It is assumed that there are no significant heat losses from the bar. It is desired to calculate 1 The temperature distribution, T(xa, Ya, ta) in the bar at a time 30 minutes after the heating began (i.e. ta = 30 min.). 2. The rate of heat transfer, Btu/hr, across the heating face into layers 1 and 2 (individually) at time ta = 30 minutes. 35 The total heat input to the bar, Btu, at time ta = 30 minuteso For this bar La 24 inches L - 0 = 50 (V-266) H 0.48 inches Xa x x — H = 48 for xa in inches (V-267) H 0.48

-125Cl /= o24" y.48 1 LAYER I / / Y"'~. r AY I" / Figure V-4. Heat Transfer Model for Example Problem 6. 300 250 w 200... —-..... L, 1o50 -- 50 2 100 __ 0 2 4 6 8 10 12 14 16 18 20 22 24 Xo I INCHES Figure V-5. Temperature Distribution for Example Problem 6.

-126y Ya Ya for Ya in inches (V-268) H 0.48 k2PlCpl (160)(500)(0.1) K2 k1C2 ( 16) (500) (01) K - 10 (v-269) Layer 1 is defined as 0 < Ya < 0.24 inches Layer 2 is defined as 0.24 < Ya < o048 inches At ta = 30 minutes klta (16)(0o5) PCp H2 (500)(0.1)(.04)2 100 (dimensionless) (V-270) To determine T(xa, Ya, ta) the table in Appendix J will be used, for L = 50, K2 = 10, c = 0.5, and Q = 100. It can be seen from this table that there is no significant temperature variation in the Ya or y direction at this time. The dimensionless P(x, y, 0) or analogously T(x, y, 0) for all y at 0 = 100 is given as x T(x, y, 0) 4 0094 7 0o163 10 0.231 20 0o.438 50 00757 The corresponding actual temperatures as a function of actual position are found using xa = 0.48x (inches) (V-271) T(xa, Ya, ta) = 270- (270 - 70) T(x, y, 0) 270 - 200 T(x, y, O) (V-272) The results are shown in Figure V-5.

-127The individual instantaneous fluxes across the heating face into the two layers can be determined using the tables in Appendix L. From the table for L = 50, K2 = 10, c = 0.5 at - = 100 ql = 0.01309 q2 = 0.01308 To convert these dimensionless fluxes to real flux in Btu/hr, it is necessary to make use of the equation Btu ki jiWT (V-110) where: w is the width in feet AT is the temperature difference driving force, 0F. Then for this model /Btu = (16)(0.01309)( (200) = 3.4907 tu (V-273) \hr j\12 /hr /Btu) = (1.60)(0.01308) ('1 )(200) = 34.88 Btu (V-274) hry 12' hr It is of interest to note that the fluxes are in this case nearly directly proportional to the respective thermal conductivities. The total cumulative flux across the heating face is found from the tables in Appendix N. From these tables, for L = 50, K2 = 10, and c = 0.5 at -= 100. Q(t) = 23.79 The applicable equation for converting this dimensionless flux to a real flux is Q(Btu) = H2wAT[cplCpl + (l-c)p2C.2]Q(t) (V-127)

-128so that for this problem. - 04)2(12 )(200)[(o.5)(500)(o0.) + (0o5)(500)(0ol)](23579) 31o7 Btu total heat input (V-275) 2. Example Problem 7 Consider a circular gas reservoir surrounded on all sides by a uniform aquifer. It is known that this aquifer is composed of two intercommunicating layers. The pertinent dimensions and physical properties are as follows rb = 2000 feet, radius of reservoir re 200,000 feet, exterior radius of the aquifer h = 50 feet, thickness of layer 1 h2 50 feet, thickness of layer 2 kl * 5 millidarcys k2 100 millidarcys =0 10% 4 lo0 centipoise water viscosity c = 7 x 106 vol/vol psi water plus formation compressibility The initial reservoir and aquifer pressure is 1100 psia. If the reservoir pressure is maintained at 1000 psia, it is desired to calculate lo The pressure distribution in the aquifer at a time of ta 88 days, assuming the gas-water interface to be stationaryo 2o The influx of water, ft3/hr, into the reservoir at ta = 88 days from each of the two layerso

-12935 The total cumulative flux of water in cubic feet into the reservoir at 88 days. 4. The actual change in the radius of the reservoir, rb, assuming the porous rock matrix of the reservoir to be the same as that of the aquifer, based on the foregoing calculations, The pertinent dimensionless ratios for this problem are: R e 200,000 ft 100 (V-276) rb 2000 ft rbh rb 2000 A -- - == 20 (V-277) H hl + h2 50 + 50 * k2 0l (lo { 0.1 j K K2 - (5 l - K2 = 20 (-278) c h = 0 5 (point of division on y (V-279),+h 2 ~coordinate) ra = r rb = 2000 r feet (V-280) ya = Hy - (h.l + h2)y = 100 y feet (V-281) = klt (o005) (88 x 24 x -600) kltlCrb2 (1)(0ol)(7 x 10-6 x 14.7)(2000 x 30.48)2 0.1 (V-282) Since A = rb/H > 10, the tables of Appendix K may be used to calculate the pressure distribution in the aquifer. The first step is to determine whether the effect of the e exterior aquifer boundary is significant at ta = 88 days, Q = Oolo From the table for R = 100, K2 = 20, c = 0.5 it

-130can be seen that the exterior boundary will in this case have no significant effect until @ = 50. Since in this problem 0 = 01o, it is necessary to use a table for a smaller value of Ro In this case the table for R = 5,0 should give a very close approximation to P(r, y, @) since the value of P(5, y, 0) is 0~994 (nearly equal to lo000) at 0 = 0olo As is true of all the tables in Appendix K, the variation in pressure with respect to y is too small to be noted with only three place accuracy. Reading from the table for R = 5~0, K2 I- 20, c O- 05 at Q O= 0l the dimensionless pressure distribution is found to be (for all y) r P(r, y, Q) 2 o0641 3 0,899 4 o0980 5 0994 These dimensionless values may then be converted to real values using the equations ra = 2000 r feet (V-285) P(ra, Ya, ta) 1000 + 100 P(r, y, 0) psia (v-284) The real values are thus ra P(ra^ ya, ta) (feet) (psia) 4,000 l064 1 6,000 1089o 9 8,000 l098o0 10,000 109904

-131Next, the influx of water from each layer into the reservoir may be determined using the tables of Appendix M. From these tables, for R 5.0, K2 = 20, c = 0.5, at Q = Ool l = q2 = 0e.4845 The applicable equations for converting these dimensionless fluxes to real fluxes are (ft3 = 01654k)H Pql (V-195a) f t3 = 0.1654 (k2)H A P (V-196a) \hr J 2 \ For this problem ft3 _ (0.1654) (005) (100) (100) (0.4845) \ hr 1 (1.0) 4.0 ft3/hr for layer 1 (V-285) /ft3> (0.1654)(o.100)(100 loo)(100)(o.4845) ur -- 2 (1.0) 80.0 ft3/hr for layer 2 (V-286) The total cumulative flux from both layers may be found using the tables in Appendix 0. From these tables, for R = 5.0, K2 = 10, c = 0.5 at Q = 0.1 Q(t) = 1.085 The applicable equation for converting to real flux is Total Q(ft3) = 2trb cw H A P[cl' + (l-c),2]Q(t) (V-199a) where: cw = water plus formation compressibility

-132For this problem Total Q = 2T(2000)2(7 x 10-6)(100 )(l00 )(0ol)(1.085) 1.91 x 105 ft3 total water influx (V-287) Now the original volume of the reservoir was Vi = trb2Hj I t(2000)2(100)(0ol) 1.25674 x 108 ft3 (V-288) The total water influx was 0.00191 x 108 ft3. The reservoir volume at 88 days is thus 1.25483 x 108 ft3. The corresponding change in the reservoir radius is only a decrease of 1.52 feet. The assumption of a stationary gas-water interface for the problem is therefore entirely valid.

VI EXPERIMENTAL WORK, MODEL STUDIES As previously noted, the majority of the information heretofore available on the flow properties of heterogeneous and/or anisotropic systems has been obtained experimentally using models of some sort. In many cases, particularly where highly irregular geometries are involved. such as in simulating an actual reservoir, model studies offer the only means available for studying the overall flow patterns. This is also true in the case of many multi-phase flow problems, with which this dissertation is not concerned. The purpose of obtaining experimental data in this investigation was primarily to verify the analytical work. Since it has proven possible to obtain solutions to the problems of flow in stratified systems herein considered by a purely mathematical approach, it was not necessary to use model studies in determining the required solutions. However, since there is nothing in the literature to directly check the validity of the analytical solutions obtained, it was deemed advisable to verify the solutions by comparison with experimental data. It should be emphasized that the two methods of attack on the problem were completely independent. No experimental results were utilized in obtaining the analytical solutionso The experimental data were used solely to verify the validity of the mathematical work. Many different types of models have been used to study fluid flow problems. The most obvious type is simply a porous matrix of the desired size and shape, in which any desired liquid or gas may be used as the flowing liquid. Such models may be constructed by packing sand or glass beads, -133

-134graded so as to obtain the desired permeability and porosity, into a restraining containero By varying the size distribution of the sand or glass beads, it is possible to construct several kinds of heterogeneous modelso Flow models of this type have been extensively used to determine flooding patterns for water floods in both homogeneous and heterogeneous systems (6,16, 41) The use of permeable fluid flow models with single phase flow is usually restricted to studies of steady state behavioro This is not true for multi-phase studies, such as in studying the flow patterns for water displacing oilo When the entire free pore volume of a porous model is initially filled with the same fluid throughout, unsteady state flow is dependent on the compressibility of fluid to provide the driving forceo In an underground aquifer, which extends for many miles, the compressibility of the water, though small, is sufficient to provide a driving force for water to enter zones of low pressure (such as an oil or gas reservoir which is being depleted). In a laboratory model, however, the compressibility of liquids is clearly insufficient to instigate measureable phenomena of this type. Even with a gas as the flowing fluid, a very large model would be required to allow accurate measurement of the transient pressure distribution. Unfortunately, steady state models are of little use in studying inter-layer fluid flow of the type with which this dissertation is concerned. Consider a model packed with sand such as that shown schematically in Figure VI-1. At first glance it might appear that by operating this model in the steady state, using either gas or liquid as the flowing fluid, some information with regard to cross-fl;ow pressure distribution could be

-155INLET HEADER //x INTER-LAYER INTERFACE; NO INTERFACE RESISTANCE A x X OUTLET HEADER Figure VI-1. Schematic Drawing of a Heterogeneous Steady-State Flow Model. P LIQUID FLOW w (/) Cn g= GAS FLOW O L x Figure VI-2. Pressure Distribution for Heterogeneous Steady-State Flow Model.

-136obtained It can easily be shown, however, that in the steady state this model will involve no net inter-layer flowo Let the pressure at the inlet header be maintained at P2, and. that at the outlet header at Pp 3 with P2 > P!. Then the pressure distribution in either layer is that shown in Figure VI-2. This may be clearly seen by considering an impermeable barrier to be initially present at the interface, with the model permitted to reach steady state. Then the pressure distribution in each layer will become that of Figure VI-2o Upon removing this hypothetical barrier, there exists no pressure gradient tending to cause inter-layer flowo Such a steady-state model is thus useless for studying crossflow patternso The lack of crossflow in the steady-state also eliminates consideration of electric potential models, such as those used by Fatt9,.n this investigationo Such models are very useful in studying steady-state flow in systems of irregular geometry, and may also be used advantageously in studies of many heterogeneous systems in the steady stateo They are not, however, well suited to the problems of this investigation. The model best suited to studies of transient effects such as those herein considered is the heat transfer modelo In this model the flowing "fluid" is heat, and pressure difference driving force is directly simulated, by a temperature difference. As previously shown, a heat transfer model simulates the flow of a, slightly compressible liquid in a porous media A. Design Considerations for Heat Transfer Models One of the first decisions which must be made in designing any heat-transfer-analog model is that of selecting the material of constructiono

-137The major factors to be considered in selecting the heat transfer medium are (1) the thermal diffusivity, k/pCp, of the material, (2) the ease of fabrication of models from the material, and (5) the availability of the material in the desired size and shape at a reasonable cost. The thermal diffusivity of the heat transfer medium is important in several respects. First of all, it is of major importance in determining the size of the model. As a general rule, the lower the diffusivity of the heat transfer medium, the smaller is the size of the model required for a given experiment. As model size decreases, problems of handling and generally problems of insulation also decrease. However, below a certain size, fabrication of the model, particularly with regard to placement of thermocouples, becomes very difficult. Secondly, the thermal diffusivity of the heat transfer medium directly affects the type of insulation technique required. Transient response of materials of high thermal diffusity is usually sufficiently rapid that only superficial insulation is required in most models. However, with low thermal diffusivity materials, particularly those where the diffusity is of the same order of magnitude as common insulating materials, heat losses so serious as to invalidate the experiment may be all but impossible to prevent. Thirdly, the variability of thermal diffusivity with temperature may be a factor in selecting the temperature level and temperature difference driving force at which a model may be successfully run. A considerable range of materials could be used in building modelso At the upper limit on thermal diffusivity are silver, with thermal conductivity, k =- 240 Btu/hr ft2 (~F/ft), and pure copper, with k 220. At the

-138lower limit in this respect are such material as hard rubber, k = 0:..1 Btu/hr ft2 (~F/ft) and pyroxylin plastics, with k = 0.075. Many materials with low diffusivities are, however, not suitable for the majority of models due to heterogeneity or anisotropy. For example, although various types of wood might seem attractive from the standpoint of ease of fabrication, the heterogeneity and anisotropy inherent in almost all types of wood eliminates the possibility of using it in any but highly specialized models. The "ease of fabrication" involved with various heat transfer materials involves, to a large degree, the complexity of the model to be built. There are, however, certain important features which are involved in nearly all models. One very important factor which must be considered in alli.heattransfer-analog models is the means by which the model is to be heated (or cooled). This directly leads to the problem of bonding the heating (or cooling) medium to the model without introducing appreciable unwanted resistance at this interface. The two most common methods of heating a model are electrical resistance heating and convection-conduction heating (by passing a hot fluid along some boundary). In electrical resistance heating, unless the entire model is heated internally by utilizing its own resistance, the medium in which heat is generated must be bonded in some manner to the medium to be heated. This is by no means a simple problem. It must be borne in mind that the models under discussion are all unsteady-state models, in which a given resistance has a variable temperature difference across it as a function of time. For example, if a heating tape, containing

-139wires as heating elements, were attached to some part of the heat-transfer medium with epoxy resin, the temperature drop across the tape insulation plus resin would, in nearly all instances, be an unknown function of timeo This could frequently prove troublesome, especially if it were desired to maintain the boundary or region in question at some fixed temperatureo Analogous problems are associated with convection-conduction type heatingo With some materials, it might be possible to maintain the heating (or cooling) fluid in direct contact with the heat-transfer medium under investigation. This, however, is frequently impractical if only a small, isolated area is to be heated or cooled, with the remainder of the model being insulated. The alternative is to pass the fluid through some channel which is in turn connected to the model proper. Since contact resistances associated with pressure-type joints are nearly always variable with such factors as temperature difference, either a bonding agent such. as solder must be used, or the two surfaces must be directly joined. by welding. Another "ease of fabrication" factor comes into play in comparatively large models, namely the ease of actually shaping the model. For example, the two-layer radial model which was studied in this investigation was fifty inches in diameter by one-half inch thicko The model was cut to circular shape on a Doall band-saw, starting with a rectangualr slab, 50" x 60", which weighed about 425 pounds. The weight of the slab, added to the fact that stainless steel is not easily cut, made this shaping a difficult task. The third major factor in selecting material —that of cost and availability —also depends to a large extent on the type of model desiredo

-140Little difficulty should be encountered: in obtaining materials for small models, such as the single-layer linear model studied. However, surprisingly few satisfactory materials are readily available for building large two or three-dimensional models. This situation becomes extremely acute in the fabrication of multi-layer models, as will be explained later. After selecting material and heating medium, the next major consideration is the means of temperature measurement. It seems apparent that the most satisfactory temperature sensing device in models of the type under discussion is the thermocouple. The type of thermocouple material which would be most satisfactory is usually apparent from consideration of the temperature level and temperature difference driving force at which the model is to be operated. Thus, the major problem in this area is the means of utilizing the thermocouples so as to obtain accurate temperature measurements. The third major consideration is that of insulation. In simulating fluid flow by a heat-transfer-analog, a fluid flow boundary through which no fluid passes corresponds to a completely adiabatic boundary. In many models, particularly those fabricated of materials of high thermal diffusivity (e.g. copper), the short operating time required precludes serious heat losses. However, with models fabricated of steel, such as in this investigation, or other materials of comparably low thermal diffusivity, heat losses frequently prove to be the limiting factor in determining the length of time during which the model will yield useful data in a given run, It is of interest to note the difference in the insulation criteria between steady-state and unsteady-state models. In steady-state

models, the heat capacity of the insulation is unimportant; the purpose of the insulation is fulfilled if negligible heat losses to the surroundings occur, regardless of how much heat energy is present in the insulation itselfo In unsteady-state models, however, the heat absorbed by the insulating medium itself may cause significant error9 even though. essentially no heat loss to the surroundings occurso The principal additional factor involved in the design of multilayer models is that of bonding at the interface between adjacent layerso This problem is closely associated with that of bonding the heat source (or sink) to the model in that any additional resistance introduced at the interface has a variable temperature difference associated with ito Thus, if a flux or bonding film (such as solder) is present at the interface in sufficient quantity to cause a measurable temperature difference across the bond, it is imperative to know the resistance of this bonding layer. A much superior alternative is, of course, to eliminate this resistance entirely, as was done in the two-layer models studied in this investigationO The range of satisfactory materials which are readily available in a bonded condition is extremely limited. It was originally planned to fabricate the two-layer models in this investigation from copper and cupronickel. Depending on the type of cupro-nickel used, this combination of materials would have a thermal diffusivity ratio of five to eight, Moreover, the comparatively high thermal conductivity of both media would minimize insulation problems. Also, such materials would minimize physical problems of construction, since both copper and cupro-nickel are relatively easily cut and drilledo Unfortunately, however, it was not possible

-142to obtain such materials in an already-bonded state without the presence of flux and other bonding materials at the interface. In addition to the factor of interface-bonding, multi-layer models in general tend to make the other problems discussed even more critical. Consider, for example, the problem of insulation in a linear model. In the single-layer case, the only temperature gradient which should be present is in the direction of the length dimension of the model. Heat losses at the surface of the model naturally cause some gradients in the transverse direction, particularly near the surface of the model. By placing the thermocouples near the center of the model cross-section, it is possible to minimize the error caused by these heat losses. In multilayer models, however, there should be temperature gradients in the transverse plane. Thus, the position of thermocouples with respect to the transverse axes of the model cannot be chosen arbitrarily. It is usually desired to place thermocouples near the surface of the model in order to measure cross-flow gradients. The heat-loss-induced surface temperature gradients, and consequently the insulation problems, are therefore more critical in the multi-layer case. B. Description of Heat Transfer Models Studied in This Investigation A total of five heat transfer models were studied in this investigation. These were 1. Single-layer linear model 2. Single-layer radial model 3. Single-layer radial "fault-plane" model 4. Two-layer linear model 5. Two-layer radial model

-143Schematic diagrams of these models (not drawn to scale) together with. pertinent dimensions are shown in Figures VI-3 through VI-7. The steam fitting used at the heating end of the linear models is shown in Figure VI-8. Figure VI-9 shows two photographs of the "fault-plane" modelo All models were designed. to simulate the "constant terminal pressure" case of liquid flowo For heat transfer models, this case corresponds to introducing a step function in temperature over some surface of an initially isothermal conducting mediumo 1. Features Common to All Models All models used in this investigation utilized steel as the conducting medium. For the single-layer homogeneous models, the entire conducting medium was mild steelo The two-layer models utilized both mild steel (k = 26) and 316L stainless steel (k = 903), bonded intimately at the interface. These two-layer models were cut from a 50" x 60" slab of clad steel in which 1/4'" of 316L stainless was bonded. to 1/4." of mild steel by hot rollingo This slab was generously donated by the Lumens Steel Companyo The process of hot rolling produced a completely resistance-free interface for heat transfer. All. models were heated with steam at from 2-20 psig, whrich was drawn from a 60 psig steam line. With the models initially at room temperature, this provided a temperature difference driving force of approximately 170 ~F Since the steam line which was used had a tendency to give considerable condensate in the steam, it was necessary to drain all possible condensate from this line before starting an experimental run. As an additional precaution, the line was continuously vented to the atmosphere at a point just ahead of the experimental models during all runso

-144LINE AIDNG WHICH THERM()COUPLES WERE \ / INSTALLED 7 1 4I"~4T STEAM-HEATED FACE Figure VI-3. Single-Layer Linear Experimental Model. THERMOCOUPLE WIRES TO MULTIPOINT RECORDER STEAM THROUGH In 0___ _ _AMILD STEEL L SILVER SOLDER.044"jK p- BOND.275".319 24" 48" Figure VI-4. Single-Layer Radial Experimental Model.

-145450 STEAM PIPE OD:0.638" D:0.551" c 450 450\ E D 6.31" ~ 6.31" A = C D = 23.68" B =5.99" 45~ E =8.62" Figure VI-5. Single Layer Radial "Fault-Plane" Experimental Model.

-1460 cu. U Eoc I, ('0-: r 0 k 0~. oo UI^ W\< 4, -lJ U).-'0 0 0: A * 43AV z W LU. luj 1~ T'C.) W LU 1

-147STEAM THROUGH C PIPE PIPE / X COPPER PIPE -.044' SILUER SOLDER.044-' - / BOND MILD STEEL ( II STAINLESS a ~______________~~~_ ____ STEEL f ~- ] —.275" Figure VI-7. Two-Layer Radial Experimental Model.

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-150The steam pressure within the models was manually controlled by regulating the inlet and outlet valves on the model. A high steam velocity through the model was maintained in all cases. This high steam velocity was principally designed to prevent condensate build-'up in the model, It had the additional effect of preventing the build-up of non-condensable gases within the model, and of increasing the steam condensing coefficient. In all models, a thin layer of copper separated the steel conducting medium from the heating steam. In the linear models, a flat copper plate,.042 inches thick, was brazed across one opening of a two-inch pipe cross, and the steel bars were silver soldered to this plate (see Figure VI-8). In the radial models, a copper pipe with an outside diameter of 0.625" was used to contain the steam, with this pipe again being silver soldered to the steel conducting medium. A twelve-point Leeds and Northrup Speedomax recording potentiometer was used in conjunction with copper-constantan thermocouples to record all temperature measurements. This recorder was a one second full-scalebalance type, and recorded a temperature every two seconds. A complete traverse of all twelve thermocouples was therefore completed every twentyfour seconds. The scale read directly in degrees Fahrenheit, with a scale range of 60-260~F. A chart speed of 60 inches per hour was used in order to separate the points recorded. A total of six different colored points were used to distinguish between thermocouples on the chart. The chart could be read within + 0.5~F. All thermocouples were calibrated at room temperature and in boiling water before installation in the models. The multi-point recorder was used as the sensing device in the calibration, so that the temperature

4151measurement would reflect errors in either the thermocouple itself or in the recordero No measureable difference between thermocouples was detected in any of these calibrationso The comparison between the thermocouple readings and that of a calibrated mercury thermometer was within 0.5~F at both room temperature and boiling water temperatureo Since the recorder chart could only be read to + O05~0F it was concluded that no correction due to thermocouple or recorder error was necessaryo This does not, however, mean that the temperatures measured were exact, since this is a function not only of thermocouple and recorder error, but also of errors induced in the installation of the thermocouples in the modelo This factor will be discussed in more detail at a later point in this dissertationo The means of installing the thermocouples was essentially the same for all models. First, a very small thermocouple bead was formed, either by electric arc welding, or by careful solderingo A hole slightly larger in diameter than this bead was then drilled. at the desired spot in the model. The depth of drilling was immaterial for all but the two-layer radial model, in which it was carefully controlled. The thermocouple wires near the bead were electrically insulated by coating them repeatedly with a solution of Duco cement thinned by acetoneo The thermocouples were then placed in the hole, and held. in. place both by filling the void volume in the hole with additional Duco cement, and by using Scotch brand electrical tape to anchor the wires near the holeo 2. Individual Features of the Models For the purposes of this section9 the models wilt be discussed in the order in which they were actually fabricated and runo

-152a. Single-Layer Radial Model As indicated, this was the first model to be fabricated and run. The diagram of this model appears in Figure VI-4. This model was principally designed to check out the experimental technique to be used on the multi-layer models. The analytical solution for single-layer radial flow was unquestionably valid, so that there was no necessity to verify the solution using experimental data. The thermocouples used with this model were 24 gauge copperconstantan. The thermocouples were installed with Duco cement as previously described. The thermocouple wire was then taped to the steel plate, leading in a direct line from the point of measurement to the multipoint recorder, as indicated on Figure VI-4. Two quick-opening three-way valves were used with this model in an attempt to raise the heating face of the model to the desired temperature as rapidly as possible. The upper of these two valves can be seen in the photographs in Figure VI-9. A line diagram of the steam system for the model is shown in Figure VI-10. The steam was initially routed through the bypass line until the system became fairly well stabilized with respect to pressure variations. Both three-way valves were then turned simultaneously, to reroute the steam through the heating pipe of the model. The valve in the bypass line was used to equalize the pressure drops in the bypass line and in the heating pipe of the model. It was thus originally planned that the pressure equilibrium of the model should not be much disturbed when the steam was rerouted through the model itself. This did not, however, prove to be true, due to the considerable condensation of steam occurring in the heating pipe of the model in the first few seconds after steam was admitted to it.

-155~~~~~~iiiii~~~~~~~ g | II t)d r H cI E4d ~ a _,~ 7\ 0 84 e\| f T - — V^ ^ | l

-154This model was insulated by a combination of glass wool, styrofoam, and aluminum foil. The insulation actually touching the steel plate was glass wool, except for small styrofoam blocks used to support the plate. Outside one thickness (about four inches) of glass wool, a layer of aluminum foil was used to minimize radiation losses. Another layer of glass wool followed. The top and sides of the model were then covered with styrofoam, and the crevices between the styrofoam blocks were sealed' with tape. The entire model rested on a plywood base. The insulation shown in the photographs in Figure VI-9 is nearly identical to that used for this model, except that in the fault-plane model the top was not covered with styrofoam. Thermocouples were placed at various points in the insulation of the model in order to find points of maximum heat loss, and gain a qualitative idea of the efficiency of the insulation. The results thus obtained were entirely compatable with the deviation between analytical and experimental results which were found. b. Single-Layer Radial "Fault-Plane" Model This model is simply the single-layer radial model, with a slice cut from it in such a manner as to simulate a reservoir fault plane. The principal reason for running this model was to illustrate the application of heat transfer models to the study of irregular geometries. The thermocouples used with this model were the same as for the single-layer radial model, but were installed in two different arrangements in order to gain a complete picture of the temperature distribution. The major difference in temperature measurement technique was the fact that

-155the thermocouples were in this case taped, to the model along an arc of approximately constant radius for at least six inches from the thermocouple bead. This technique was designed to decrease the error caused by conduction of heat by the thermocouple -wire The insulation used for this model, along with the thermocouples used, are shown in Figure VI-9. As previously noted, the only difference from that used for the single-layer radial, model is the absence of a styrofoam top layero The two three-way quick-opening valves, in conjunction with. a. steam bypass line, were also used, with this model. co Single-Layer Linear Model This was the third model to be fabricated and run. The primary purpose of the model was to refine the experimental, techniqueso In, particular, the method of insulating linear models was thoroughly investigated.L Also, this model made use of 30 gauge copper-constantan thermocouples, rather than the 24 gauge wires previously usedo The model was first constructed and run with no insulation in order to establish a limit on possible heat losseso For the next set of experimental. runs, the bar was first wrapped with asbestos sheeting, then with aluminum foil (over the asbestos), and then covered, on all sides with glass wool This insulation was roughly comparable to that used in the two previous models. T'he results obtained checked moderately well with the analytical solution, but still. indicated significant heat loss from the bar after a short period of timeo As previously me:ntioned, the problems of insulating unsteadystate models are often qui.te different from those for steady-state modiel.so

-156Heat lost from the model due to heating of the insulation in an unsteady state model is frequently a major source of error, even though no heat is lost to the atmosphere beyond the insulation. The ideal situation for unsteady-state insulation is to have the insulation immediately around a model heated by some external means to a temperature exactly equal to that of the model. This can be most accurately done by using a feedback loop, whereby thermocouples sense the temperature of the model, and the model temperature is used to actuate and regulate some electrical heating device. Such a system is usually extremely accurate, but is also quite expensive. A much simpler, but very satisfactory, device was used in this investigation to heat the insulation surrounding the linear models. As shown in Figure VI-8, the linear models were heated at one end, at which they were silver soldered to a steam-heated copper plate. In order to heat the insulation from an "external" source (i.e. not the bar itself), a galvanized sheet steel "channel" was also soldered to this copper plate, as shown schematically in Figure VI-11. This sheet metal channel was then also heated by the copper plate, at approximately the same rate as the model. This is, at any time, the temperature at a given.distance from the copper plate would be nearly the same in the bar and the sheet metal channel if there were no heat losses. The bar was wrapped first with asbestos sheeting, then aluminum foil, and was supported inside the channel with styrofoam. The remaining void volume inside the channel was then filled with glass wool and/or powdered asbestos. The entire assembly was surrounded with approximately six inches of glass wool.

-157STEAM -HEATED COPPER PLATE LINEAR HEAT TRANSFER MODEL (STEEL) CHANNEL \:I Figure VI-ll. Sheet Metal "Channel" for Insulating Linear Experimental Models.

-158Since the sheet metal channel had a considerably greater surface area than the model proper (i.e. the steel bar), the insulation received the majority of the heat necessary to raise its temperature from the channel, rather than the bar. Heat losses from the bar were therefore minimized. The thermocouple wires in this model were withdrawn vertically from the point of measurement. Since the wires were thus perpendicular to the direction of heat flow, they were thus approximately along an isotherm. This tended to minimize errors in temperature measurement due to heat conduction along the thermocouple wires. Experience with the two single-layer radial models had indicated that the steam system incorporating quick-opening valves (see Figure VI-10) was not entirely satisfactory. The steam system was therefore modified to that shown in Figure VI-12 for the three remaining models. In this system, the steam was vented to the atmosphere immediately after the surge tank until the amount of condensate in the steam decreased to the minimum possible. To initiate the run, the steam pressure in the surge tank was allowed to build up to 20-35 psig, and the 3/811 globe valve was then rapidly opened. The resultant surge of steam through the model raised the heating face to the desired operating temperature in less than ten seconds. It was then necessary to simultaneously close the 3/8" globe valve and open the 1/4" needle valve, in such a manner as to maintain the temperature of the heating face constant. The temperature of the heating face was then controlled for the remainder of the run using the 1/4" needle valve.

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-16od. Two-Layer Linear Model This was the first of the two two-layer models to be run. The model was nearly identical in design to the single-layer linear model, except for the bi-metal character of the bar. The insulation methods used, including the galvanized sheet steel channel, were identical. As indicated in Figure VI-6, the thermocouples (30 guage copperconstantan) were inserted from the "edge" of the bar. In this way, it was unnecessary to carefully control the depth of the thermocouple bead, as was true for the two-layer radial model. The thermocouple wires were again withdrawn perpendicularly from the model in order to minimize the effect of heat conduction along these wires. e. Two-Layer Radial Model This model was the last to be run, and as such incorporated most of the improvements in experimental technique which were found from experience with the other models. This model was run first with the mild steel side up, with all thermocouples in the mild steel side, and was then inverted and rerun with thermocouples in both layers. The fabrication of this model was the most difficult in several respects. First of all, the sheer weight of the metal plate (over 300 lbs.) made handling difficult. Second, and more important, this was the only model in which the depth to which the thermocouple bead was inserted had to be carefully controlled. This in turn required that the thermocouple bead be very small. The mean of inserting thermocouples in this model is illustrated schematically in Figure VI-13. The ceramic insulation had a diameter of

COPPER CONSTANTAN 30 GUAGE 30 GUAGE CERAMIC INSULATOR -THERMAL MODEL (STEEL) THERMOCOUPLE BEAD Figure VI-13. Thermocouple Installation in the Two-Layer Radial Experimental Model.

-1620.058 inches, and was predrilled to accept two 50 guage bare thermocouple wires (diameter OoO10 inches each). The thermocouple beads in this case were formed by electric arc welding under Xylene. The beads obtained were all of the order of o.010 inches in diameter. By carefully controlling the depth of the hole drilled in the steel, and by inserting the ceramic insulation (together with the thermocouple bead.) to the bottom of this hole, it was possible to position the thermocouple bead within + 0.020 inches of deptho It should be noted that the thickness of each of the two layers of the model was 0.260 inches, so that the uncertainty in position of the thermocouple bead represented approximately 8% of the total thickness. The ceramic elements were held in place in the model using Duco cement and Scotch brand electrical tape. The thermocouple wires were withdrawn perpendicular to the model surface in order to minimize errors in temperature measurement due to conduction along the thermocouple wire. C. Experimental Data The original data obtained from the experimental models in this investigation were obtained in the form of multi-point recorder charts, It should be noted that the pressure gauges associated with the experimental apparatus were used only to facilitate the manual control of the steam pressure in such a manner as to maintain a constant temperature at the heating face of the models. The experimental data will be reported in two formso First, the basic data, as obtained directly from the recorder charts will be presented, These data are in the form of actual temperatures as a function of actual

-163time and actual position. Tables will also be presented of the dedimensionalized data, in the form of dimensionless temperatures as a function of dimensionless position and real timeo This second set of tables is much more useful from the standpoint of interpretation, since it is difficult to readily note such factors as reproducibility using the tables of basic data, As previously noted, the recorder chart speed used for the majority of the experimental runs was 60 inches per houro In most cases the initiation of the temperature step function at time zero was accomplished sufficiently well to obtain a time scale of + 0~25 minuteso The conversion of actual temperatures, as recorded on the multipoint recorder chart, to dimensionless temperature was accomplished as follows: 1. Before each experimental run, the temperature at all points in the model was recorded using the multi-point recordero The maximum gradient occurring as an initial condition in any of the runs was 2~F over the maximum dimension of the model. In most cases, no detectable initial gradient was present. The values reported in the tables of basic data represent the average initial temperature of the modelo 2. The time at which the temperature step-function was initiated was estimated from the recorder grapho This estimate was facilitated by always starting the steam flow through the model at a time 4 to 8 seconds before the first heatingface temperature measurement was due on the recordero The time thus obtained was designated as 0Q ta - O o

-164 3. The actual temperature measured by all thermocouples was determined as a function of time, ta, from the recorder charto Near the beginning of the experimental run, during which time the temperatures near the heating face were changing fairly rapi.dly (io.e up to 30OF during the 24 second recorder interval), it was necessary to draw a curve through the recorder points to facilitate interpolationo The temperatures were read from. the chart within O o5F in most caseso These readings of actual temperatures are presented i.n Tables VI-1l VI-49 VI-6, VI-8, and VI-10 of this dissertationo 4. The ciL.uai temperatures, Ta, were converted to dimensionless temperatures, T, using the equation Ta Ti Ts - Ta T - 1 lT -- I ) Ts - Ti Ts - Ti in which: Ta is the actual temperature,^ F obtained from the recorder chart Ti is the corresponding initial temperature at that point, ~F Ts is the mean temperature of the heating face of t.he model over the course of the experimental runm The heating face temperature was maintained within + 2'F (after the first minute) in nearly all caseso Any extreme fluctuations in this temperature (as frequently occurred due to steam line surges, which. in turn were due to the large amount of condensate) caused the run to be

-165discontinued. The experimental runs reported herein constitute perhaps 30% of the runs which were actually begun on the various experimental models. The remainder of the runs were discontinued at a sufficiently early stage as to be totally worthless, or had some other distinct defect (such as a steam leak within the model insulation) which would. negate their value. The experimental data for the models in the form of dimensionless temperature as a function of real time, with parameters of dimensionless position, are presented.in Tables VI-2, VI-5. VI-7, VI-9, and VI-ll1

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-169TABLE VI-4 BASIC EXPERIMENTAL DATA FOR THE SINGLE-LAYER RADIAL FAULT PLANE MODEL Heating-Face Run Temperature Initial Condition Number (~F) (~F) 1 235. 85. 2 240. 101. 3 24o. 74. 4 24o. 75. 5 240. 70. 6 240. 78. 7 24o. 74.5 Actual Temperatures from Recorder Chart, ~F First Set of Thermocouple Locations Thermocouple 2 Thermocouple 3 Thermocouple 4 Thermocouple 5 Thermocouple 6 ta Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (min) No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 1 129 140 126 124 122 134 114 118 88 104 78 79 88 104 78 79 88 104 77 ~ 79 2 141 153 135 137 135 107 129 131 96 n1o 85 87 96 n0o 85 87 95 109 84 86 4 152 162 147 148 146 157 141 142 105 119 96 98 105 119 96 98 103 117 94 96 7 16o 169 156 156 154 164 151 151 114 128 107 107 115 128 107 107 112 126 104 105 10 163 173 16o 161 159 169 156 157 120 133 112 113 122 135 114 115 118 131 110 111 20 170 182 168 169 168 181 166 167 130 145 123 124 136 151 130 131 129 145 123 124 4o 179 186 176 176 179 186 176 176 140 151 135 135 149 16o 145.45 141 152 136 136 70 182 181 182 182 181 182 147 142 143 158 154 155 149 145 146 100 184 185 148 159 151 200 Thermocouple 7 Thermocouple 8 Thermocouple 9 Thermocouple 10 Thermocouple 11 Thermocouple 12 ta Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (min) No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 2 No. 3 No. 4 No. 1 No. 3 No. 4 4 88 104 77 79 90 106 80 82 88 104 77 79 7 93 109 84 84 99 114 90 90 93 108 83 84 10 97 112 88 88 107 120 97 98 97 111 87 88 20 106 122 98 98 122 139 115 116 6 123 98 99 86 102 76 77 86 102 76 77 75 40 116 129 108 109 138 149 132 132 119 132 111 112 90 105 80 81 91 106 82 82 86 76 77 70 124 117 118 147 142 143 128 121 123 95 86 87 98 89 90 89 80 81 100 123 148 128 92 96 86 200 Second Set of Thermocouple Locations Thermocouple 2 Thermocouple 3 Thermocouple 4 Thermocouple 5 Thermocouple 6 Thermocouple 7 ta, Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (min) No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 1 112 120 115 107 115 110 74 82 79 74' 82 79 72 80 76 72 80 76 2 128 132 129 120 127 124 82 90 86 82 90 86 76 85 81 77 85 81 4 137 143 141 133 139 137 94 100 97 93 100 97 86 93 90 87 93 90 7 146 151 150 142 147 146 102 109 106 103 110 107 94 101 98 95 102 99 10 151 156 154 148 153 151 109 115 112 ill 118 114 100 107 io4 102 109 o106 20 16o 165 164 158 163 162 120 126 124 127 133 131 112 119 116 118 124 121 40 169 173 172 169 173 172 132 137 135 142 147 145 124 130 128 133 138 136 70 179 178 179 178 145 143 157 155 139 137 148 146 100 200 Thermocouple 8 Thermocouple 9 Thermocouple 10 Thermocouple 11 Thermocouple 12 ta Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (min) No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 No. 5 No. 6 No. 7 4 71 74 81 77 71 7 73 81 77 80 87 83 75 82 79 10 75 83 80 86 92 89 79 86" 83 20 83 90 87 101 107 104 72 80 76 92 99 96 40 93 100 97 117 123 120 77 84 81 108 114 112 72 80 76 70 109 106 134 152 91 88 125 123 85 81 100 200

-170o c 000 p^.. r-1 r<^ fQ C! i^\ C - al I rr\ CC ) CM 0 0 H PC - Co 0 o 0 (\ 0 OCO H 4-t^> 0 C* 0 0.. s 00'.0 a o HO\CT\ChK\^-^OC\1^0 Q t- rCo 0\t\.0 H\...\C. 0 O Coj. 0 C C-P00 ON. N K 0C\ C0 X Cc to-.0'\ r- C0\ P (............. 0 14~~~~~~ ~ Po 0 0o o o\0 H a o CH'( C OC4 EC\ (00 0'. CO o — ICoCo (0 Cod^r 00 \0 -i \C\ co t-, 3\ o\ 6 pP- V o5C \ Q C \O'c\o p0 OfT~~ooo'.~o'.OK^ rt rr\0f^ u wo P o~oo[ o &H0-\ 0 ( ( 0. % si ^..0...... PI~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~po t ( 0 0\ 0\ OCO C \O J~ ^ irEt~-o~ot —ou~ir\ Q C- (0\ 0 - 0 CU<S C0 UO t- 0 Lr\ i.... 0.CooCH, 0(0 C C-o- (0 a"&(2'I 0 o c\D~~0 ( O OH DC0\ ('0 IO C <.a......... (0 V\~c\L —0 \J' p - 0 - Od otoo a ci coip\o fMir~oir\ ~l a\ QJH u 0oa\o'o5aoo:', v, O, u~ - i o\ i-lo c-~-rco P; Co\\ols o. C' * tQ 0....t a co \Co\0\ o \U-\o U\ Pi 0 0 - \ ta ~ ~ ~ ~ ~ { Q-o (j oc 8i 8I f'.'(-O (0 0c H (Q0 U'\o0\0\coo~t( 0 OC\ c o0 (0 00 a' ~ CoC-Coo (0, a\ C, 0 a\ t C \D C-r\ \ ""i iozU'U............ 41........ o H (0c~~~~~~~~~~~~ o U,0 =o P4 L —\ oC P- o(.M (3\ H O 0\ r'I It 0m H 00 " 0t- \ l0 UN U0 C k > ooo 0 0 o-C —Clt'. i o g o o 0 a C r-C 0 (0 — 0' -- g;...... p<....... | U.\ L8 — u\ 5 ( o \D a\1 K\ S0 (So c Co J'~1 ^'\ i t \ O ^ ^P ^ ^ d "1 U~o r ^ K^ O \0P 9 i-l?-I\ C 4C L O- o oU C- 0 a' * 0O 0 0 (Lr \ \"0 - O \ H pC ) — 0 (00Kot- o \ -a tDP 0 -- -a- 0 o 0..... (0 0, o~o io-uo- ~- ( \&.~ 0.l (0 C- c C-oil'. to. P. COOO1t co c.CCra'o Po.o Co ri K\_D I' 0\.OO\ -H o- f P \- t>o^0 ir\ r -t t- 0\C CC)COI...\ o.-..c\co \\ \ c Co ta'.co C-C —Co.. u'~kc 0 0 0~\i0r^_ K~\ t \j0cINd \00;....... ^ a - - - - - Q k...... I.~0R ~ ~ E 05 0 (0 C oC ~ H\CO\D \O \0 in in in(0 006 \.4S I 0 1 is co 0 I; \. ~.H(0oH- oH.. 0'. 0j~.g ~ ~.-oa.- H\0H^.0.O...'. C-\ Co o 0. L-r\ o i (00 C) — I C - N - (00 C-Cou~ o~~o4 —4o ko (C- Co-_ - 0 a 0 H\\... H0...-00 (0\ o -4- H0 8 -4\ C.).... - Lrl\ UN Cl.)...

-171-' - P; -....O..C C.. Cuir1 H H H H 0\.. Cu * i Hl-^..,(* rC\ t-H r- C.- C. O... Cu( y- H H H H ~C l\ V\LnV\V\ U\ C- \ 0\... U\.S\ 0 LC.LC J' O - J O8 K 0o ir u\rU L0 ~ r\ 0\ *0.0 CO ~ 0 0 -o _ cO 41^iPl J 4,U ~. C. C........- 0. C. C.~..r C.4a'Cu C-a' a C vIu-5O H a'a' 0~o 3o'''D H \~ ~ ~ ~~~~~0 M I 5 oy ^o^HoCu - 0 HHHH H HH &< uy a <a h -^-irai~ir ~ m ir~i~ ir~ icO CuCc-K —^0~ 0t v Cu 0 C-\ tCi.-C-dil 0 Cl Hu HHHH Cu H WH - 0,.,,-~'0-Ca'a 0H 1ii CVJ K\ tt O c iIKC a-t' C- -a Orlrl- r CIC J0 H M 0OO ~ r-lH~iJ\0 Cu C Cu HH HH HCM H J g H 0.o C-. CuO d K'\0C-rn n CO 0 Cu\Cu Cu\ OOHH u u C ur\ CN CucC o -H HHHHH u uu1 cu -MH H HH.0 U 0 L \ Ur Cu 0l 0K K\JS t- r l8'\0C-COa'H H H CO -- c r\ I Cu CO HHHHHH CuCu Cuj Cu H-l~ HH H LC.H\D 0 — c CO IC 0\ Cu O =~HHHHHHHH H H 4 HHr H CC gr8 1HCU H HHCu

-172TABLE VI-7 DIMENSIONLESS TEMPERATURES FOR THE SINGLE-LAYER LINEAR MODEL Run No. 1: No Insulation or Bar Runs No. 2 and 3: Asbestos, Aluminum Foil, and Glass Wool Insulation. No Sheet Steel Channel Runs No. 4 and 7: Asbestos, Aluminum Foil, Sheet Steel Channel, and Glass Wool Insulation ta Run Run Run Run Run Run Run Run Run -Run Run Run Run Run Run (min) No. 1 No. 2 No. 3 No. 4 No. 7 No. 1 No. 2 No. 3 No. 4 No. 7 No. 1 No. 2 No. 3 No. 4 No. 7 x = a 0.05 x =.10 x = 1.5 La 1.734.786.774.788.768.967.982.976.982.976 1.000 1.000 1.000 1.000 1.000 2.594.627.621.622.616.884.902.900.900.899.979.988.985.988.985 4.469.477.474.475.470.752.768.764.770.765.913.927.921.926.923 7.394.376.376.369.366.645.639.641.640.640.827.829.829.829.827 10.358.322.324.315.310.585.566.565.566.560.770.755.756.758.750 20.308.242.241.228.223.495.431.432.421.416.663.609.606.593.593 40.284.187.184.168.164.451.334.332.313.310.603.480.476.454.453 70.275.154.150.135.131.430.279.274.254.250.579.401.397.369.366 82.144.264.383 100.139.116.247.220.362.325 110.112.215.315 150 225.124.093.215.176.309.259 x =.20 x =.30 x =.40 4.973.982.976.976.979 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 7.916.924.921.920.923.985.991.991.988.991 1.000 1.000 1.000 1.000 1.000 10.866.869.865.864.863.967.972.971.970.973.993 1.000.997.994.997 20.758.728.724.714.714.899.893.888.882.884.961.967.965.959.964 40.692.591.585.563.560.830.768.764.743.744.907.884.879.867.869 70.663.495.491.460.461.794.664.659.631.631.878.792.794.764.765 82.474.636.762 100.444.405.680.565.726.696 110.395.548.678 150.404.547.668 225.374.328.506.455.615.571 x =.50 x =.60 x=.70 20.985.990.988.988.991.991 1.000.997 1.000 1.000.994 1.000 1.000 1.000 1.000 40.949.942.938.935.935.970.976.974.970.973.982.989.988.985.988 70.919.869.865.849.851.946.927.924.914.917.961.957.956.950.952 82.847.908.945 100.808.786.876.863.918.908 110.770.847.894 150.750.818.865 225.691.649.759.726.803.774 x =.80 x = 1.0 20.994. 1. 000 1.0.000 1.000 1.000 1.000 40.988.997.997.994.994.997.997 70.970.979.976.970.973.991.991 82.969.985 100.947.940.968 110.926 150.900.926 225.838.809.868

-1735co0 I'D 0 t0- H p;~~0 oru 0... oC A... 1,0' ^^~^~~ S I t s 02I 02 0 K'a, 0 0 4 - A...I...... r......oD s co.\ o.. 0 N i-li -4 r \'IoC r-o co E1fl 0\\t ot. to- ('sri N d) i icc \o rc (o cu C- c H o 00t cyUCUCUCUCUU \' _t tc c — O C ) -cc 0 - H _ -C -C-t -IO-I - 0 (SkO - r1 ir-4r-T C -0a00-S- 0i Hi H. HH H CU ooNM CM ~-oo CM CM o CM i. i 0 XCUl r 0 r f r-l yH vM M 0} ^""0 H -'... C-0. (1 HH HH H Cci CO cc rC-r U- -rc Ln 0Iip i\ iH\ Lr\ 0 0 0 0 H -. )r-I CCL0 0% C C) HHHHHl-l-aiMCJ HH r-H 0 tr _r(7 Ui U/ Lr\ L) 0\ r; O~ 6 I ON L 0 0R H 0 S ~ f<<R\^t-0% 30000 a zd. 0 Kt^t K^ lO r II -~lnlu~c^-:td.^o ^ Sa C 11 o -— t- UDt-0\O — r- (0 CcO-C\o-HjLr\ -t —CY\ CC rj 0 O'\ C\J, 01 r-fi-l ri-' -I(SA\|<MC\J > t- I r-4ci\0cO H H H H O. r -l-t \0 L-c- C\ - H -o t- c c CM UHNkoC O D 0 0%H0r-4r-ir-Cr-0OHai<X - r- rO —C-r-O ^0 a O. C r-4l-0J-ltC O r —cC-C 00\0a-O c'J t0 0- cOOcO r t0 rHl. -COO- r-1 -CO ic - Ct-OCOc H tf-iCcj- tC — 0 00 CcOOJ-0C —00't O oO 0000 0 0 x - r cy rl H c- 0c C- -- oHo\ 0C-0o)\ e-f-o; ~ H HH HH~; t Cf d 0

\o r4 0 * _- (HDN N L' C —0 0H C'JCr0- HCO 0 t —'DOD Lr\ 0 \;............d;o o- Ho o o H O 0 O C11 - O - on O CDN o 0 t \ \0 - I r; P h.\..D \vo \ o-t c(n Cu cu Cvl o o \ O\ c N'D....... o.... O II oN\Hn ncOLi\OnLNLn\o C Orl CO0. C —o0Q cOHr\C-^ j- u 0 * OO H or HO pO * 0\NaCO O'-OJ-=mOcuOu 0 0\o \ CO\o Y - UN \O0 LC \ l \ LrNj-H 0 t CO; r — 0 O 00 N l- t- \ CC)Cq OH C 0 A x;..........''HO......'CJ 0~C~' o CO ~ Pi. t1 0\ 0\ \o \H -- oCU CU o O M 01 \\ CC)\O CE;..........6.... Oc Co COC o\ H \o u~- CCU 0 o O\ C O C AO Cu-l\0 Hc \0 o CO NOC —O 0 1 U 3 O, c t a - LCn.... o O.... H \1 # t0 nfW f - t N rl\ H H e O 0....L. \1 \O 0 N M n 0 O H C O CO *O CC- O L NY 0. cu U. d ixi - - m C o C C -1r CJH O NCO (L0 0N t- Lr\- -4- Ot 01 O N NC I II.. ON c~N Cu.~-tlC HP ^ CO C.......... 1..... HFf ~ ~ c P O — M M 0 \O D O N C y;d -N O t — \L o m C Nr\ r\- -c;E; 1 * HC a H OH O -H O cfi II O CQ H 7;r CO 00 -4 1 YNCUHH ON OC O C- O ON 0\ 0 ~ CO O r- q a H \IOo A r o\ \oCO C-1-'O ONO\ NO E-1 0 -40 Lr\ N \0 ^ O 00 rCC) NO D 0 J f l\ 00 r 0 0 n 0 o 0 t- L6\ d — I~ OJ0H rH H 0 0 ON OOO C-V - \0 0C 0C C'I CO.. H..H.,... 0 _ CC 0 ( ~.. C(x]) - h CU d * v! O ID C OO O N t- L O\ \ 0 c r. I 0 ~; ZO COO 0 F I 0 0;NN> O O0JO O\ 0 0 p oo co^-L^^^-oioji-iLf^ ooocr~oooo'<oLT\-=LC c(oC a,ii a N F0 0 g 0 O\ NN M O M ffi O H 0 0 O _ 0 O ON ^....... I I.................. UN 0 -C — C>-d- 0 N O\ 0 -C C0 0UOONC\ NC COH I C\OO CO O O N4 - - N)CU 0CU H HH O ON \' \- (' 0 ONO OOD a; zO 0 % L\ -J- J N N H 0 0 \ O) 0 - M M O ON C - C.......*II o N O c-O O- O ON CU N - o0 o o N C CO M O,O O O, — O o COO Lc_::- H 0ONO CNCO O J' 0 O\m0 M- Lr \ ON ON ONrl ^ - H, CuHU-\ —4 _: ~ M O \NCO kOC\ C - \O \ C-MO' w EH C- HH ON - OHU4-Cu(' O 4-O 00\000\0 LIN H —-c'NO 0 ~ ~ ~ HC4C-OooOHU —000000 H~~u4-C —OON H ~o4 —0N~ 4-C-oON HH- HH- HHIr

-175TABUI V1-10 BASIC EIDO IMQTAL DATA FOR TEB TWO-IAYR RADIAL MOIL Run No. 1, 2, and 3 ver made with th* mild *teol side up, all therboouples in the aild *ide. Runs No. 4, 5, 6, and 7 vwor.ad with the stainles teel ide up, with th troooupl e in both layers. Iotting-face Initial Run Temprature Condition Numb.r (r) (IF) 1 224. 73.5 2 224.5 81. 5 225. 82. 4 222. 79. 5 222. 78. 6 225. 80. 7 225. 78.9 Actual Temperatureo from the Recorder Chart. F ta Run Run Run Run Run Run Run Run Run Run Run Run (oin.) No. 1 2 No. 2 No. 5 o.. 1 N No. 1 No. 2 No. 3 o. 1 No. 2 No. ra-1.26in.; ya.0.59in. r -r.20in.; y-O.59in. ra-5.13in. iy-~0.539n. ra-4.39in.. ya-0.39in. 1 107. 115.5 114. 81.5 88.5 90. 75. 82. 83.5 2 120. 125. 125.5 91. 98. 99.5 79.5 86.5 88. 75. 82. 85. 4 152. 134.5 135. 103. 108. 109. 88. 94. 95. 79. 85.5 87. 7 159.5 142. 142.5 112. 116.5 117. 96. 101.5 102.5 84.5 91. 92. 10 144. 146. 147. 117.5 121.5 122. 101.5 106.5 107. 89. 95. 96. 20 151. 155.5 154. 127. 151. 151.5 112. 116.5 117. 98.3 104. 105. 40 157. 161. 160.5 155. 140. 140. 120.4 126. 126.5 107.5 115.5 114.5.70 162.5 165. 165. 141.5 145.5 145. 128. 152.5 133. 116. 121. 121.5 100 164. 167.5 167. 144. 149. 148.5 131. 136.5 136.5 119. 125. 125. 200 172.5 155. 143.5 135. 500 176. 159.5 149. 139. re,6.26in.;ya-O.59in. r9=12.l51in.Iya-0..9in. r,1-8.76in.;ya-0.39in. r^d5.001n.I;ya0.39in. 4 74.5 81.5 83. 7 77. 83.5 85. 10 80. 86. 87. 20 87. 95. 94. 74.5 82. 85. 40 95. 102.5 IM.5 77. 85. 85.5 81.5 82.5 70 105.5 108.5 109.5 85. 89.5 90. 76.5 85. 84. 75. 81.5 82.5 100 106.5 115.5 113.5 83.5 95.5 93.5 78. 85.5 86. 75.5 83.5 8,.5 200 122.0 102.5 96. 91. 500 128. 109.5 101.5 98.5 t Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (in.) No. 4 lo. 5 no. 6 No. 7 No. 4 lo. 5 No. 6 No. 7 No. 4 No. 5 Xo. 6 No. 7 No. 4 o. 5 o. 6 Io. 7 No. 4 No. 5 o o. 6 o. 7 ro-1.26in.;ya,0.l1ln. r-1.26in. ya-.039in. r-2.20in.;yl-0.135n. ra-3.13in.;ya-0.15in. ra-3.llin.;y-0.359it. 1 107.5 15. 1. 105.5 13. no. n.5 112. 86. 84. 85.5 5. 80. 78.5 81. 80. 80.5 79. 81.5 80.5 2 118.5 116. 119. 118. 125.5 121.5 124. 125. 94.5 92.5 95. 94. 81.5 8. 84. 85. 83. 85.5 84.5 4 129. 128. 150.5 129.5 153. 152. 155 4. 104.5 103. 105. 104.5 90. 88.5 91. 90. 92.5 90.5 95. 92.5 138. 156.5 139.5 158.5 140. 139. 142. 141. 113.5 112. 114. 113. 97. 96. 98. 7. 1 00. 98. 100.5 100. 10 12.5 141.5 1544.5 143.5 144.5 143.5 146.5 145.5 118.5 117. 120. 119. 1021.1 01. 1. 1. 104.5 103.5 1l6. 105. 20 150.5 149.5 153. 152. 151.5 150.5 154. 155. 128. 127. 130. 128.5 112. 110.5 115. 112.5 114.5 113. 115;5 115. 40 157.5 156. 159. 158.5 158.5 157. 160. 159.5 137. 135. 158. 157.5 121.5 120. 123. 122. 123. 122. 124.5 124. 70 161. 163. 162.5 161.5 164. 163.5 141. 145.5 145. 127. 129.5 128.5 128. 130.5 150. 100 165. 166. 165. 165.5 166.5 166. 144.5 147. 147. 131. 135.5 155. 132. 134.5 154. 200 169.5 170.5 152.5 139.5. 140.5 r,-l8.76ln. y^-0. 13n. r^-6.26.yin O. y,.0.5.10i n. ya-^..00i. y10in. r 1in.;. ra12.51in.; y —0in.59in 4 80. 78.5 81. 79.5 7 81.5 80.5 83. 81.5 10 84. 82.5 85. 84. 20 90, 89. 91.5 90. 80. 78.5 81. 80. 80. 78.5 81. 80. 40 80. 80.5 79.5 98.5 97. 1oo. 99. 83. 81.5 83.5 82.5 85. 81.5 8. 85. 70 80. 82. 81. 104. 106.5 105.5 79. 80.5 79.5 85.5 87.5 86. 86. 88. 87. 100 82. 84.5 8. 108.5 111. 110. 80.5.5 81. 8. 91. 89.5 89.5 91.5 90.5 200 90.5 118.5 87.5 98. 99.

-176TABLE VI-11 DIMENSIONLESS TEMPERATURES FOR THE TWO-LAYER RADIAL MODEL Runs No. 1, 2, and 3 were made with all thermocouples in the mild steel layer. Runs No. 4, 5, 6, and 7 were made with thermocouples in both layers. ta Run Run Run Run Run Run Run Run Run Run Run Run (gin.) No. 1 No. 2 No. 3 No. 1 No. 2 No. 3 No. 1 No. 2 No. 3 No. 1 No. 2 No. 3 r=4.0; y=.75 r=7.0; y=.75 r=10.0; y=.75 r=14.0; y=.75 1.777.773.773.947.948.943.990.993.989 2.691.693.691.884.882.876.960.962.957.990.993.993 4.611.627.624.804.812.809 4.904 09.908.964.969.965 7.561.575.571.744.752.752.851.857.855.927.930.929 10.532.547.539.768.718.716.814.822.823.897.902.901 20.485.495.490.644.651.649.744.752.752.834.840.837 40.445.442.444.591.589.589.688.686.684.774.773.770 70.409.414.411.548.550.553.638.641.638.718.721.720 100.399,397.397.532.526.529.618.613.613.698.693.695 200.362.484.565.637 300.338.453.526.595 r.20.0; Y=.75 r=40.0; y=.75 r=60.0; y=.75 r=80.0; y=.75 2 1.000 1.000 1.000 4.993.997.993 7.977.985,979 10.957.965.965 1.000 1.000 1.000 1.000 1.000 1.000 20.910.916.915.993.993.993 1.000 1.000 1.000 40.857.857.855.977.972.975 1.000.997.996 1.000 1.000 1.000 70.801.808.805.937.941.943.980.986.986.990.997.996 100.781.773.777.920.913.918.970.969.972.987.983.989 200.714.850.895.930 300.672.801.857.878 ta Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run Run (minJ No. 4 No. 5 No. 6 No. 7 No. 4 No. 5 No. 6 No. 7 No. 4 No. 5 No. 6 No. 7 No. 4 No. 5 No. 6 No. 7 r=4.0; y=.25 r=4.0; y=.75 r=7.0; y=.25 r=10.0; y=.25 1.801.819.821.818.762.778.776.774.951.958.962.959.993.997.993.993 2.724.736.731.732.689.698.696.698.892.899.897.897.983.972.972.973 4.650.653.652.654.622.625.621.624.822.827.828.825.923.927.924.925 7.587.594.590.592.574.576.572.575.758.764.766.767.874.875.876.877 10.556.559.555.559.542,545.541.545.724,729.724.726.839.840.841.843 20.500.504.496.500.493.496.490.494.657.660.655.661.769.774.772.770 40.451.459.455.455.444.451.448.449.595.604.600.599.703.708.704.706 70.424.427.429.420.420.421.563.562.562.660.658.661 100.410.406.411.406.404.404.539.538.534.632.631.630 200.380.374.496.586 r=10.0; y=-75 r=60.0; y=.25 r=20.0; y=.25 r=80.0; y=.25 1.990.993.990.990 2.958.965.962.962 4.906.913.910.908.993.997.993.997 7.853.861.859.856.983.983.979.983 10.839.840.841.843.965.969.965.966 20.752.757.755;753 1.000 1.000 1.000 1.000.923.924.921.925 40.692.694.693.692.993 1.000.997.997.864.868.862.863 70.653.651.651.986.986.986.819.817.819.993.997.993 100.632.631.630.625.624.623.972.969.973,788.786.788 200.579.921.729.938 r=40.0; y=.25 r=40.0; y=.75 20.993.997.993.993.993.997.993.993 40.972.976.976.976.972.976.972.973 70.948.948.952.944.945.945 100.924.924.928.920.921.921 200.870.863

VII DISCUSSION OF RESULTS A. Comparison of Experimental Data and Analytical Results The comparison between the experimental data and the results obtained from the mathematical analysis can be made most conveniently using graphical methods. Three of the thermal models which were studied can be compared directly with an analytical solution by plotting the dimensionless temperature distribution of the models as a function of time, with parameters of position, and superimposing the corresponding curves from the mathematical analysis. For the single-layer radial and single-layer linear models, the comparison of results serves to indicate the accuracy of the experimental technique. In these cases, the analytical solutions are known to be valid. Any deviation from the analytical solution which is observed is thus due to experimental error. A knowledge of this experimental error is important in interpreting results for the two-layer linear modelo The data obtained on the two-layer linear model serve as a verification of the corresponding analytical solutiono The data for the single-layer "fault-plane" model have no direct bearing on the subject of this dissertationo As previously noted, data were obtained on this model solely to illustrate the use of heat transfer models in studies of systems of irregular geometry. No analysis of the data, beyond the process of dedimensionalization, was therefore attempted. The dimensions of the two-layer radial model, in particular the ratio of the radius at the heating surface to the thickness of the model -177

-178(rb/H), unfortunately fall in a range where the analytical solution obtained for two-layer radial flow does not readily yield valid numerical results. As discussed in Section V of this dissertation, the convergence of the analytical solution for values of (rb/H) less than 1.0 was extremely slow. For this model, (rb/H) was 0.61. Comparison of results was therefore not possibleo It should be explained that all of the experimental data which appear in this dissertation were obtained before valid mathematical analyses were completedo The model dimensions were chosen in an attempt to simulate an actual reservoir systems since at that time the range of convergence of the analytical solution was unknown. For the single-layer linear model, the dimensionless temperature distribution from the analytical solution could be plotted directly from the values presented in Appendix F. without interpolationo For the singlelayer radial model, the dimensionless temperatures from Appendix G were interpolated by plotting dimensionless time, @G as a function of dimensionless radius, r. The analytical, solution for the values of r used in the model was then read from these plotso In the case of the two-layer linear model, values of dimensionless temperature at the values of x in the model were computed directly from the analytical solution, in order to increase the accuracy of comparisono These results are presented in Table VII-lo The comparison of results for the single-layer radial model is shown in Figure VII-l. The average of the experimental data, (dotted line) for this case appears to indicate a lower dimensionless temperature at a given time and position than is obtained from the analytical solution (solid line). This is true in all cases except for r 4 ol13o This discrepancy is believed to be primarily due to the following two factorso

-179s \ _~ Oh C- t N O CO t — \ C\j rkO t — rlQll~~~~~~~~~~~~~~~ H Oo (o~a o' o' c~ o~ ~'0 L'~?~ K' " C\ J, X e H * H * v v................. CO HO OCO L Or t CU C ot O l t O CO J - co n 0H 0 ON O \0 0 - 0 Cs l \ n Q - \ r-lH II 0d O ON O, ON ON ONCcO CO \ O Lr_ t N\ Cj CM rH X H C 0 \.t oQ o\ o\-:~ 0 0N t ^ mO N o cO N S L O Vf\ ir\ H. o 0..a...H. L... H.... HO O 0Uo \ O O (o 0\ co HO 0C 0-t CUN CP H Ht HL _ N 0\ C o o Ir\r 4 C. o\oMo 0 Oc M H ND, \ t-O H UNo i 0 - co NH- ~ ON oM N 0~ rO Lr ~ WQ oi O m h ~ \ _ Co 0 K \a n \ r C t > 0 r- -p- O r- t -:Hd- H HO Oo Or o._~ 0 0'\(7\0 CT a\COOOO0^-~ C - ^ O' \ C HH-CCM COUNr O r ii CONONO.......... Ct rQ *' i l O Ol, 0 Q 00 Co \O H \ t - C L, r \ o rl - - CO CM H H C- L HM~~~~~~~, H.?- l CO 11*O CMI 0!\ O\MO \ CO Ht M t- t\,C MUN| -\NO prN 0 N HUN O NOH — H 0 0 0HU' —I FI CH 00\m c tC-\ 4 K\ f CM \) (0 M\ OH H HV- O\ U _:I OD N 00E G ON 0 0< r\ONONaON ONO OOtN \COCO U - I- UN N- MMNCMrC HO OO O 0 rIe.. M. O CO ONM. 0 O,, 00 QONONON ~ H~H0ONH~CM00 \- rVJ 0 COOCOC Lr\ 11 oo 0 C ) co~ -, t- \\ O o o t- co o —- oi c\ o o o|0 O N ON ON O ON O N C t- t- c - N —t -4 M \ID N CM C - M M N oCM H H~ H H 0000 ii OON OO O CO-C MCNC CMCMrl O C \) c H HHHOOOOO OIJ. \ 0\ O\ 0\ ONNCO CO^o HO HLn tnf C L UONH O - lri CM 0 W\ r \ tO ONr0 CO -CO - 0 n0 Hl H 8 -CM LMn -O hC 0 H

-1801.00 l-.95 4.90 -^\ --- - ^^ -- - -^ V — \ - - - ------ \ 0.s.65 \ 0..50 8 00 --— 0 \ I I I Ii^i, \. r..O.45 ~ 4.7~~~~~0 ~ tot~~~~~~~~~~~~~~~ 0100.4D0 TIM, _ — - ---- EXPERIMENTAL DATAr (AVERAGE) ~ I50 In I0nI.f\ ______ ___ __ _ _ _____ ___ " _ _ \. ______ ___ _ _I

-181First, the introduction of the step function temperature change at time zero for this.model was not precisely accomplished. The valving system used for this model was such that a period of at least 20 to 30 seconds was necessary to bring the heating face temperature to that desired. It was thus difficult to locate the ta = 0 point on the recorder chart. This factor would be of importance only at low values of 9 in Figure VII-1. Secondly, and more important, the 24 gauge thermocouple wires used in this model were evidently too large in diameter to obtain very accurate temperature readings, due to conduction of heat along the thermocouple wires. The thermocouple wires for large values of. r passed over a hotter spot of the model at a short distance from the thermocouple bead. Conduction of heat along the thermocouple wires thus tended to increase the temperature reading, which in turn decreased the dimensionless temperature. This effect is particularly noticeable for r = 31.4 and aboveo It should again be noted that the single-layer radial model was the first model to be fabricated and run. On the basis of the comparison presented for this model, it was decided to use the smaller 30 gauge thermocouple wires in all subsequent models, in order to minimize this effect. It may also be noted from Figure VII-1 that the experimental results tend to cross the line from the analytical solution, so that at high values of @ the data predict too high a value of dimensionless temperature. This effect, which is more pronounced in the graphs for the other models, is largely due to heat loss from the model. Due to the definition of dimensionless temperature which was used, an error in actual temperature being too low is reflected by the dimensionless temperature

-182being too high. Thus, heat losses, which tend to decrease the model temperature at a given time and position, cause the corresponding dimensionless temperature to be too high. Three separate graphs are presented for the single-layer linear model, in order to more clearly show the effect of heat losseso Figure VII-2 shows the comparison of results for the data obtained with no insulation on the model (Run No. 1 on this model). The effect of heat losses are very apparent, since the experimental data deviate greatly from the analytical solution at high values of time. Figure VII-3 shows the data for the model from Run Number 35 with asbestos, aluminum foil, and glass wool insulation on the singlelayer linear model, but without the sheet steel channel. The improvement over the data in Figure VII-2 is very apparent, due to the reduction in heat loss. Figure VII-4 shows the data obtained in experimental runs number 4 and 7 on the single-layer linear model. In these runs, the sheet steel channel cut the heat losses from the model to a very low valueo Comparing with Figures (VII-2) and (VII-3), it can be seen that this sheet steel channel caused a significant reduction in the experimental erroro The comparison of results for the two-layer linear model is shown in Figure VII-5o In this figure, all of the experimental data points are indicated, and the dotted lines represent the weighted average of these data. The temperature gradient in the y or crossflow direction is so slight in this model that the lines of experimental data closely represent the temperature for a given x and 0 at all values of y. It can be seen from Figure VII-5 that the experimental data fall very close to the analytical

-1831.00.95 \, \!\ \\ \ \ \0.,.90.85 X:O.10 Hd i.80 ~o HH,.75 aI).70 -I..65.65 4 P4 ~ l\ \ 0).50 o.40 S.60 ---- \-\ -- -- - - - -rr ---- \ --- -\-t^- - -Jl~-.H 0 CH \ 00ri.40 s -- -- \ - V —1 s ANALYTICAL SOLUTION~ ---— O- -EXPERIMENTAL DATA 10 10\ MINUTES, to

5-1841.00.95 \\ \\ ^ \\ \V\ \ a).85 ~rr -I0.80 --- _ —- -- c) 4Di.\o\~~~~~~~~~ \~~~,%\~ \ \ %\'~~d \\ O t-H 4t) -r~ r-P 4- rq *65 V^- 0.60 V"' i63 \ \ ~I I 1. I 1~UII1~ \\ \ \\ - - \ \rd 0 0 OH-P..O \ \ \ \'SO I \ \' \ \ \y~ 0-9.40 ---- -- _ _ - - -— 0_^^ _ ^ -- ^ _ _ _ 3 M I N -......ANALYTICAL SOLUTON\

-185-.95 \\ \ ------- ANALYTICAL SOLUTION - — O — EXPERIMENTAL DATA. r-' _so -T —- -- PiU.80 0) i\ \\I I \\ I~~ tt I \IMI <"ri i.75 1\ I I I\\ \ I I II~ \\ I \ \Y I I I I I I rll I 14-) C.70.- - S.. 65.60 \ I4 \r60~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Q0 Q.55 ----- -- 1 -- _ _ _ _ - ---- a _ -- _ _ _ _ V _ -- -- _ _ _ _ _ pq 0 __ ^ _\_^, L \\ \ ^.s -0.50 o oi OAO H.30 2 3 1.0 10 10 10 MINUTES, to

-1861.00.95.90.85 \ \ \ \, |\ \..80.70 cd.65, w.5X 46.60 C, o.4,,, vt o\.550 to — 0 —EXPERIMENTAL DATAo u \\ V \| \ \~ A.O \\.350 \ \ \ -\ 1X P.5 ----- -— EX TDIMENSIoNLESS TIME, 8_ DIMENSIONLESS TIME,

-187solution, with the normal deviation due to heat losses at higher values of QO On the basis of this comparison, the experimental data are thought by the author to constitute an adequate experimental verification of the corresponding analytical solution, Bo Comparison of Two-Layer Flow With Single-Layer Flow In order to show some of the ramifications of the solutions for two-layer flow which have been obtained, this section will be devoted to comparisons between the flow as predicted by these solutions and the flow which might be predicted using only single-layer theoryo The principal emphasis will be placed. on flow of liquids in underground. reservoirs. Analogous comparisons to those presented herein could also be made for heat conduction. lo Comparison of the Form of the Analytical Solutions The solutions for the two-layer flow systems which were obtained. in this research are largely based on an extension of the technique of separation of variables which was used for single-layer flowo The form of the two-layer solutions would therefore be expected to be similar to that for single-layer flowo This has proven to be trueo A comparison of the form of the solutions can be made for the pressure distribution in single-layer and two-layer radial flow of slightly compressible liquilis, for the case of constant terminal pressureo For single-layer flow~ c~o P(r,G) X= C(U(br)eb (IV-82) mz=

-188For two-layer flow P(r,y,@) Cmne YmnF(b)U(br) (V-188) Now in both cases the eigenvalues b are roots of the characteristic equation Jo(b)Yl(bR) - Yo(b)Jl(bR) = 0 (Iv-65) In both cases the function U(br) is defined as U(br) = JO(br)Yl(bR) - Yo(br)Jl(bR) (IV-69) The expression for Cm for single-layer flow is given by C = 2bV(b) (IV-81)' bb2V2() - (V-1) This is the same function as that defined for two-layer flow as F(b) 2bV(b) (v-183) bV2(b) - 4 jt2 It can thus be seen that the solution for two-layer flow contains many of the same functions as for single-layer flow. The added complexity of the two-layer solution is due to the fact that the pressure distribution with respect to y must be represented by an infinite series in the twolayer case, whereas it is constant for single-layer flow. The behavior of the two-layer solution as it approaches a condition of single-layer flow is worthy of mention. Such a circumstance would occur as K2 approached one (i.e. kl approached k2), or as the dividing point, c, on the y axis approached either zero or one. It has been determined that the solution for two-layer flow loses its validity at these limits. For the sake of brevity the complete mathematical analysis showing

-189the breakdown of the two-layer solution to that for single-layer flow will not be presented here. The reader may however note that at K2 = lo0, 7 = Y. In the two-layer solution, it was assumed that neitheir 7 nor P was equal to zero, It can be shown that for K2 = 1.0 the only term in the "n" series must be for 7 = f = O That is, for two-layer flow, the one term which is neglected (n = O) is precisely that one term which gives. the'solution for single-layer flow. The two-layer solution thus becomes invalid at this limit. 2. Comparison Between Separated and Intercommunicating Two-Layer Systems The effect of crossflow or interlayer flow can be easily seen by comparing the fluxes obtained from a given two-layer system with and without interlayer fluid communication. For single-layer unsteady-state flow of slightly compressible liquids: We(ft3) = 2cwrb2hAPQ(t) where: @Q i kta -ltcwrb2 The corresponding flux for a two-layer system is We(ft3) - 2trb2cWHAP[c[1 + (l-c) 2l]Q(t) where Q(t) is a function of dimensionless time, klta Piljcwrb For 01 = 02 this becomes We(ft5) = 2iccwrb2HaPQ(t) (VII-1)

-190Then for 1 =0 2 the ratio of the fluxes in cubic feet is equal to the ratio of the dimensionless fluxes, Q(t) Consider a two-layer radial system where R " 20~0 and c - 0.50 The dimensionless flux from this system will be computed both for separated. and intercommunicating layers3, for K2 = 5 and K2 = 100. Let 92L represent the dimensionless time for the composite interconnected two-layer system) and let 91 and 92 represent the corresponding dimensionless times for layers 1 and 2 respectivelyo From the definitions of dimensionless time 91 0 2L (VIi-2) 92 K202L (VII-3) The total dimensionless flux assuming separated layers is given in this case (c = 0o5) by Q(t)SL -= 0 5 Q(t): + 05 Q(t) (VII-4) Values of the dimensionless cumulative fluxes Q-(;) and Qft) for the separate individual layers may be obtained from. the tables in Appendix I of this dissertation and. the paper by Van Everdingen and. Hurst (40) Values of the total dimensionless cumulative flux, Q(t)2L assuming intercommunicating layers may be obtained from the tables of Appendix 0. These dimensionless fluxes are shown in Table VII-2o By comparing the valu.es of Q(t)2L for the intercomnllmunicatirng system with the values of Q(t)SL for the system of separated layers, it can immeditey e seen tat e separate system always exhibits less flux at a given time than the corresponding intercommunicating systemo The increased flux from the intercommunicating system is due to crossflo.wO It can also be seen that the amou=nt of iincrease n the flux is greate:r for the greater K. o

-191TABLE VII-2 DIMENSIONLESS CUMULATIVE FLUXES FOR SEPARATED AND INTERCOMMUNICATING TWO-LAYER SYSTEMS K2 = 5.0 32L = Q1 {2 Q(t)L Q(t)2 Q(t)2L Q(t)SL 5 25 4.499.14.57 9.958 9.535 10 50 7.417 24.84 16o 75 16o13 20 100 12o32 42.91 2 866 27.62 50 250 24.84 86.22 58~97 55.53 100 500 42 91 133 52 97 91 88o17 200 1000 73537 177-15 146o4 125o26 400 2000 117.67 196o93 185o0 157530 800 4000 165o05 199.47 198o4 182 26 1000 5000 177-15 199o50 199.2 188o33 K2 = 1000 Q2L Q31 e2 Q(t)L Q(t)2 2(t)2L Q(t)SL 0.2 20 o.6o6 12o32 70463 6.46 Oo5 50 1.020 24 84 14.69 12o93 1. 100 1o571 42 91 25o04 22o21 2. 200 2o442 73537 43 25 37090 5o 500 4o499 133o52 86.92 69. o 10 1000 7.417 1771.5 134o3 92.28 20. 2000 12o32 196o93 1776 O14. o65 50. 5000 24.84 199o50 198o7 112o17 80. 8000 35596 199 50 1.99~5 11 73

-192It can be stated unequivocally that crossflow between layers always increases the flux from the system over that which would be obtained from the separate layers, in systems of the "layer-cake" type which were studied for this dissertation. This is due to the fact that a part of the liquid present in the layer of lower permeability is carried to the producing face through the zone of higher permeability. Note for example in Table VII-1 the fluxes for K2 = 100 at 02L 80. In the system of separated layers, the layer 2, with higher permeability, has been depleted completely at this time, but only approximately 18% of the liquid in layer 1 has been produced. If the layers were intercommunicating, a good share of this liquid could be produced through the high permeability zone. Indeed, for complete communication, both layers would be completely depleted and depressurized by this time. 3. Approximation of Multi-Layer Systems by a Single-Layer Homogeneous System Having Mean Physical Properties This section will be devoted to a study of the question of when and how the behavior of two-layer (and in general multi-layer) systems in the unsteady-state may be approximated by a single-layer homogeneous system having some mean physical properties. The concept of "millidarcy-feet" will be examined in conjunction with this question. Maximum and minimum limits on flux from a system where the degree of interface resistance is unknown will also be shown. The concept of "millidarcy-feet" has frequently been applied to heterogeneous systems of a stratified type by reservoir engineers. With this method, for a stratified system of N layers, the layers of which each

-193have some permeability kl and thickness hi, a mean permeability km is defined as N Z. (kihi) km-.... (VII-5) N Z hi i-1 That is, the mean permeability by which the system is treated as a homogeneous single-layer system is the total millidarcy-feet of the system divided by its total thicknesso Systems of separated (ioeo' non-intercommunicating) layers may first be considered. Several papers have recently been published on such systems, such as that by Lefkovits, et.al.(29) It can easily be shown that there is no single constant mean permeability by which a system of two or more separated layers may be represented by a single-layer systemo Consider the equation for cumulative dimensionless:.flux in the radial flow of slightly compressible liquids in a single layero 2~)-^ — a......^J -- (IV-87) Q(t ) 2 eb2J 2(bR) (IV-8) a b2 [J (b) - J12(bR)] where kta ~ = -k (IV-48) Now consider two layers with the same physical dimensions rb, re, and thus R, which have different values of (k/0)o Then the only difference in the solutions for these layers appears in the dimensionless times, 9G Define ( )m crb2 (vII-6) \ t -crb2

-194(I-Y k) ta - (VII-7) -~ 0 ~ Jl 4 crb2 _ k ) ta (VII-8) \2 PI 4 tcrb2 where ( ) is some mean property by which it is hoped to approximate v m the total cumulative dimensionless flux from both layers using a singlelayer system. In order to accomplish this, it must be true that e-b2 = C [eb2@l + e-b292] (VII-9) where C is some constant. But this equation cannot be valid for all values of real time ta. It would be necessary to either allow the constant C or the value (b) to be a function of real time, ta, in order to make this equation valid for all values of ta. It is thus impossible to represent a system of separated layers of different physical properties by a single-layer system with some constant mean value of the physical properties. This in turn means that the concept of millidarcy-feet, Equation (VII-5), is not valid for systems of separate and distinct layers in unsteady-state flow. In considering the question of whether intercommunicating layers may be represented by a single layer with some mean physical properties, it is necessary to define what is meant by "complete crossflow". Since the crossflow is dependent upon the pressure gradient in the y direction, it would seem at first glance that complete crossflow might well be defined in terms of this gradient. The interlayer flux or crossflow is, however, also a direct function of the flow resistance in the y direction, which isn tn a fu n f in turn a function of the interface area, the thickness of the layers, and the permeability of the layers. The author thus proposes a definition

-195in terms of the flux at the producing face In this dissertation, the term "complete crossflow" will be used to represent that condition for which the instantaneous flux in cubic feet per second per square foot of the producing face from each layer of a multi-layer system in unsteadystate flow is directly proportional to the permeability of that layer for all values of time. That is, regardless of the initial distribution of fluid in the system, the flux at the producing face is dependent only on the permeabilityo This necessarily implies that a part of the fluid being produced from a given layer initially may have originated in another layer of the systemn This definition of complete crossflow leads to consideration of Appendices L and M of this dissertation, which present values of the instantaneous dimensionless fluxes, ql and q22, for linear and radial unsteady-state flow, respectively. The application of the definition of complete crossflow to these tables indicates that for complete crossflow the values of q. and q2 should be directly proportional to the relative thickness of the layerso That is, for c = 0o5, the crossflow between layers is complete if ql. q2. Inspection of the tables of Appendix M thus indicates that (except for the single case of R = 5 and K2 100) the crossflow in the two-layer radial systems for which values are tabulated is complete. The values tabulated in Appendix L for linear flow, however, indicate that the crossflow in most of these systems is not complete, although this limit is approached for large values of L = La/H. The reader may well. at this point question why crossflow is complete for the radial systems, but not for the linear systems It must be remembered that all of the radial systems for which values have been

-196tabulated in the Appendices are comparatively thin, so that (rb/H) is greater than 10.0. This means that the distance from a given point in the lower permeability layer to the interface with the high permeability layer is in most cases much less than the distance to the producing face. There is thus very little resistance to crossflow. Indeed, although the crossflow in the radial systems is complete, it has been previously noted that the pressure gradient in the y direction is so slight that the tables of P(r,y,Q) in Appendix K do not include parameters of y. In the linear case, however, the systems for all but very high values of L = La/H are quite thick in comparison with the radial systems. The resistance to crossflow is thus higher in these systems. The tables of P(x,y,Q) for two-layer linear flow in Appendix J show a significant pressure gradient in the y direction although crossflow in these cases is not complete. This factor of the pressure gradient in the y direction being smaller for the case of complete crossflow than for only partial crossflow again serves to show the merits of defining crossflow in terms of fluxes rather than pressure gradients. Now consider the implication of the definition of complete crossflow with regard to the concept of millidarcy-feet. According to the definition for complete crossflow the flux from a given layer is independent both of the position of the layer in the system, and of the thickness of the layer. Then in this case the concept of millidarcy-feet, and the use of Eguation (VII-5) to define a mean permeability for the system should be entirely valid! This in turn implies that when a multi-layer system exhibits complete crossflow, it should be possible to accurately predict the

-197behavior of the system using a single-layer homogeneous system with mean physical properties, where km is defined by Equation (VII-5), and the mean. porosity, m, is given by the analogous equation N (Oihi) m -.. —- (VII-10) hi i1l 1 The foregoing discussion indicates that it should be possible to predict the behavior of the two-layer radial systems for which values appear in Appendices K, M, and 0 by using a mean permeability as defined by Equation (VII-5) in conjunction with the tables for single-layer flowo The validity of this assertion will be examined by comparing the fluxes from a two-layer intercommunicating radial system as predicted using twolayer theory to the fluxes predicted using a mean permeability in conjunction with single-layer tables. In order to use Appendix 0O it is necessary to consider 01 = 02~ The preceding section of this dissertation has shown that in this case the ratio of the real cumulative fluxes, in cubic feet, is equal to the ratio of the corresponding cumulative dimensionless fluxes, Q(t)' In terms of the nomenclature used for two-layer flow, Equation (VII-5) in this case, indicates the use of a mean permeability km c + (l-c) K2 (VII-11) The definition of dimensionless times, 9, is such that in approximating the behavior of the two-layer system by using a single layer of mean permeability, GSL km2L (VII-12)

-198where: 0SL dimensionless time for'the single-layer approximation Q2L: dimensionless time for the two-layer systemo Values of cumulative dimensionless flux for the two-layer system, Q(t)2L' may be obtained from Appendix O. Values of Q(t)SL for the single-layer approximation may be obtained from Appendix I plus the tables of Van Everdingen and Hurst.(40) For this example, the values in the tables were interpolated linearly where necessary. Table VII-3 shows the values of flux obtained for a system. of R B 20, K2 5 and. 100, c ~ 0o.L 0o5, and 0o9, for rb/H > 10o Examination of Table VII-5 clearly shows that the fluxes predicted using a mean permeability and single-layer tables, Q(t) SL are the same as those from two-layer theory, Q(t)2L. within'the limits of accuracy of linear interpolation of the tables. Thus the use of a m:ean k in conjunction with single-layer theory yields a perfectly valid representation of the total cumulative flux from this two-layer system, as predicted, due to the complete crossflowo As additional verification, an example showing the comparison between the pressure distribution P(r,y,g) for a radial two-layer system as computed from two-layer theory and, that computed, using a mean permeability and single-layer tables will. be showno Values of P(r.y,@) from twolayer theory will be obtained directly from Appendix: K for R B/20, c 0- 05^, K2 = 5, and rb/H > 10o Corresponding values from single-layer flow may be obtained from Appendix: G. In. this case, the values for single-layer flow were interpolated graphicallyo The comparison is shown in Table VII-4.

-199TABLE VII-3 DIMENSIONLESS CUMULATIVE FLUXES FOR THE APPROXIMATION OF A TWO-LAYER RADIAL SYSTEM BY A HOMOGENEOUS SINGLE-LAYER SYSTEM 92L gSL Q(t)2L Q(t)SL Q2L GSL Q(t)2L Q(t)SL R = 20, c = 0.5, K2 = 5 R = 20, c = 005, K2 = 100 5 15 9.958 9.95 0.2 10.1 7.463 7.46 10 30 16.75 16.74 0.5 25.25 14.69 14.68 20 60 28.66 28.65 1. 50.5 25.04 25.03 50 150 58.97 58.93 2. 101. 43.25 43.24 100 300 97 91 97 81 5. 252.5 86.92 86.82 200 600 146.4 146.40 10. 505. 134.3 134.23 400 1200 185.0 185.00 20. 1010. 177.6 177.54 800 2400 198.4 198.42 50. 2525. 198.7 198.67 1000 3000 199.2 199.20 80 4040o. 199.5 199.47 R 20, c = 0.1, K2 = 5 R 20, c = 0.9, K2 = 100 5 7 5.740 5.749 0.2 2.18 2.625 2.559 10 14 9.459 9.439 0.5 5.45 4.816 4.700 20 28 15.88 15.88 1. 10.9 7.878 7.872 50 70 52.56 52.55 2. 21.8 13.14 153 4 100 140 55.90 55.89 5. 54.5 26.57 26.57 200 280 93.41 93.41 10o. 109o 45 93 45,89 400 560 141.6 141.61 20. 218. 78.19 78.08 800 1120 182.3 181.86 50. 545. 139.7 159.64 1000 1400o 190.1 190.09 100o 1090.o 18.1 1 80.68 2000 2800 199.0 199o 04 200. 2180. 197.8 197.74 R = 20, c = 0.1, K2 = 5 R = 20, c = 0.1, K2 = 100 5 23 13.70 13.68 0.2 18.02 11.41 11.38 10 46 23.29 23.26 0.5 45.05 22, 91 22.90 20 92 40.18 4o. 16 1. 90.1 39551 39.50 50 230 81.31 81.21 2. 180.2 67.87 67.86 100 460 127.6 127.56 5. 450.5 126. 1 126o08 200 920 172.9 172531 10 901. 171. 8 171.16 400 1840 195.9 195 82 20. 1802. 1.95.6 195.6 800 5680 199.4 199.42 50. 4505. 199.5 1.99.5

-200TABLE VII-4 DIMENSIONLESS PRESSURE DISTRIBUTION FOR THE APPROXIMATION OF A TWO-LAYER RADIAL SYSTEM BY A HOMOGENEOUS SINGLE-LAYER SYSTEM R - 20, c = 0.5, K2 - 5 Dimensionless Time Dimensionless Pressure Distribution for all Values of y From Two-Layer Theory From Single-Layer Tables 02L @SL r = 4 r = 10 r a= 20 r = 4 r = 10 r = 20 5 15.664.963 1.000 0663.962 1.000 10 30.585.906 0992.585.908.990 20 60.518.829.948.515.828.948 50 150.420.678.784.416.671.785 100 300.304.490.567.305.490.567 200 600oo.159.256.296.153.245.288 500 1500.023.037.042.022.042.052

-201The pressure distribution from the two methods checks within the accuracy with which the single-layer tables could be graphically interpolatedo It may thus be concluded that the Equations (VII-5) and (VII-10) in conjunction with tables for single-layer flow may be used as an alternative to the use of Appendices K and 0 in determining dimensionless pressure distribution, P(r,yQ) and dimensionless cumulative flux, Q(t) for two-layer radial systems of intercommunicating layers, for unsteady-state flow, and for rb/ > 1iOoO Moreover, Equ.ations (VII:-5) and (VII-10) and single-layer tables may be conveniently used for values of K2 between 2 and 1.00 for which. valu.es do not appear in Appendices K and 0, and for cases in which.l 2~o Most importantly, it can be seen by direct extension of the foregoing discussion that it should. be valid. to use the rethodJ of single-layer approximation for any multi-layer radial system in the unsteady state, under the restrictions that 1. rb/H > 10 2. The ratio of (k/P,) of any two layers not be greater than 100 3 R > 10 These restrictions are estimations on the basis of the results for two layers These approximations by a single layer which are valiS.1. for radialflow would not be expected to apply to the cases of two-layer linear flow for which values appear in Appendices J, L, and N, since for these cases the crossflow is not completeo The author has verified. this by attempting to compute dimensionless pressure diistribution. P(xy',O), for the two-layer

-202case of L = 10, c -= 0.5 K2 2.0, by using a mean permeability and tables for single-layer linear flow. The results obtained were very poor approximations. The validity of the application of the concept of millidarcy-feet, i.e. the use of Equations (VII-5) and (VII-10) and single-layer tables to represent the behavior of multi-layer systems, may be summarized as follows: 1. Millidarcy-feet may be used for true steady-state flow, with or without impermeable barriers between layers. 2. Millidarcy-feet may be used for multi-layer systems in which there is complete crossflow. It is usually necessary to use multi-layer theory to determine whether or not the crossflow for a given system is truly complete. 3. Millidarcy-feet is not a valid concept for use with systems of separate and distinct layers, i.e. where the layers are separated by impermeable barriers. For stratified systems of more than three or four layers, or in systems where there is some finite but unknown interface resistance, the use of the multi-layer theory as presented in this dissertation to determine the completeness of crossflow is very difficult. This is also true in anisotropic stratified systems in which the vertical permeability is not equal to the horizontal permeability. If the crossflow is not known to be complete, the validity of approximation by a single layer is questionable. It is, however, possible to establish maximum and minimum limits on the total cumulative flux at the producing face for multi-layer systems of the type studied in this dissertation. The minimum flux is

-203that which is predicted treating each layer of the system individually assuming no crossflow (see Table VII-l)o The maximum flux is that assuming complete crossflow, for which the flux may be calculated using singlelayer theory and. Equations (VII-5) and (VII-10) For partial crossflow, the flux should. fall between -these limitso C. Steady-State and Unsteady.-State Permeabililt.ies of Stratified, Systems The term, "permeabi.l.. ty" as applied to heterogeneous systems must be carefully d.efined. By choosing a sufficiently small vorlime element, nearly all heterogeneous systems can be resolved into a combination of homogeneous elementso In each of these homogeneous volume elements, the original concept of permeability as defined. by Darcy applieso That is, the permeability of these homogeneous elements is a function solely of the physical properties of the porous matrix, and. is independent of whether the system is in steady-state or unsteady-state flowo On the macroscopic scale however, the term "permeability" is frequently used to indicate the average physical properties of a heterogeneous system. as in "in-situ permeability" In such cases, the permeability which is measured in the steady-state is not necessarily that which would accurately predict the unsteady-state behavior of the system. Consider, for example, a stratified system of separate and distinct non-intercommunicating layers. A macroscopic mean permeability of this system could be determined by measuring the flux from the system in steady-state flow. This permeabi.lity would not, however, be valid for predicting the performance of the system in unsteady-state flowo As shown

-204in the preceding section, in order to use a single mean permeability for representing the behavior of such a system in the unsteady-state, the mean permeability would necessarily be time dependent. The difference between steady-state and unsteady-state macroscopic mean permeabilities must also be considered in predicting the performance of a heterogeneous reservoir on the basis of core analyses. It is for this reason that measurements of in-situ permeability by unsteadystate draw down and/or buildup tests usually indicate a much more valid macroscopic mean permeability (and porosity) than can be obtained from core analyses, even though a very large number of cores are tested. In stratified systems of the type considered in this dissertation, it has been shown that the use of a macroscopic mean permeability as obtained by using the concept of millidarcy-feet as applied to core analyses would indicate too high a flux for a system that did not exhibit complete crossflow. D. The Flow of Gas in Stratified Systems The non-linearity of the equation governing the flow of gas in porous media (Equation III-29) makes it impossible to obtain a complete analytical solution for gas flow in stratified systems, using mathematics of the type used in this dissertation. It is possible, however, to investigate certain aspects of the flow of gas in stratified systems on the basis of the known solutions for the flow of slightly compressible liquids. For low pressure differences, the density of a gas is relatively constant, and may be approximately expressed by Equation (III-23) which was used for slightly compressible liquids. Under these conditions, the solutions in this dissertation for the flow of slightly compressible liquids

-205should provide accurate approximations to the gas flow problem. With high pressure differences, on the other hand, the gas density may no longer be validly approximated by Equation (III 23). In this case, the predictive accuracy of the liquid-flow solutions may become completely void' Another problem encountered to a great extent in gas flow, but generally of little significance for liquid flow, is the phenomenon of "turbulent" flow. However, with a sufficiently high pressure gradient to induce turbulent flow, the density variation of the gas in the system is usually too large to permit use of the liaquid flow solutions even without this additional complication. For gas flow with high pressure drawdowns, it is possible to make only qualitative statements regarding the flow in stratified systems. First of all, in porous systems which. would exhibit crossflow for liquid flow, the systems will also exhibit crossflow in gas flow. Secondly, this crossflow must tend to increase the flux at the producing face above that which would be obtained for separate and distinct (ioe. non-intercommunicating) layers. One factor with regard. to gas flow which is worthy of mention here is the behavior of a stratified gas reservoir in an isochronal backpressure test, In these tests a gas well is opened to flow for a given length of time for several different flow rateso The instantaneous flux at the end of this time period, is plotted versus the difference in the squares of the pressures at the well. bottom and far out in the reservoir at this time The term "isochronal" refers to the fact that the same time period is used for all test pointso Since the pressure in the reservoir

-206is allowed to return approximately to equilibrium after each point is obtained, the same "radius of drainage" is obtained for all points. It was thought at the outset of the research leading to this dissertation that the slope of the isochronal back pressure curve might be a function of the degree of stratification of the system. The results, however, indicate that for stratified systems of the types studied herein this would not be true. In all systems studied, the solutions obtained for the constant terminal pressure case are independent of the pressure difference applied to the system. That is, doubling the initial pressure difference driving force in all cases exactly doubles the flux obtained from the system at a given time. The behavior of the stratified systems in this regard is exactly the same as for a homogeneous system. It is therefore concluded that the slope of the isochronal back pressure curve is not a function of the degree of stratification of the system. It seems very probable that this slope is entirely independent of all types of reservoir heterogeneity (since the same radius of drainage is applicable for all points), except insofar as the heterogeneity affects the turbulence level of the gas. Attempts to show that turbulence is the cause of the variation in the slope of the back pressure curve have heretofore been unsuccessful in that values of the reciprocal slope below 0.9 cannot be predicted, but do occur in practice. In all such attempts, an average permeability as from Equation (VII-5) was used for the system. The use of an average permeability inherently implies the assumption of a mean gas velocity. Since turbulence level is not a linear function of velocity, the linear average

-207velocity which is thereby calculated would give inadequate predictions of the actual turbulence effects for that portion of the zone actually delivering the bulk of the gas.

VIII RECOMMENDATIONS FOR FUTURE WORK The investigation of fluid flow in heterogeneous and/or anisotropic systems is certainly of great importance in understanding the complex flow problems associated with underground formations. To date, the surface has only been scratched. Opportunities for both analytical and experimental work are abundant. This author will not pretend to indicate which problems are of the most importance in this field, but will indicate some problems of particular interest to himself. In the field of mathematical work: 1. As indicated, the solutions presented in this dissertation for multi-layer radial flow are very slowly convergent for small values of rb/H, so that accurate numerical values are very difficult to obtain. The author believes that a "pointsource" solution to the multi-layer radial flow problem would yield valuable results in this regard. It is also possible that resolving the problem using some transformation technique, possibly a combination of a LaPlace transformation and finite Fourier trigonometric transformations, might yield pertinent approximations not apparent from the solution using separation of variables. 2. Although solutions are presented in this dissertation for more than two layers, no numerical values have been obtained for these cases. The computation of a complete set of tables of pressure distribution, etc. for three- and four-layered systems would require a prohibitive amount of time, due both -208

-209to the complexity of the solutions and the multiplicity of parameters. It should, however, be feasible to compute values for certain particular caseso It would be of considerable interest to determine how the arrangement of layers in three- and four-layered systems affects the flow patterns and flux across the producing faceo For example, given three layers, of 1, 10 and 100 millidarcy permeability respectively, three separate and distinct arrangements of these layers may be made (ioeo 1, 10, 100; 1, 100, 10; and 1.0, 1 100). The author believes that the arrangement of layers should in many cases have a distinct effect on the flux obtained. 3. All of the work in this dissertation has been done on systems which are completely isotropic within a given "layer" of the system, Actual underground formations frequently are not. A multi-layer solution for vertical permeability (i.e. parallel to the y axis in the models) being some fraction of horizontal permeability would therefore be of great value. 4. The "layer-cake' types of models used in this research are poor approximations to many formations which involve continuous permeability distributions as a function of depth, Investigation of the flow patterns in systems with random permeability distribution would be very informativeo Such an investigation would probably be based on some statistical distribution, and woul.d most likely require finite difference techniques to obtain a useable solutiono

-210In the field. of experimental work: 1o The use of heat transfer models is very convenient in simulating irregular geometries This is true of both. homogeneous and heterogeneous systemso The linear and radial cases have been considered in the research leading to this dissertationo The next logical step would seem to be an elliptical. model. 2. Types of heterogeneity other than the "layer cake" type of model could'be simulated without undue difficulty using thermal models. The first step in this direction might be the introduction of an additional resistance at the heating surfaceo This could be easily done by using a material other than copper to contain the heating steami It should also be possible to investigate systems of "random" permeability using thermal modelso Such models could be built up of several block:s of different materialsso It would. probably'be necessary to use a solder bond between blocks, which w01ould make the resvuts less quantitative than for the models used in the research leading to this dissertation. By using high conductivity materials such as copper and various cupro-nickels the effect of bond resistance could probably be minimized sufficiently to obtain results of adequate accuracyo

BIBLIOGRAPHY 1. Bruce, G. H., Peaceman, D. Wo, Rachford, H. H.., Jr., and Rice, J. D. "Calculations of Unsteady-State Gas Flow Through Porous Media." AIME, Petroleum Trans., 198, (1953) 79. 2. Brunot, A. W. and Buckland, F. F. Trans. Am. Soc. Mech. Engrs., 71, (1949) 2530 3. Buxton, T. S...Pressure Distribution in Radial, Infinite, SinglePhase Reservoirs Produced at Constant Flow Rate.. -Masterls Thesis, Oklahoma State University, Stillwater, Oklahoma, 1958. 4. Cardwell, W. T., Jr. and Parsons, R. L. "Average Permeabilities of Heterogeneous Oil Sands." AIME, Petroleum Transo, 160, (1945) 34. 5. Carslaw, H. S. and Jaeger, J. C. Conduction of Heat in Solids. Oxford at the Clarendon Press, 1947. 6. Caudle, B. H. and Witte, M. D. "Production Potential Changes During Sweep-Out in a Five-Spot System." J. Petr. Tech., December, 1959. 7. Chatas, A. T. "A Practical Treatment of Nonsteady-State Flow Problems in Reservoir Systems." Petro. Engr., May, 1953. 8. Churchill, R. V. Operational Mathematics. Second Edition, McGraw-Hill Book Co., New York, 1958. 9. Churchill, R. V. Fourier Series and Boundary Value Problems. McGraw-Hill Book Co., New York, 1941. 10. Coats, K. H., Tek, M. Ro, and Katz, D. L. "Unsteady-State Liquid Flow Through Porous Media Having Elliptic Boundaries." AIME, Petroleum Transo, 216, (1959) 460. 11. Coats, K. H. Prediction of Gas Storage Reservoir Behavior. Ph.eD. Thesis, University of Michigan, April, 1959. 12. Coats, K. H., Tek, M. R., and Katz, D. L. "Method for Predicting the Behavior of Mutually Interfering Gas Reservoirs Adjacent to a Common Aquifer." AIME, Petroleum Trans., 216, (1959) 247. 13. Craig, F. F., Jr., Sanderlin, J. Lo, Moore, D. W., and Geffen, T. M. "A Laboratory Study of Gravity Segregation in Frontal Drives." Trans, AIME, 210, (1957) 275. 14. Crawford, P. B., Landrum, B. L., Flanagan, D. A., and Norwood, B. C. "A New Experimental Model for Studying Transient Phenomenao" Trans. AIME, 216, (1959) 533 -211

-21215o Fatt, Irvingo "A New Electric Analogue Model for Nonsteady State Flow Problems." AIChE Journal, 4, (1958) 49. 16. Ferrell, H., Irby, To L., Pruitt, Go T., and. Crawford, P. B. "Model Studies for Production-Injection Well Conversion During Line Drive Water Floods." AIME, Petroleum Transo., 21.9, (1960) 94. 17. Gaucher, D. H. and Lindley, D, C. Water-Flood Performance in a Stratified, Five-Spot Reservoir —A Scaled Model Studyo Paper No. 1311-G, Presented at AIME Meeting, Dallas, Texas, October, 1959. 18. Guy, W. T., Lesem, L. B., and Crawford, Go W. LaPlace Transformation Solution of Simultaneous Linear Flow in Two Regions Separated by a Fixed Boundary. Proceedings of the Sixth Annual Conference on Fluid Mechanics, University of Texas, Austin, Texas, September, 1959. 19. Hicks, A. Lo, Weber, Ao G., and Ledbetter, Ro Lo "Computing Techniques for Water-Drive Reservoirs." J. Petr. Tech., June, 19590 20. Hodgman, C. D., editor. Mathematical Tables from Handbook of Chemistry and Physics. Ninth Edition, Chemical Rubber Publishing Co., Cleveland, Ohio, 1952. 21. Hutchinson, T. S. and Sikora, V. J. "A Generalized Water-Drive Analysis o AIME, Petroleum Trans., 216, (1959) 169. 22. Jahnke, E. and Emde, F. Tables of Functions with Formulae and Curves. Fourth Edition, Dover Publications, New York, 1945. 23. Jakob, M. Heat Transfer. Vol. I, John Wiley and. Sons, Inc., New York 1956. 24. Katz, D. L., et alo Handbook of Natural Gas Engineering. McGraw-Hill Book Co., New York, 1959. 25. Katz, D. L., Tek, M. R., and Coats, Ko H. "The Effect of Unsteady State Aquifer Motion on the Size of an Adjacent Gas Storage Reservoir." AIME, Petroleum Trans., 21.6, (1959) 18. 26. Katz, D. L., Tek, M. R., Coats, K. H., and. Katz, Mo Lo Engineering Studies on Movement of Water in Contact with Natural Gas. Annual Report for The American Gas Association, Project N031, University of Michigan Research Institute, September, 1960. 27. Landrum, B. L. and Crawford, P. B. "Transient Pressure Distributions in Fluid Displacement Programs." J. Petr. Tech., November, 1959. 28. Law, J. "A Statistical Approach to the Interstitial Heterogeneity of Sand Reservoirs." Transo AIME, 1559 (1944) 202,

-21329, Lefkovits, H. C., Hazebroek, P,, Allen, Eo Eo, and Matthews, C. So A Study of the Behavior of Bounded Reservoirs Composed of Stratified Layers. Paper No. 1329-G, Presented at AIME Fall Meeting, Dallas, Texas, October, 1959. 300 Mickley, H. S., Sherwood, To K., and Reed, Co Eo Applied Mathematics in Chemical Engineering. McGraw-Hill Book Coo, New York, 1957. 31. Mortada, Mohamed. "A Practical Method for Treating Oilfield Interference in Water-Drive Reservoirs." AIME, Petroleum Trans., 204, (1955) 2170 32, Muskat, Mo The Flow of Homogeneous Fluids Through Porous Media. Jo Wo Edwards, Inc., Ann Arbor, Michigan, 1946. 335 Muskat, M. "The Effect of Permeability Stratification in Cycling Operations." AIME, Petroleum Trans., 179, (1949) 3130 34. Sangren, W. C. Differential Equations with Discontinuous Coefficients. ORNL 1566, Oak Ridge National Laboratory, October, 19535 350 Sneddon, I. No Fourier Transforms. McGraw-Hill Book Co., New York, 1951. 36. Standing, M. B., Lindbald, Eo N,, and Parsons, Ro L. "Calculated Recoveries by Cycling from a Retrograde Reservoir of Variable Permeabilityo' AIME, Petroleum Trans., 174, (1948) 165. 37. Stevens, Wo F. and Thodos, G. T. "Prediction of Approximate Time of Interference Between Adjacent Wellso" J. Petro Tech., October, 1959. 38. Stewart, F. Mo "Simplified Water Influx-Pressure Calculations Above the Bubble Point." Jo Petro Tech., September, 1959. 39. Tsarevich, K. A. and Kuranov, I. F. "Calculation of the Flow Rates for the Center Well in a Circular Reservoir Under Elastic Conditionso" Extract from Part I of Problems of Reservoir Hydrodynamics and Thermodynamics, edited by E. M. Minskii, Leningrad, 1956. 40o Van Everdingen, Ao F. and Hurst, Wo "The Application of the LaPlace Transformation to Flow Problems in Reservoirso" AIME, Petroleum Trans., 186, (1949) 305. 41. Van Meurs, P. "The Use of Transparent Three-Dimensional Models for Studying the Mechanism of Flow Processes in Oil Reservoirs." Trans AIME, 210, (1957) 295. 42, Warren, Jo E. "The Unsteady-State Behavior of Linear Gas-Storage Reservoirso" The Petroleum Engineer, November, 1956o

-2l443. Weills, N. D. and Ryder, E. A. Trans. Am. Soc. Mech. Engrs., 71, (1949) 259. 44. Wylie, C. R., Jr. Advanced Engineering Mathematics. McGraw-Hill Book Co., New York, 1951.

APPENDIX A DERIVATION OF THE ORTHOGONALITY CONDITIONS FOR THE "Y" EQUATIONS IN TWO-LAYER FLOW -215

-216The orthogonality condition (V-65) for the equations + Y2y = < y < c (A-l) Y", + p2y 0 c < y< 1 (A-2) y = a2-b2; = - b2 (A-3) Y' () = Y' () = O (A-4) with the interface conditions Y(c-) = Y(c+) (A-5) Y' (c-) = K2Y' (c+) (A-6) will first be derived. The more general case of K being a continuous function of y will then be considered. Consider Equation (A-l) through (A-4) for some given value of the parameter bm. At a = al,(i.e., n = 1), Y Yl, and P = P1. There will then be some solution Y1 satisfying the equations Y1 + y2YY =0 0 < y c (A-7) O2y~c (A-7) Y" + i Y = - c < y 1 (A-8) Similarly, for a = a2 Y + Y2 =O 0 y< c (A-9) Y + Y2 c < (A-10) Y2 + 2Y2 <y <= Multiplying both sides of (A-7) and (A-8) by Y2, both sides of (A-9) and (A-10) by (-Y1), and adding 2 2 YY, - YY - ( - Y)YY2 0 y < c (A-ll) Y1Y -Y Y1Y = ( 2 - P2)YY2 c < y (A-12) 1 2 1 2 11-12 Integrating both sides of (A-ll) and (A-12) with respect to y over the intervals on y in which they are valid [YY2 - y = ( - Y)YY2 dy (A-1) 1 1 f [Y1Y2 - Y1Y2]dy = f (D2 - 1)Y1Y2dY (A-14) c c

-217Now Y12 YY2 = [YiY2 - YY1 (A-15) Therefore c c [YY2 - Y2Yl]o = (oY Y)YY2 dy (A-16) [YY2 Y- 2Y1] = (2 - P)YY2 dy (A-17) From Equation (A-4), the quantity [YIY - Y'Yl] [ 1 2 21] vanishes at y = 0 and y = 1, but not at y = co Define ((c)= c)Y2(c) - Y0(c)Yl(c) (A-18) Then C 2 2 ( - 7,) f Y1Y2dy =- (c-) (A-19) 2 2 1 (2 - 1) Y1Y2dy -- (c+) (A-20) Multiplying (A-19) by 0(c+), (A-20) by 0(c-) and adding c 1 (Y2 - Y2) (c+) f Y1Y2dy + (2- P1)0(c-) YYdy 0 (A-21) Rewriting Euation A-8) in terms of the limits approaching y c from Rewriting Equation (A-18) in terms of the limits approaching y ~ c from either side (c-) = Yi(c-)Y2(c-) - Y(c-)Yl(c-) (A-22) (c+) = Yi(c+)Y2 (c+) - Y2(c+)Yl(c+) (A-23) But the interface conditions (A-5) and (A-6) are valid for all solutions, Y. Therefore, substituting Z(c-) - K2Yi(c+)Y2(c+) - K2Y2(c+)Yl(c+) = K2 (c+) (A-24)

-218Substituting into (A-21) and cancelling c 1 (72 y1) f YlY2dy + (p2 )K2 fJ Y1dy - (A.-25) o c But from the definitions (A-3), (72 _2) K - 2) (A-26) Substituting (A-26) into (A-25) and cancelling c 1, 1 f YlY2dy + f YY2dy - Y1Y2dy = 0 (A-27) o c 0 Since the entire foregoing analysis holds equally as well for a,gP anr as for a1, a2, it can be written in general that 1 f Yn= Yn=r dy = 0 (A-28) o for i 7 r. This is the Equation (IV-65) which was to be derivedo Now consider the more general case where K is a continuous function of y, so that 6k/7y is not zeroo In this case, the separated equation becomes k(y)Y" + kI(y)Yt + [a2 - b2k(y)]Y 0 (A-29) where 2m-.lh iT b m ( 2-~) L m " 123,o (A-3O) This may be written I (kY') + a2Y b2kY (A3l) dy Consider the eigenvalues am, an with corresponding eigenfunctions YM9 Yno d (kY~) + Ym - b2kYm (52) d-(kYA) + anYn - b2kYn (A-3) dy

-219Then multiplying (A-32) by Yn and (A-33) by (-Ym) and adding Yn dy(kYm) - Ym dy(kYn) + (a - a)YmYn (A-34) Rearranging (a - a2)YmYn = Ym d(kYn) - Y (kYm) dy [(kYn )Ym - )Yn (kA-5) dy Integrating both sides with respect to y over the range 0 < y < 1 1 1 (a - an) f YmYndy = [k(YmYn - YmYr )'o (A-3)6 0 But from (A-4) Y (0) = Y(0) = Y (1) Y (l) = O (A-57) Therefore 2 1 (am - an) 2 YmYndy - 0 (A-38) or for m n n 1 f YmYndy = 0 (A-59) This is the same condition as was found to apply for two separate and distinct layers. It should be noted that it is very important that the relationship of the eigenvalues a and b in Equations (A-4) - (A-4) and (A-29) be clearly specified. In all problems in this dissertation, the value b is merely a parameter for the problem in Y, leading to a set of Sturm-Liouville systems with eigenvalues a (one such system for each value of the parameter b). The reverse situation for the same equations, treating a's as the known

-220parameters, leads to an orthogonality condition 1 f kYmYnd- 0 (A- 0) o in which the k(y) appears as a weight functiono The Equation (A-40) is not valid for any problems in this dissertationo

APPENDIX B EIGENVALUES, b, FOR SINGLE-LAYER AND MULTI-LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE Characteristic Equation: Jo(b)Y(bR) - Yo(b)JY(bR) = 0 (IV-65) Solved by: Half-interval method on IBM 704 Estimated Accuracy: + 2 parts in 106 Note: The values shown are photographed from the IBM 704 computer outputO The eigenvalues are listed by rows rather than columns (ioeo, row number one contains b1 through b5, row two contains b6 through blo, etc.). The format used is one of the "floating point" output formats used by the Fortran System on 704, and can be easily interpreted as follows: 6.321872EOO = 6.321872 6.521872E-01 = 0.6321872 6.321872E01 = 63.21872 6.321872E02 = 632.1872 That is, the number.f.ollowing: the."E" in each case represents the power of ten by which the number preceding the "E" should be multiplied to obtain a normal, fixed point number. -221

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APPENDIX C EIGENVALUES C:FOR SINGLE-LAYER RADIAL FLOW, CONSTANT TERMINAL RATE Characteristic Equation. Jl (c)Y (CR) - Y1(O)Ji (o) O (Iv-120) Solved by: Half-interval method on IBM 704 Estimated Accuracy- + 1 part in 105 Note: See Appendix B for explanation of formato -239

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APP EI\TD IX D EIGEIVAL'uES, a, FOR TWO LAYER LaIEAR FLOW, CONSTANT TERMINAL PRESSURE Characteristic Equation: PK2 sin P1(-c) cos e0 + y sin yc cos p(l-c) 0 (o Pv-6o where: 7 i a"a2 -'b (-46, / = la( v-.46) /a. P =, - b (v'-47) * k201 K2 -= 1 (liquid flow) j 4l) k2P1Cpl (heat conduction) kl2Cp2 K2 = kl All values shown are for K K2 =2 L 2 b = (2-~. m 9 = 1,2,5... ~V=)-tSolved by: Half-interval method on IBM 704 Estimated Accuracy: + 2 parts in 106 Note: See Appendix B for explanation of format. -250

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",:, r -', I-:,!,:'.,4,:'.4 II r-:i -:i r-: r-:i -:i r-: r-:i -i r-:ir-:i -.- *.t II r-: 1.r:")r-,,r-:i -:ir-:i -:*r-:i'.: ** *:o*,..| r-:i***:r-:,*:'*r-:.t *.... * II*,.LO 0 I,,'' r:''O: r*-:i,"*'*,,*.:** **0 1* 0-If Lir-; 1-** C,) 4i:'',.........,:'.' 4. c _ r.' ICZ, r:, r-.- cj,:'.. "' I'-.t-,:;", *,:-j )- - T' -: -j Ij ":.,.-'".,-* r," 1'' "',.g rL - 1 1 1 % "~ C —..t:'..t I I.U. U' It I I LL'.:I - r:'.o~ rr I.~i i:T-4 r-:i **t rr-.C, -- L 0 * - t* * 1*:i: *Ji!i 7O: Ll *- 7.i7- t r-: i: -i:O * - CO' N C-4 L, - * 1:'' *t -' 11~~~~~~~ ~~~ rU4, *~: *t 0 7-'':: -:** 1'- ** *: -i.: *11 *4- -- E *!: 1 i7 — 0 LI:I: *f **::i 10 -*- IO — 11... - 7 — *t 1.: 7- 7-'*4 - UJ oI -r, - -'**'rE 6- * r -- **t ** r-i -- **r - jj -i -- -i.<i * r"i *4 )*- r-0 i -: LJ -* r-r.:'*7-:***l':'-'**t' 0: JJ'- r:i 01:Or*t - -j -: 0 -.::i 0 ***4 r-:i -.t U~~~~Yi C`- I-! o r: ~.:* -ii-.Jii- - r:~ -: -.,~ 7. 0 C - r-.: 0C LI —-. I *-A r-: t - 0 -- D r-: 11r-:O() ** -i* C.................- O.... F........... J J........ 13: *r- -:i LO r — 7 i- ) - **- -.T T-i C -4::':,,, - 4 -4~ 4 ij;:^i..... r:, *':....: i'.., ": v:. ~,.,',; If ".' —'-'- El' = "- r —4 r.:".,':O'' - -. - CIU'.'.,-?. -.UJ'. I' - e~~~~~~~~~~~~~J-4.......0-9 C C' ~ ~ ~ ~ ~'.. CO'"", 0", %.t..TC rjc. (-C-:.4 i, -,t-:, i II' ~: b-i' J:- o r- - -,.*"4:7 ~:I D IC.': ~A.2- rV j, b'3. C".,:, 4 Li j o 7',L. r -) Y 1 -',:F- - - I' —. - G.C -, CC- C): ~ q'.2:,'.1'"- O",,~''7"4.2:: P.t U! - O 7,._")'.::i" C,..O U-) r~~ C', -,r!1 r,- rj,- r.-,r,),.,. r, r,.. -t,- r-, r-" rj r-.,;:...I r..t -- r.:!'.i, 4. If I i C, D I'D J! C....... 1CA -t 0'-ft C:. 1'.0 CC, ---:.711 I,1,",4C -4 ";'t,, —r r.;, -,'IIr-.-0 U.,,=..t I-.. u I:.r-.- r~:C'-.t L~~~~~j __ -- -C, I"~~~~~~~ ~ ~.- b~~ i.: rr'o p r', -F?-. -.- -'-.-1 r-) -,. -) - C-,. I- ~ r,-j~~~~~ ~ ~ ~ ~~~~~~~ c _j r,, r-....U'..J~ -.4~ ~ ~ ~ ~~~~~~~~~~~~[,F r',. ".,:-., " - r-., 2:3 I!:'"M' -:'- r -o:-1:': "'.'-..,:-' U)~~~~ ~ ~~~~~~~~~~ -."'.!~,j r-.- "om - - -.! ~:':1 J" =, r o. -, r- o r — J-3 -..' 0 E, r I

APPENDIX E EIGENVALUES, a, FOR TWO LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE Characteristic Equation: PK2 sin p(l-c) cos ye + y sin ye cos p(l-c) = 0 (V-177) where: rb 7 = -H a2 b2 (Nv-17') =rb K' K2 = kb2 (liquid flow) (V-174) k2P1C1 (heaat conduction) klP2Cp2 k2 K2= kl All values shown are for K2 = K5 Jo(b)Yl(bR) - Yo(b)Jl(bR) 0 ( 70) Solved by: Half-interval method on IBM 704 Estimated Accuracy: + 5 parts in 106 Note: See Appendix B for explanation of Format. All values are for rb/H = 50 -266

-2670 4 - 0 c00 0 0 00 0 0 0 0c' 0 c - N 0 4 0 C 0 4 0 0 0 0 0 C - 0 0 0 0' 0 C 0 0 0 C t l l II I III 1I I1 I I 1 I I I I W WLJwuwwWWwlW Ww uwWLLWw WW WLWLiWWWWWWu W W Wui4ww WIIII W WWWWLL u n 0} Vin I ) r-nii- on- o i o 0 in c~ if'< r in o o *< rf o~ as r4 -\o< *or o o c r- o r u3 in 04 - ur r s r c3 r 6 o - i o 1 r. - i, 04 Lo- N a cr- M? FD a N 0 c e 10 ~- n s Q 0 on c o,n o < io'3 rn ) o r' - r NI* s N *O n oC< - N o iLo-.3 0 r ~ ~ 4 a nl "'i- no iC -s o 0 4 O 0 N o C sro or'T~'I i3 r r-o) a 7rt- in 3-*V O 0 -04 o c r'I4 o Or oo 0o 0 Q inr L 3 0 10 II ri 0 Id co - nin o v- LO rino 04 N nc? T r i Cn m N - q o -4 cr*.- 0 r4 I YI oono. oo - - Ncoc r' a 0N -4 r —-*<. rI *4 a T0 a0c 0 -4 r r- Pe r — * - r4- r t 04 in o 7,. r-ri c o in 04 -- r. o t in' *<r r rn r C r a m < c r- ar-, o0 4 rl —. *. 4 * * ^.4. * *. * * *.*. J.i.....4 4 r.~...4.'. C *<** ur co o- - C, oo oom o o 0 V oo oo c^ r^ t -O7 o n O? c< r- *- in n 4 u- C- r *'0 n C co - -s - ^ C r r? - T- N 04 N c C a~* in a 0 Cc* o <T <r cC Iill:, t *^ r^ in m I N ~ 0 in "a ro \+ u o C i * r e co i <f. o * - co *0' * 0-o N -n 1 3 rCO a C It " - in r -- 1, O Or o Mr oO I C - o,~ ry r — C- 0 o o' (4 S i n i. n -N ^ c i n)'< l c o- ir r 0 0- c' *o o' r 1rr c r~ <C c C 0 Cjo4 n 4 o ~ *4 o-'4 M 0 - - O c M i' ro; N r j Lo o ~ 4 * * n * < a - i UCrFrI'<-F I *Li- o LC 0n cr n oUra r — 0 - Cp vr I)Di 0) VCO rf J n r, r,'. C PII - n C' c r,':,. O4 q I I. rI Ir 0 "'o.f......r.'. r..... r N i-...-Coe*..l.c. 1' 0< M o 1co ~- ~ M T - (\ - I Nc -~U c r -t V c^ *^) 10 in r O, r CO ~~ v, co r- On O Co r''<! ** ~~ ~ r c' n - r - a- ~~'- N r t) 0 CO N ) -.. rM.,- 6' ( i CSa o o " o 0^ 0o o - - o o e, c O N) I~ %O o o d r r-0 I 1: 0't -t -, -- LO -_0 * c - VSi 0 r 0c4 CO *4 c ^' co r. r- o o 0) 4- - In o o o r4) o- o o V- Co-.. N) Co 02'^ ~~ in inin ^-0.a'. *r-0 ~.O-c Q0QN. CTN'n f 0TIn o',.r-' r- c r'-''c'0O U aM I~~~~t *sI"^ *~i-0~ ~ C.0 ~ o ~ o r"-~ o_4~ yQ 4- - N., 4",. LO:,O - -4 - - N r) -. *' T c c r 4 LUl LU LUU LU LU LU LU LU lJLU UlW LUlUUU U LU LUU LU LU L LULU LU U LU LU L L L L LU L LU L L L L UUUUUU UUU LOf -,t 6n r, co o c^ O 0 PI r- o 1 r r — rr cc' NJ IC co 4 i T r — CO (A On o o, %D'n ot *0 o' CO r- I -'- 1' o VI r- c in T 000C- 2 r I I I'' I 11 I' I' I I' 1 I* l i I > I <'I I t I~ I * I 1 I I' I' I I. 1 1 1 i I I' I I I lj I I l l I l I ) l l t IIf I I I I I I I I I II I I I I I I I I I I I I j * T T-' t CO'7' r^ *a, M %r * " *' 0, 0 N I y TO n 0 * r- r, y- 4 i qn 1 r7 ) r-' r- - k-*4.O ic kD *r r- c.. ra it 0 C;% CO u 03 TO - r 4) -~ In PI<) C) rC" 0 t~~ * t ~- lf.5 a 04 < 4a r 1"- co r M GS cri o o a r,- to u) o r'- c. rn % r-, u") a.r t v ( c - c4 r i *< r o a c4 o 3 Ur-o o - C I r,-..O 4 o M 4 C N - r- t - co'ti v3 0 - o = O v 6 r* n L LU LU LU L'U3 L U L O L rU- - - CO L - L LU 0L rUj L LU LU L L rLi n 17 0 IQ LU'L rLU LU LU 0 LU VL LU LU L U 0 -- L UZ 0LU C-4 LU LU Nr- L W N 0 L!3 0 (M *-* in *^" c-4 r r - I-n -N Tr) r- C^- f rC r~C ~ N - V M -- -'It r — a 3 0 - (4 r n c 4 in G^ r — b Li co'4-. *< o M 1 r- 3 y3 in r- C. o r)' t %o'sf o *D * % t- ON Gs r - r- T- In CN O- Nr U - I - r ) N) -3 i 1-0'- N O V* *' t -'y in N) o Gr,3 *<t" ~ - r,) %r co 1 C0 g.. r," T- r,, rq r-o r'-'N o r -- Cr7, to Jr, c7i f-. *- 0 r'"', *< - v ) T co rO P4'7t'~t N, LO' 04r'. in T ) co 0 0 h C h r4 n-0' 3 n r c *0COU4 "O' u-c C r-4 Lo uin i r-1 cr. <' C4 040 4 Ci *4:0 04 -in Is U1 1 I' I C,, Vn.** r'- ln %3 CSJ o *<t- r ) 0 C Oi o ) o rn -a Lo ^ r j cM o TO * co r- ) r Ms 0I r Lb r-, )a r -4 r — %t c4 r *4 r - 1 - o co's cO CN co C O' in 01 -r u~/ *t in o T i- ri) o 04 In, N O c^ N3 r- 03 o LU I U' ) 0) T % c 0 y:- C^o 04 o r^- *' _ a o v3 m^f ~m' rp N' un 0 to s o4-' o o P- - c T, 4 0 CO r L Y r- r- 04 Ni * t CD - n f o I' o 0 I co Cr < a:> n- U') ON - 3 0 ~- T It UN 0 - C ( r- GC0 k N 1 U ( O- 1- b o * O t r- o ) *t - c C * 0 n - TO f n T r- 00 C % 0~ M ) 0 ) 0 r — 0 N - 0 ) Oi P- I n r 4 ID. 03 C* n ~o 04 Q, c^'cI t3 C N t- C ~'t'F- "-O 04 0) 1Ci r'') It- -rTO a -eo C 04 ^,O *< I: Co o c- 03 G CO03 - I b" Ij - %C — % Go C 4 in r t- Ci 1n %D * - o LO GCTl'- - Co Z! r 0 I- ~- - y3 * LL 0 co - c^'i -a q o D m ro co in o 6-T D TO * m v ^o N 0 n c i cu r- o 0 cTO r. r-~ o TO o - in C O r 3 o TOr p o ir o'l T ^ a n 3co t^t cN com r- mo 04,ocm af r6 p- N'6- N o 0 c',, U) Co C, 0 d n 0 CO r',( Cr~ n C 0 a v r ) o ) - r - 0 A r- *.f TO 0,4 r., i r - O- N - Gi'-< r- %3 l M F3 V %D r q r- 0 t- c —4 ~-* si3 o u-n o j, cO rj< 0 0 r -,P- 0 a- ~- 0 N i~..... 4...... cj.......M......... V,.~., C1....... t^....... C, % 0, L ~ L W L............... U.. W W.... W W...L.. W.. L....... r........L. inC)\il OOO O 000 0000 0 00o O OOOOO,'O O300 OOOO oOO OO 000 00oO 000 LA, U-) IJ,;~ *~ U~ 8 O O'D 0 0 OC30 C 0 0 0 0 0 C 8 C3 8 8 8 I=') O' O' O' 8 C~ 8 C; I _' 8 C; C; C; u.. C O O Ct -nnn ihoo o d o o c jo-I- b —-o - nOiOniOnino o 00000 UU.IL 51 n000 M aNo 0~ N-o NWn 0 O C NLIOO 0: NO O C Ni4 nOin Nc) 0 NinO O OO M a N 00 NOO C0 8 j 10 0 111 a C1 If, N LO C U ( c4 Ir, 0 C- 4 IY1 a N, - r', I~ - I) l II.I II "' =' Y~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-,o

APPENDIX F DIMENSIONLESS PRESSURE DISTRIBUTION, P(x, Q), FOR SINGLE-LAYER LINEAR FLOW, CONSTANT TERMINAL PRESSURE Calculated from 2 _a2 P(x, g) = Z 2 sin a x e (IV-24) m=l a Eigenvalues, a, from: a = ( 2 ); m = 1,2,3,. o (IV-19) Sum Truncation Test: [Term < |Su| Estimated Accuracy: + 0.00002 Definition of Dimensionless Variables: xa (IV-5) La kta ktp-aa (heat conduction) (IV-25) pCpLa2 -268

-269O \ CM \O r- \ - r- 0 c- OJ - \ [ t- 0 o0OH CM HO~r-lK~?Cf^=t.-00\1-0 L- r- H 0tI ~ ( ~ (CU \ t — - _ U C -H r-I 0 O O O O o cr\ CU L\C\ oCM - ur\ o -Lrn H \ C o0 H O\7\t — [-t — C O —O\0M\OOO II I I * OO\ O\ o t\< o N- \Pcu H H o o o o o Cr\ o ~0 003\ (I\ i r\ 0\D \D Ir\ L (O N O H O H Lr\ 0\ \ CC CQ H- Qo HO "0 0 0tII 1 0 7\ 0\ - \. C\ M t[- _' OJ I — H OOOCO H 0o m d ic a "o cm[ ~ (3 r"\t 2O o 0 Lr Cn cO H L r-LcOCJ 0 O\ cO 0 \ O\ [- U ( \ —L D o -\ 0 0 0 II I I I0 m0\Ho \ 1 o o o 0000 x............... H O K\ [ — C O CO C- H O\ r- O Cm \D CO O\ O L t\C \[ - 1 0 0r L CU HO - a O"\D t~ — 0O ON o HC0) CC)g V3 OD CM 0 0O\NC Lr-\lH[ — C \OJLr\ S LC L N OJ 0 0 0 I I I lOON CT O c -tHtHOO0O OO XO a ~iI O\ \ co \ e * H O O O O 0o a a 002 CC [ - C ON c O HK'0, O 0 \ t - ) \( L\ CO O 4 K \ UL \[ -CO * O \ ONSOH \H H M O CC O Ln\ CJ 0 0 0 IO I,\ 0\ cOC)O \O N cMHOOOOOO 0 c 0\ OCO-0 O- C. Lcr CC 0 O \DO L.n H, — "\,C\ Ln 00o O\ n i \r\O \ CD 0 K\\o o - cc L\ ir\ 0 o l cO L o r\ r\O N o CmD r.* O 0\ \ CC) l C OC) -- Lrn CM t —_t O O O II I ON ONN (N cO -- rK\ LN1U\NC 00 0 000 H 0 O O OOJ0 0 cO - DL O - O O OCO-\0 O o o L~ o o O ~,~o ~ ooDC ~ 0 to oo 0 ~o O0\ ON L\ r\ 0 O M -0\0D H OOO II I O0\C\- 0) n ) \ 0 -1 C H 10 00 OOO..*............... H 0 0 t —,H 0\- 0 OJ C C\ \ 0 0 0 M K C) r-I Qt- HK HLr\t-Lr\ COC L\Lrl\ <" O\ CM rHojN " O a\ CM n L r- F ~ ~ a H a O -CM n 0o CO \o \1oo r- i o0 t t —0 n r\lOo\ t_:- r~ i0 OOOHH4~>i-HK\-0 2Lr ).oMHOOO o ONa CC) D \ - D O8 g H r Hl O H O OC OC O Ir 01\ 01\ \10 0\ -i 4 "0 0 0n P O 0 0( N a t O OCM \OO — 4 OO-4 2OK\O 0000t 0II I 00 C OC NP C OOOOOOO H 00 0 - H 0\ -H ON O 004 \ H CM ICOHD ONC 04C 02 ON O O\OCMCMHH[-t > -H-C~OLr <>CNONDOJO\t K\Hr rHI O 0 CS 0 0 _-H Q o- -o Ho r N CMHOOOO II I OI O\ 1\ O -\ \t- L - tr Cj ro 0 0 0 0 0 0 0 0 H........................*.* H o o- o- o -, U C ~ O o o o o Co 0 OL n ON C r- \ID rl 1 0 Lt! oD a l\ \ 0-: CU - oJ CC 00 O ONONO\Lr CC)O CMH K\ > C t-O- \CM[ — NH\ CM \ COCO H0 0 O -CA kD\\I \O N r C tO. - ( La- O- O OH a oc o\ C cDO O c \ - -t \rl H \ID - J- CM ~ 0 0 a H 0 I I 0 0m' O\. —- - tHrO C O CM r-I r [ — O O H 00...................,.. H _ O ON CC)O LO Oc L L c 0 K L ON NCO t- 1 — S-[ COkl CO P~A O \ O N OUN K Lt~ c N D r0 CO -0'- -' r t ( o 0 P (\ 0 0 o -\- aco CC \O C t.-> - C — 0 lH 0 — 0 0 II ONOO OOc t — cmcc)\[ t- \o-L —ONO M ( M\ OC) OOCQHOOOO I O O\ ON CO 0- L r' —Kn NC N H l rHI O O O O O O O O O H CM -=- It - 0~ 0 0 0 0 H0 H OCM- 0~ O-c O OOOOOOOOOOOOH OO O O O O CtONOOO H H H CM CM KC

APPENDIX G DIMENSIONLESS PRESSURE DISTRIBUTION, P(r, @), FOR SINGLE-LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE Calculated from: _b20 P(r,Q) - Z Cm U(br)e (IV-82) m=l Eigenvalues, b, from: Appendix B Truncation Test: ITermi < ISum Estimated Accuracy: + 0.0002 Definition of Dimensionless Variables: r - ra (IV-46) rb R = r (iv-47) rb kta 9Q - ~kt2 for liquid flow pcrb (IV-48) kta for heat conduction pCprb2 -270

-271R = 1.5 R = 2.0 R = 2.5 R = 3.0 8 r=1.2 r=1.5 Q r=1.5 r=2.0 r=1.5 r=2.0 r=2.5 r=1.5 r=2.0 r=5.0.005.9584.010.9997.05.9066.9989.10.7831.9819 1.0000oooo.006.9380 1.0000.015.9968.06.8777.9972 1.0000.15.7019.9514.9997.008.8960.9999.020.9898 1.0000.08.8263.9911.9998.2.6453.9185.9981.010.8562.9993.03.9662.9999.10.7831.9819.9990.3.5702.8582.9880.015.7730.9935.04.9368.9994.15.7019.9512.9918.4.5213.8088.9684.02.7097.9792.05.9066.9977.2.6452.9171.9763.5.4860.7678.9425.03.6198.9303.06.8777.9943.3.5690.8505.9277.6.4585.7323.9131.04.5564.8686.08.8261.9817.4.5166.7887.8693.8.4159.6717.8505.05.5061.8044.10.7824.9624.5.4749.7315.8099 1.0.3818.6192.7886.06.4631.7422.15.6964.8973.6.4389.6786.7527 1.5.3127.5083.6494.08.3905.6293.2.6290.8249.8.3769.5841.6485 2.0.2571.4179.5540.10.3302.5327.5.5202.6884 1.0.3243.5027.5582 3.1738.2826.3611.15.2175.3509.4.4320.5724 1.5.2229.5455.3837 4.1175.1910.2441.2.1432.2311.5.3589.4757 2.0.1532.2374.2637 5.0795.1292.1651.3.0621.1003.6.2983.3953 3.0723.1122.1245 6.0537.0873.1116.4.0270.04355.8.2059.2729 4.0542.0530.0588 8.0246.0399.0510.5.0117.0189 1.0.1422.1885 5.0161.0250.0278 10.0112.0185.0233.6.0051.0082 1.5.0563.0747 6.0076.0118.0131 15.o0016.0026.0033.8.0010.0015 2.0223.0296 8.0017.0026.0029 20.0002.0004.0005 1.0.0002.0003 3.0035.0046 10.000oo4.0006.0007 30.0000.0000.0000 4.0005.0007 5.0001.0001 R = 4. R = 5.0 0 r=1.5 r=2.5 r=4.0 r=1.2 r=1.5 r=.0 r=2.5 r=3.0 r=4.0 r=5.0.10.7831.9995.010.8591.9968.15.7019.9961.015.7735.9960.20.6453.9886 1.0000oooo.02.7098.9896.3.5702.9660.9999.03.6209.9662 1.ooo000.4.5214.9395.9992.04.5609.9368.9997.5.4864.9132.9971.05.5171.9066.9989.6.4597.8887.9954.o6.4834.8777.9972 1.0000oooo.8.4210.8454.9807.08.4541.8263.9912.9999 1.0.3937.8089.9624.10.3993.7831.9819.9995 1.0000 1.5.5486.7364.9042.15.3433.7019.9514.9961.9998 2.3177.6776.8410.2.3088.6453.9183.9886.9991 1.0000 3.2703.5789.7220.3.2670.5702.8582.9660.9942.9999 4.2314.4958.6186.4.2415.5214.8095.9595.9850.9996 1.0000 5.1982.4246.5299.5.2239.4864.7699.9132.9730.9986.9999 6.1697.3637.4539.6.2107.4598.7373.8887.9596.9968.9998 8.1245.2669.3331.8.1920.4211.6866.8457.9519.9908.9985 10.0914.1958.2444 1.0.1790.3958.6486.8100.9055.9822.9955 15.0421.0903.1127 1.5.1584.3499.5840.7434.8490.9540.9792 20.0194.0416.0520 2.1458.3226.5418.6963.8040.9217.9535 30.0041.0089.0111 3.1298.2876.4856.6289.7331.8549.8904 40.0009.oo19.0024 4.1185.2628.4444.5769.6745.7904.8250 50.0002.oo04.0005 5.1091.2419.4093.5317.6220.7501.7625 6.1006.2231.3776.4907.5741.6742.7045 8.0858.1902 3218.4185.4894.5749.6005 10.0731.1621 2744.3566.4173.4902.5120 15.0491.1088.1842.2394.2801.3290.5437 20.0329.0751.1236.1607.1880.2208.2307 30.0148.0329.0557.0724.0847.0995.1039 40.0067.0148.0251.0526.0582.0448.0468 50.0030.0067.0113.0147.0172.0202.0211 60.0014.0030.0051.oo66.0077.0091.0095 80.0003.ooo006.0010.0013.oo6.0018.0019 100.0001.0001.0002.0003.0003.0004.0004

-272R= 6.o R = 7 R = 8.0 9 r=2.0 r=4.0 r=6.0o r=2.0 r=4.0 r=7.0 9 r=2.0 r=5.0 r=8.0.5.7699.9986.5.7699.9986 1.0.6486.9978.6.7373.9968 1.0000.6.7373.9968 1.5.5840.9902 1.0000.8.6866.9910.9999.8.6866.9909 2.5422.9785.9996 1.0.6486.9824.9996 1.0.6486.9824 1.0000oooo 3.4892.9508.9966 1.5.5840.9562.9965 1.5.5840.9563.9996 4.4557.9234.9893 2.5422.9289.9887 2.5422.9292.9978 5.4318.8981.9778 3.4890.8788.9611 3.4892.8812.9878 6.4136.8750.9635 4.4545.8346.9247 4.4556.8421.9702 8.3865.8335.9298 5.4282.7943.8855 5.4314.8093.9478 10.3659.7961.8934 6.4061.7568.8459 6.4125.7804.9230 15.3255.7128.8033 8.3678.6877.7702 8.3827.7298.8710 20.2916.6391.7207 10.3342.6252.7005 10.3580.6846.8195 30.2344.5139.5796 15.2634.4929.5523 15.3057.5853.7018 40.1885.4133.4662 20.2077.3886.4354 20.2615.5009.6006 50.1516.3324.3749 30.1291.2415.2706 30.1915.3668.4398 60.1220.2673.3015 40.0802.1501.1682 40.1403.2686.3221 80.0789.1729.1950 50.0499.0933.1045 50.1027.1967.2359 100.0510.1118.1261 60.0310,0580.0650 60.0752.1440.1727 150.0172.0376.0424 80.0120.0224.0251 80.0403.0772.0926 200.0058.0127.0143 100.0046.0087.0097 100.0216 o0414.0497 300.0007.0014.0016 150.0004.0008.0009 150.0046.0087.0105 400.0001.0002.0002 200.0000.0001.0001 200.0010.0018.0022 500.0000.0000.0000 R = 10.0 R = 12.0 R 14.0 o r=2.0 r=4.0 r=7.0 r=10.0 8 r=3.0 r=6.0 r=12.0 9 r=3.0 r=7.0 r=14.0 1.0.6486.9824 1.0000oooo 1.0.9055.9998 2.8059.9989 1.5.5840.9563.9998 1.5.8493.9983 3.7435.9942 2.5422.9292.9989 2.8059.9947 4.7004.9862 3.4892.8814.9942.9998 3.7435.9820 1.ooo0000 5.6682.9762 1.0000 4.4557.8430.9862.9990 4.7004.9659.9999 6.6430.9653.9999 5.4319.8120.9761.9968 5.6682.9491.9997 8.6053.9435.9993 6.4137.7865.9651.9932 6.6430.9329.9990 10.5779.9230.9977 8.3874.7464.9422.9816 8.6053.9033.9960 15.5324.8799.9885 10.3686.7157.9195.9658 10.5779.8778.9905 20.5031.8459.9730 15.3369.6599.8650.9174 15.5322.8276.9678 30.4647.7933.9324 20.3141.6170.8139,8658 20.5021.7889.9381 40.4372.7501.8880 30.2772.5450.7207.7676 30.4589.7261.8731 50.4139.7112.8439 40.2453.4825.6382.6798 40.4240.6718.8097 60.3926.6749.8015 50.2172.4272.5651.6020 50.3926.6223.7503 80.3537.6083.7226 60.1924.3783.5004.5330 60.3636.5765.6952 100.3188.5483.6514 80.1508.2967.3924.4180 80.3121.4948.5967 150.2460.4230.5025 100.1183.2326.3077.3277 100.2679.4247.5122 200.1898.3263.3877 150.0644.1266.1675.1784 150.1829.2899.3496 300.1129.1942.2307 200.0351.0690.0912.0972 200.1248.1979.2386 400.0672.1156.1373 300.0104.0204.0270.0288 300.0582.0922.1112 500.0400.0688.0819 400.0031.0061.0080.0085 400.0271.0430.0518 600.0238.0409.o486 500.0009.0018.0024.0025 500.0126.0200.0241 800.oo84.0145.0172 600.0003.0005.000oo7.ooo8 600.0059.oo93.0112 1000ooo.0030.0051.006 800.0000.0000.0001.0001 800.0013.0020.0024 1500.0002.0004.0005 1000.0000.0000.0000.000 1000.0003.ooo4.0005 2000.0000.0000.0000

-273R = 16.0 R = 18.0 R = 20.0 o r=4.0 r=8.0 r=16.0 o r=4.0 r=-8.0 r=10 r=18.0 r.0 r==2.0 r. 5.8120.9897 5.8120.9897 5.8121.9985 6.7865.9832 1.0000 6.7864.9832 6.7865.9968 8.7465.9684.9999 8.7464.9684 1.oooo 8.7465.9915 10.7162.9530.9995 10.7162.9530.9999 10.7163.9843 1.0000 15.6641.9175.9963 15.6641.9175.9989 15.6641.9634.9997 20.6296.8877.9891 20.6296.8878.9959 20.6297.9424.9986 30.5847.8411.9653 30.5850.8423.9830 30.5851.9058.9920 40.5544.8043.9353 40.5558.8083.9636 40.5561.8761.9802 50.5304.7724.9033 50.5339.7805.9411 50.5350.8511.9649 60.5095.7431.8712 6o.5159.7562.9172 60.5182.8293.9476 80.4720.6890.8091 80.4853.7130.8689 80.4914.7909.9105 100.4379.6394.7511 100.4583.6737.8221 100.4689.7563.8730 150.3634.5306.6233 150.3983.5856.7150 150.4201.6782.7840 200.3016.4404.5173 200.3464.5093.6218 200.3770.6087.7037 300.2077.3033.3563 300.2619.3851.4702 300.3036.4093.5668 400.1431.2089.2454 400.1981.2912.3556 400.2446.3949.4566 500.0985.1439.1690 500.1498.2202.2689 500.1970.3181.3678 600.0679.0991.1164 600o.1133.1666.2034 600.1587.2562.2962 800.0322.0470.0552 800.0648.0952.1163 800.1030.1663.1992 1000.0153.0223.0262 1000.0370.0545.665 1000.0668.1079.1247 1500.0024.0035.o406 1500.0092.0135.0164 1500.0227.0366.0423 2000.0004.0005.0006 2000.0022.0033.0041 2000.0077.0124.0143 3000.0000.0000.0000 3000.0001.0002.0002 3000.0009.0014.0016 4000.0000.0000.0000 4000.0001.0002.0002 5000.0000.0000.0000 5000.0000.0000.0000 ~ _...' ~_ =___ — = _..,-. =__. R = 50.0 R = 75.0 0 r=4.0 r=10.0 r=20.0 r50.0 r —4.0 r=10.0 r=20.0 r=7 10.7163.9844 1.0000 20.6297.9424.9993 15.6641.9635.9999 30.5851.9059.9963 20.6297.9424.9993 40.5562.8766.9909 30.5851.9059.9963 50.5353.8528.9841 40.5562.8766.9909 60.5192.8331.9767 50.5353.8528.9841 80.4952.8020.9614 60.5191.8331.9767 100.4780.7782.9468 80.4952.8020.9614 1.0000 150.4491.7364.9154 1.0000 100.4779.7782.9468.9998 200.4305.7083.8905.9999 150.4491.7364.9154.9982 300.4o64.6710.8536.9992 200.4304.7083.8905.9942 400.3909.6463.8270.9969 300.4064.6709.8531.9798 500.3795.6282.8066.9929 400.3904.6455.8250.9605 6oo.3707.6139.7901.9875 500.3781.6255.8014.9391 800.3574.5923.7644.9736 600.3675.6082.7801.9170 1000.3474.5759.7442.9575 800.3488.5774.7412.8731 1500.3281.5442.7041.9138 lo00.3317.5490.7049.8308 2000.3119.5172.6694.8703 1500.2928.4846.6223.7335 3000.2825.4686.6064.7888 2000.2584.4278.5493.6475 4000.2560.4246.5495.7148 3000.2014.3334.4281.5047 5000.2320.3848.4980.6477 4000.1570.2599.3337.3933 6000.2102.3487.4513.5870 5000.1224.2025.2601.3065 8000.1726.2863.3706.4820 6000.0954.1579.2027.2389 10000.1418.2351.3043.3958 8000.0579.0959.1231.1451 15000.0866.1437.1860.2419 10000.0352.0582.0748.0881 20000.0529.0878.1136.1478 15000.0101.0167.0215.0253 30000.0198.0328.0424.0552 20000.0029.0048.0062.0073 40000.0074.0122.0158.0206 30000.0002.0004.0005.ooo6 50000.0028.0046.0059.0077 40000.0000.0000.0000.0000 60000.0010.0017.0022.0029 80000.0001.0002.0003.0004 100000.0000.0000.0000.0001 150000.0000.0000.0000.0000 200000.0000.0000.0000.0000

-274R = 100.0 R = 200 @ r=4.0 r=7.0 r=10.0 r=20.0 r=50.0 r=100.0 9 r=10 r=20 r=50 r=100 r=200 1.Oxlo.4778.6658.7782.9468.9999 gx103.5769.7459.9343.9957 1.0000 1.5.4490.6273.7364.9154.9992 1.5.5502.7130.9055.9879.9999 2.4303.6020.7082.8905.9973 2.5326.6910.8837.9788.9996 3.4063.5691.6710.8536.9908 1.0000 3.5096.6617.8524.9611.9973 4.3907.5476.6463.8270.9825.9999 4.4943.6422.8303.9454.9925 5.3794.5319.6281.8066.9736.9995 5.4830.6278.8132.9316.9857 6.3706.5197.6139.7901.9649.9987 6.4741.6162.7993.9192.9776 8.3574.5014.5925.7648.9486.9956 8 4600.5980.7768.8969.9593 1x10.3478.4880.5768.7458.9341.9907 1x14.4485.5831.7578.8765.9397 1.5.3314.4650.5499.7125.9037.9729 1.5.4237.5509.7163.8294.8905 2.3200.4491.5312.6888.8780.9511 2.4011.5215.6781.7852.8432 3.3024.4244.5019.6512.8326.9051 3.3596.4-676.6080.7040.7561 4.2870.4028.4764.6181.7907.8601 4.3224.4192.5451.6312.6779 5.2726.3825.4525.5871.7510.8170 5.2891.3759.4887.5660.6078 6.2589.3634.4398.5577.7134.7761 6.2592.3370.4382.5074.5449 8.2337.3279.3878.5032.6438.7004 8.2084.2709.3523.4079.4381 1x104.2108.2959.3500.4541.5809.6320 1x10.1675.2178.2832.3279.3522 1.5.1631.2289.2707.3513.4493.4888 1.5.0971.1262.1641.1900.2040 2.1261.1770.2094.2717.3476.3781 2.0562.0731.0951.1101.1182 3.0755.1059.1253.1626.2079.2262 3.0189.0245.0319.0370 0397 4.0452.0634.0750.0973.1244.1354 4.0063.0082.0107.0124.0133 5.0270.0379.0448.0582.0744.o8o1 5.0021.0028.0036.0042.0047 6.0162.0227.0268.0348.0445.0485 6.0007.0009.0012.0014.0015 8.0058.oo81.0096.0125.0159.0173 8.0001.0001.0001.0002.0002 1x10.0021.0029 0034.0045.0057.0062 xl06.0000.0000.0000.0000.0000 1.5.0002.0002.0003.0003.0004.0005 2.0000.0000.0000.0000.0000.0000 3.0000.0000.0000.0000.0000.0000 R = 500 R=1000 o9 PSr=10 r=20 r=50 r=100 r=200 r=500 o r=10 r20 r=50 r=100 r=200 r=500 r=1000 4 5x103.4831.6278.8134.9322.9935 1xlo.4511.5866.7631.8864.9746 1.0000 6 ".4742.6165.7998.9206.9900 1.5.4342.5647.7357.8584.9568.9996 8.4609.5993.7789.9016.9823 1.0000 2.4230.5502.7171.8387.9419.9987 1x10.4511.5866.7631.8864.9746.9999 3.4080.5308.6923.8116.9186.9953 1.0000 1.5.4342.5502.7171.8387.9419.9972 4.3981.5178.6757.7929.9011.9906.9999 2 ".4230.5502.7171.8387.9419.9972 5.3906.5082.6632.7788.8873.9855.9997 3 ".4080.5308.6923.8115.9184.9897 6.3848.5006.6533.7675.8760.9803.9993 4.3979.5176.6753.7924.9001.9790 8.3759.4890.6383.7503.8581.9702.9976 5 ".3900.5074.6621.7773.8846.9666 1x105.3692.4804.6271.7374.8443.9611.9948 6.3833.4987.6508.7643.8706.9536 1.5.3577.4654.6076.7147.8196.9413.9841 8.3716.4834.6308.7409.8446.9269 2.3496.4548.5938.6986.8018.9242.9706 1x105.3607.4692.6124.7193.8201.9004 3.3370.4385.5725.6736.7734.8936.9411 1.5.3353.4362.5693.6687.7624.8372 4.3260.4242.5538.6517.7482.8650.9115 2.3117.4055.5293.6217.7089.7784 5.3156.4106.5361.6308.7243.8374.8825 3 ".2695.3506.4575.5374.6128.6729 6.3055.3975.5190.6107.7012.8107.8544 4.2329.3030.3955.4646.5297.5817 8.2864.3726.4865.5724.6573.7600.8009 5.2014.2620.3419.4016.4579.5028 1x10.2685.3493.4560.5366.6161.7123.7507 6.1741.2265.2955.3472.3958.4347 1.5.2284.2971.3879.4564.5241.6059.6386 8.1301.1692.2208.2594.2958.3248 2.1943.2527.3300.3883.4458.5154.5432 lxlO.0972.1264.1650.1938.2210.2427 3.1406.1829.2388.2809.3226.3730.3930 1.5.0469.o61o.0797.0936.1067.1172 4.1017.1323.1728.2033.2334.2699.2844 2.0226.0295.0385.0452.0515.0565 5.0736.0957.1250.1471.1689.1953.2058 3.0053.0069.0090.0105.0120.0132 6.0533.0693.0905.1064.1222.1413.1489 4.0012.0016.0021.0025.0028.0031 8.0279.0363.0474.0557.0640.0740.0780 5.0003.000oo4.0005.000oo6.0007.0007 1xl07.0146.0190.0248.0292.0335.0387.0408 1.5.0029.0038.0049.0058.0066.0077.oo08 2.0006.0007.0010.oo011.0013.0015.0016 3.0000.0000.0000.0000.0001.0001.0001

APPENDIX H DIMENSIONLESS PRESSURE DISTRIBUTION,' P(rQ), FOR SINGLE-LAYER RADIAL FLOW, CONSTANT'TERMINAL RATE Calculated. fromr2 + 4Q R2Rn r p(r,0) 2(R2_l) (R2 1) ( + 2R2 + 4R4.n. R 3R4) r(R2 -, )2 + e- J122 (lR)[Jl(C)l(o(a) (r) - Yi(a)Jo(cr)] + I t. 2.....( j -(-) —- (IV-150) nl-l C[J2 (oR) - J12 () ] Eigenvalues, a, from~ Append.ix C Truncation Test: ITermi < ISum Estimated Accuracy~ Four significant figures Definition of Dimensionless Variables~ r a I'V..6 rb R re (IV-47 rb Q= kta2 for liquifd. flow P.crb2 (IV'-48) --. for heat conduction ~pr~b ~275 -275

-2760O i 0 oLn Ir N N N N O O aln ao N o~ CY, N o O O, —.,O N \ "' 0, — N'O V O0 O O O rt O' O'. O O O' - _ _' o o00o o "so.O N O o O _,. N _ O N O,0 _,,. u~ cn,D N 00 Ln Lr 00, O * c c O C _ C 00 00 N O c O N oo Io s 1 O O O N,.O N L 11 I O O _ ~ " O ~ C' "O * * * * * * * * * *a * * * * * * 4'_ N.ON P4 _ e o in -, On ~n,. v 1..'..- O O M 0 o o o o, i. _ O00 a oo O^ O O ^i'_' O m N Ln O Ln O',O N, —"O M M I -'- I I II II o,i_ bN 00 " r-' 1 1- II " N.- O r- t- o "-4 ~ o o ~i ~n ~ ~ ~4 ~ ~ ~ _. _ O _ O O C D N O _. _O O O O N N OO O n O OOO-O O 0. OO O O O O O0 Ln OP - Ln.O _;:c "-4. o o e "n n' N Oo o II oNO O N ur') u~ N ur~ II O O O ""O 00 ~-0 " a,% 0 ~0000N~ ~ II 0000~N0 ~0I ~~0~ oo OOon ~ a%0 n Mia n i Ln o In^a Ln sDnco *'te Ln ec e N o Ln oo Ln oo0 o0 n oo O N * o oo 00 o oo< N _ o N u 0 0 N o o o Oo oN " _ 0 0 soN 00' O o o P om m n 4 o o o o o o tn n Ln "3 O O O - e0 00 o 0'00 CT rs 0 a.... _4, o*o Ln - _ a - _ n o _O-o _ Ln 0 _..O O I. ~o O. O O O O _' O O N O s0 _o _ n sr _. (Moo o > soocqO (.(Mo^ @c _- "o4 P:. Oh e os > >$ o o _ N Lo 00 0 O N as N oo Oo co N O s0 \0 ~D ~O ~O _ 0 eII -O O -e _ o....e _~ oJ e Ln > F _ _ _ ll - N ~ d~ ~0 ~1v G sC o II N 0 ~ ~0 GO ~ ~ ~ O 1~, O O O O O O O. O. -O O O O. OxOO n o, o,-,o o o o, -4~~~~~~~~~~~~~~~~, j-4 o

-277o 0s n m oo 0 oo - Imo Loo Ln C o _ N -,,O 000 oO o0 O O* * * * * * *0' " e O 00 00 N 00 - t- r oo 0 _ oo -, o Oo ~, II,, ~.0~OCO O 0 N_ ~dCI o N - 0 t 0 0~. O.J 1oooda'O 0 * 0_ 00 00 I I M >' L _ _ e %o O14 Q N' " 0 -4 N NL O N N N NO N cn v v t- v en -na' CY% (t 00 oN 0*0o 0N o 1.nri N _coO (- o o 00 00 00 O.. 4 e N _ foo no o..,..... 00. 0 0 r. ". -. ". - 00* * * " * ^ en 00 _1 _ N' (In Ln V O O o o e r Ln %O O \O -O,O LO Ln O M O - O 1 O O.-4 CO0"4~ cD NOfl OOO ~I -NLOOO L N m %O 0 CY% P 4 14oo_ N4 N CN - o o oo-4 o O f,~ c:' _ o O o 0 0% oo-0 0 o00 o 0 Ln C n o o' 0 00 P4 00 00 O C C4 Ln 0 Ln O D r4 N Ln N Ln - 0 o O -.,, b..,..,ro o o-,, o ~,,o oM o o ~ oo o DO-. 4, -; N. 0 _ n Ln _o cn -r _I,,~ ~1- 0or,0 _ ~. _. 1 0 U ~ _ol- oo 0 0. or o O C. (I oo. -. t O O 1. 1 ", _0 ~ 4W ~ ~.. II N N " - 0 - -* -'" -'' 0 0 0 * * 7- I- oO -C - moo O~ O " O CO.-'I 0 0 0 CO " 0"0OOCO no^ mo 0 O'~ 0' N -O' O, O O,-','"CO r-Oo _o d N 0 4 N o- o o 4 o o0 r o0 C_ N r I 0I N'OD0 o _ O O O 0 N r t 00 0 0 " OC' mo d ~O"N~~~~ 009e 6. 0o _ - o oO^ ^oo *n w O _4o o5 0 O n c > SO _Ln O O O - N o O, - 1 c' 1 so N00 _.'o o _ N O00' c- ~ cl 0I O O 00 *-.1nO 4 -N N.4 O 14 %1 0 a, 014C-4 N LOO e~N c r,., o o -, C'.. o 0,'cY, o *c' ~ o o o o M o,- o _ o O _0 O Ln o-o 00 O N _ O' V -, r 7,O O O 0 N 0O O o.- N LO P-4 I 1'0 0- cO oO C -- I I' N u' 1 oN O 1 C' C O' C 0 O O'ct ~-~-(o 0O (M -~~''. -~J J.J * —Nooooo oo'^oo GO14,o^Q^-~ooOL~fNC 14 o ^m -'c4 Lo -~ O O * O U"I ~ O C O O,O I' oO0 b LO O O'-O Lo O00 "- ~..0N -4COCOC'NO- 1 N 0 N 0 LAno C N o N - T^ oo on 4D O0 0N'O w 00 -'-4 00ooLAc ON.- 11 00 I'0O - C O..-O,,,'0CO.-eC"lCO0.-4C"') I CF) Ln ct'0" 0C 0'0'0 0 OLn L, LAOA0 0 * -0 nOo un^o o O noo oovom O O O O -N~ LA J'-. N L~ -~ M o -<.- ( I

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-279o o N %0 o Do oo0 N OL Ln U Ln o o 00 N - F' ro N 0000 c co 00 N' cr N'O 0o OO Ln o - o O 000 o L 000 w _ N Ln O N e' c7 N N O O N - LAm - 00 00 00 II _ e O 0' e 000000. l.. -4... m''_ *" Nc rM O 0 a'LA7000 00 0a'n No e Om LAO LO LO A N LA0 Lo A Loo ooLo h.ooso~rsi i Ln Ln itn ~o o N -~ c- Cr' r cc) so s Ln O L r " N C' _ - eA LA LA LA 1 -r Ln 00 eC 000 00 00000000 ( - r o Ln OL O O o - (N LAn o LO' N

APPENDIX I DIMENSIONLESS CUMULATIVE FLUX, Q(t) FOR SINGLE-LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE. Calculated from: R2_1 00 e-b2Q2(bR)D Q(t) = (- = -2 Z e 1 ().- (IV-87) t) 2 mal b2[J2(b) J12(bR)] Eigenvalues, b, from: Appendix B Truncation Test ITermi <IS!I Estimated Accuracy: All figures shown are significant, Definition of Dimensionless Variableso r -a (IV-46) rb R e (IV-47) rb kta kta for liquid flow Opcrb (IV-48) kta 2 for heat conduction PCprb -280

-281w O* M N _ o o O r rCI e N ooo o so ^ N o rC ^t Nh a s^o ^ O X c a oO O ^ O O *&I ~ c; ~d~ r; 06 C~t~o Co:o~; cW* c*f *4 lo * r * * *o * * C* * n * *~ e *. * ~ * *,,,,,,,4,, _ - _ -, _ -4, -, N N N N N N N N N N N N e' " " e(' c"^,, "' Ln in O n o NcoO' 4N0%NaN, Ln 0O% 4o CT(o o CyOo 0 o CY; m r' oo o..,,'" o' " u mn %O sO oO %O %O %O r t rCr -rC q: N N N~ Ln Ln Lm% o N v O-. 0M N O P N -Ln *nC% O * u Ln - % o o Ln 0 Ln Ln 0 w 00 a, m c,4 P 5 tf, L l a% Ln N.O winKd Ln o P4 Ln w ~ a O w N e N w 0" N' -O w _ N. 0 %O, - Ln"'4, 0 L O 0 O " cn N N n an - O 4 n - ~ > X O N 4J ) C^jN 6 Z Co C~ - O OZ Z _ - ooo c oo o r1 _~ Fo *)ooo NO ~ o l t _t ot rt o _ov.o v "4 o _ oi_ -or,:i O-oo co N o C ) o N o 0 Loooooo oooooo ooo CD N N N N'" o' o' L" L' oo o C L o o O O,O COO I4 N N crcn n r Ln On No,,O, II 00000000000 tIn o o oo o N v oo o cooOC co Ln 0 0o Ln c Ln o o o o, 0 0 c N L c f C) CO n c rN C 0 *n Ln L0 0 Ln Ln O O i O t; N LA O Oo -4 o _oN CoONN, O 0N, o0 o O n LSL %oC o Lno oono L ooooo00 - - -, —

-282r. N a e os6 o' o 6 o -O o r_ sOP _ 6 o o ao _ N o _ n ( - so r- _ - - oo r o- -o - — C r- _ t- n oo KO c o - N N c e + l n ur Lin ^\so oo so No No eo mo -I CD oJN, o 0ooN,.o O,o o o oo o o o o o o c st 4 W4 " oo- o oo^ C Q + _ > + 0s O N N el", ( r *. c C sN P L 00 F,,-, N ooo N O r oo o U'~ _oon,. I-' 0 _oo0 O r.'Oq-'.Oi n O _1, Cp o C N,. 0. o.o a a O' o Ln n o n o oo oCD m e 1 01 e Ln n ur i i i so r Nooo ao N so oo o U ------ - -- --— ooo^3l^~sOO~o~s~f~f~t<-^^s-oso'l~tl<^o^^-. I| 000000000000000000000000 @ Co) sO0o0ON0~,O O0NO00 00OOOO ----- ~ NN N N 10 N oO o N O Co Co O o 0 1 aN N Co - 0 00 a% o % o ^^ c^^ ^ _^ N oo c, c% co _-^ -m Ln wo: _ N v NIn o oo _ ls v a v oo M C ai r oo n o4 cmi No C u oo_ r.: C,: r- V: o Ln N O o ao f o o o _ NO o o oO Co C o0 N N N N N N N Na o o \O N N 0 7 a j a CD 0 0 7 j 0 L \ 0 0 0 000 C 0D N oo O co 0N 0\ cO Nt - Co 0 N O 0o OC 0 0 4. ci.4 10 _oO ooC - a CO t cP- oo oo W W 00 O C L C- N W 10 - O O 4 CN e N _O m N 0- rn N O W, N O w u _ N N w n O o N II 000000000000000000d 00 N g N n io w 10 o C N d'- 0 Co O d n o O o Ln O a O o CDa N (IQ N N N M M M M V dIt dv Ln Ln UO LO LnU No % oo o a o N ^ N de 00 00 rN O m r t - -n 1 N - 0 oo a O 4 M 4 cr N *n a 00 oo Lc - f e a OC O s Iy| 4 dc,O P(Mn n O u O In co N c > a _ N co Y n CT al ON a- 7 ON a% all, CD t oo N N O oo N D O O O O O O O O OCD r N v N N r C C C Lr) C dLo 00 0a N - 90 ~ ~ cr,~~~ _r _n _~ o o C c d n

-283d c N oO r r L 0'o - P- L N so 0o r r - in - oo 00 N N O ~O \s o "r-e 1t so _ m o0 -o - wo c v cr eC oN c cn ve e N _ % tn _ oo 0 o oN t- - in i O O O O O O O tO O O O O O O O O % OO OO O O O OO O LA I~ CI ~N N N3 cm N N N (N cN cN N cN cN cN en cN m (N m m en CN cN cn (n n cn n CD m l,,-,~ m m M,,- ml m,. m m.., m m. m m M, C o XXXX XXX XXXXXXXXXXXXXXXXXXXX X XX Ln Ln N 0 o Ln o Ln L L o Ln 0o o n - N 1r LtA L o r oCo 0 0,-i,, N N,. e,,4,,, Ln, Io r, C o0 o.Co' C - - " - In Co -4 V V oa nb bLA O' 0 CO n In oLA N -4 rt en a' N r r- o r - _ 4 C oo c Ain Is d< o I o o 00L so c n e o _ C _ oo P4 C o r^ _ _ o - _ n ooI -; O aoo oo o o ob o Oe ac o r e r 0 c, Ln 00 0oo - C_ co O n N N e ew e P d - 4 ml c Td %O r- I- co 00 o' 0 0 0. - r4 e N N N N N N N N N N N N NPN N N _. O O O O O O O OOOOOOOOO O OQO 0 ~ O o Coo otoNL o oo o o- Co o o o eoo o sO D o 1, o L0 o' o o 00r co coi.., o...... D N C0, { n fn ro Ln _\ 0 W dl M _o. o ono i 0 co no LA N co N N L co Co - C Ln N L n. m as ( so a ~ o n < o oo n O oo M n o O en o _? o a <^^^++t<~r0rjoo^q<mmsrXoo^O0>oi O 0 OL O O 0 O 0 V n b In O C 0 0 O C O 0 0 L 0 O 0 O CID t- 0 Ln 0 oL O _n 0 0 0 oo o 0 oO 0o X~'S' F ri uO s o _ o m un o sorfCe N o N o o~ o o N 000000000000 Ln 0000000'00 00 0 a o - - cc tn ro _ o oo a% a% a% a, %^ oe oo 0 oo o 00 0 N O O O Oc O O O O O O O O 000 0 a O 0 0 0 0 000' ) n' I) LA) LA L o t- co 0 N V.o Co 0 N 4'' Co0 0 LA 0 L0 0 t a' N L C- Co a' Co r- o 4 Co v 0 LA NL a o 0 _ r LA N ooo o0 No N Co -4 a' O LA i _- a r N ao Ln oo a' Coo c Co 0 ~0 N o'oo o~0 N Nooo C 0 ooo n on 0 n o oo o0o o000ooo C e 1cne 4 1 4 4 L Ln A Ln A L A O O i oo o a' ca o N so oo 1 oo _ _ -4_ N N N

-284Ie oa uo c) dr -^ o@ o +5( o r o- n v > ou, > ooo' o o at o~ a-I f 4co N'^i nD asoo o o oo ~ o o - _ oo~o s oo _ or N~ u soo N oo N n O o M o _ L) r tL c - oa 0 0 N r C o N o 0O OO O o a' o- c' 0 (oo - oo a No'~ 00 0000000 00000 C 0 0 0 0 ft CM N N N N N _ _- N CO N M N N N N nnn "-I 4 — 4 - -- - - - - N N- N. N. N N N- e 4 - -4 4'4 -4 P- -I Lo 0 N 4 N c; (4 ~,, Ln,; O IC; I.. 00 00 0~ O _ _ _ _ -; _ _- _; N j 4 4 N" C4 t o o In Mr oa ao o Ln Co N % V _- oo O om * -. * * X *X X *n *o -o X X X * X * * * * * * oZ S Ln c m - - o 0 N 0 o C oo O i n - o oo cr cr c cy _ _r c0 o^ m o o,. o ~ o,.n so. noo oo m o,, a,o,. 00o o', o -~.o..-;.-; V q4 1 4 t J4 j v Ln Ln L Ln o. Ln Ln C'N C'0 N L " C) a' 0 O a' N " N' 0 0 O0 N qt ( so oo o iLn o nn o Lo oLnaO L r c o o N _ _ _. _ o o n ~ e CV - ~ e cc-i ~ e n + ~ o ~ ~ d e _s n z ~ O 00000 C0000 0000000 0 04 C0 4 C o n o c0 0000 0 4_ a% V ( 00 n e Ln %O S o r O - V o d n - M _ o o %0 M oo N %O o o N M n V N CXN Cn (y L rLn -O r- 00 _) - N Cn qJ4 Ln %O r 00 00 ba- O P- ) n Ln 00 oo O _ N - -- - - -- - --- - ---- - N N N N N - -,, _M Cn Cn Cn Cn CV) C Cj M M M Ce n C C M Cn n M V 4 t t + t. o o 00 o o.0 o o 0 o 0 o0o o o o o o0 o 0 o, X X X X X X X X X X X X X X X X X X X X X X X-X X X N N X X N t# %o w o Ln o Lon o Ln o Sn o tn o Ln O Lrn o Ln - N M 14 Ln %o w0 o, o - - - N N I' e Cl ds Co CN'0 oo -oo 0cO C _O CO _ _ _ _ r N _4 o CV) l0 o s > e s O Ooo 00 rfU N cV O -'I0 C.' N - N 4O a'0-Nt e- nL r- C" 00 CY (ON r -NC 0% -'0, -4 4- M ON 00 N ON Lf CT M %o N f ) oC o 3 ca o P-4 -( - 4 I 0- 0 0- N N 0 C0 0 d0 Lf) L), O > oo00 00 00 00 000oo0o0 _ -N N LN -0 N N N 0 N 0c N N 0 N IO N C 0 0 O 4 4 4 d 4 X X 4 v 4 P-4 X 4 P4 4 P4 P4 + 4 x P-i P-4 fr- fri dr CI D X X X X X X X M X. X X X X X X X X X X X X X X a: ~tc in \ n 00 0C 0 N 4 0O 00 0 0 Lrn O n O Lfn O u- O O _ _ _ _ N N N N 4C 4I e LO L; i S O'. r 00 COO

-285-' J v 4 v v v v Ln Ln in in in in In n in in in m in -. in in in in in in _N LF co 00N N. O. o 0 N CN N - 0 0 N C N - CA DO 0 N o 000 co a ca' a%' o, ~... ~ CI,.... U,"l e~ u'"l c"1 ~,,,0 " 0"', U'" O ~1 "~ N0 N N m"l ~ N U'"'l c, el,,. ~, d,,,,,,. U' N 0 0 O v v vt LAn LLn Lm L A Ln uLAn Ln Lm Lmn.Ln o o0 o o s o s0 -so o',O o o o o r rN r r r000000000000000000000000000000000000 If P.-4,,-. - -.-,4, —4 v.4,...-4,P-4,.-,.. —.,-,~,-4,-.4,-,-.,.,., —e - - P- ----- P- - - - P GCD XCX X XXXN X X X XXXXXXXXXXX X X X X X XXX H4...XN Ln O L N 0 0 oo o n o Ln Nr 00 Lo o o mo n in m, In.Ln i Ln i m I 000000000 XXXXX34XXXXX in N,o o N r- N m oo N.O 00 oo en of un,, LAN 0AN n e,, N N N N "a LmO e o...,...... 4 n o N0 0' " o 4o _" _ _ _ _ _ __ Ln,,.,O r.- 00 00 o',,-., —,.,O o Ln t n Ln u n n in.o mo no %o o o \, No so %o so 0 00000000000000000 Ln A. " " -, P.I " - -I,-4 4 -,,. M4.,,"-4 -4 "4 4 11 CD XXXXXXX XXXXXXXXX f OL. O O O O O N O 0 0 m O O O O t + in <o i 00 o\ _ _ _ _< N oo ^ in o V t 00 DO P-. N C- N N0 N C - a' "-00 CA N N N N oo c 0 soo 0S c 0 0 c4 o 0 o 0o I,: o o oo o cn On owo O so co o n v _4 o o -r O aa' 00 in o L A 4_ o oo 00 -L t- " C % r C t a' in Ln0oo 0 O O co00 -00 o P-4O 0o o cn _ r- % _4 A Ln r- cy o o co - Ln oo _,o o: A -4 O 0o %o v 3 0v L Lrn v cn Ln %O cn a NN N N n C c c c e Ln Ln Ln 00 N Noo on o ON o N 00oo o0 o _ _ _ _ _ _ I N Ni N N Cn C n in Ln 000000000000000000000000000000000000 O, ---- -- - v v i v v i oq d v - t - - - -- v v v - v- - Ln -- Ln n "n Un U" 4 Ln PO XXXXXXXXX4XX4XXXX4XXXXXXXXX XXXXX4XX r.- o o,, o to oo o,o c,,o o L. o", o o o o o" o', o',, o " o -,, co, o o o o 00000nLA 000a'0N~0000LA0LA0 LA0LA0Looo00 Nco00Ao0 00 L a' N N o00 00 "co C' -A No N NC N 00 N N N N N N CN 00 LA -0 c0 00 ba 0 Ia LA< 0 O AOO NO a N 000L0 Ln 0 L0 0 0X 00 O L; O O I 0L A L O 0 O 4 N W C L A0 N 0 0 a' c 0O 11 u O i O n oi n oi n O o rsa ~ oo oo Ln o Loo Ln O O O O O 02 <) tt, n co - inms;so r ~:r~oo o:; -(;; N'r: r^^CoCn'^ ^m n~ ~r'OoCT^

APPENDIX J DIMENSIONLESS PRESSURE DISTRIBUTION, P(xygQ), FOR TWO-LAYER LINEAR FLOW, CONSTANT TERMINAL PRESSURE Calculated from. -a~ 2 P(xy,) = Z [l Cmne Ym] (b)sin bx (V-86) m=l n=l mn Eigenvalues b from: b = (-2- ) m = 1,2,5 oo (V-43) Eigenvalues a from: Appendix D Truncation Test (Both Series): ITerm < 1|1 Estimated Accuracy: + 0,001 for all values not in parentheses Definition of Dimensionless Variables: x -= (V-3) H y = a (V-4) La L a (v-5) H G = klta (for liquid flow) 1lcH2 (V-15) k= lta — (for heat conduction) PlCplH2 P1 pl. _ K2 k2 All values presented are for K2 K2 k2 -286

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oo "' oJ -8 o 0 oo oN 00 NoC ~~0~~ ~-04* 0 Lr\O t Lr\ ~00_ r- r-l0 U \ QJ \D K\ D 0 ND o O — IU O\,-I _O ~II O ~0 0\ _ \ CC _s r i 000 \ OKNCO \-:- CL. 0 00U 0\ t0\ t- Lr\ Ck 0\ O L\ — \O N\O 0 iOl OlO\ LNo'O LrC MO M 0 0L ONOE- O\ J x x ^i..................... 0 O l — t- O \o _( H iUN 0 Co 1 ho P C — O Co —0 \COt- -r rHlCrl\o CO II 0 - \ 0\, 1^ d r<"\c 0N 0 0 0 (\-oLP o< or0 OON ONC-o \ CM-\1 C \. -— CMOONH~s 0 OO C,\ -ONO cCO OO\n CM a\ C-tpO Nt\ 0- pO C 0 * I\ O\C t — COMC \ C- O O r OO- 00 0 K0 c 0 0 0 QJ0 0\00 Lr\ \M -o a\O Co II ON 0\ HO\ \\ C- J 0 0 0\ 0C 0O:\ L LCNo O 0 0 0N 0\ OD tC- L-\ u Oj 00...................... 0 0\ NOt-ON C4ON- 0 CMOJ o H Ooc0-O l \rNOr<OOOOONJo N 0o OCno - aa\- ON -OCo II O Co \O \HO L\N' \OiO O000 r 0 LNoCO ONC OJ 0 0 0 sON_-t = O OC\ -oo 0 ON II QONONONO\\ N\O - OJ00, 0t 0\ 00\:\CO O \ _O rN < H 00 N O O UCOC-L\N 0 0 O ~cO \O \O - \O Ln K\O 000 0\ C- -4- H, 0~ cO \0 QJ 00O 0 0\ O O- L 0-t0 O O.......,... N ~....... ~.~~. G~OO... 000 O HO o HoH 0I ON ONH tr4ON\ OCX)t0^0 rl44O OO C L ON O' OO — d-0 - O C(N O CM II. ONqp QJCM5 O4CM O C K^ ^-4^( 00 O C 0 0\ CO Ho \CNO\ NONCOHM r-C ON ON 0\CO CMC 4 _-iC \NCO\OOO ~ 0,\ -00t OJKt~, —Ot0000\0 r-olor\ 0 OOOO O - \j 01\ 0D 0 OI ON O OC- Lr OOOO O H1' 0 0 0\0 \ - h r- O O 0\ O' ONONON Cb- L CU\OJ JOO o- oNCM L\ o r\-:-O0ONH 0\Qoc o HO H C-4ooHC ONO \ON \-o4O ONCO\ \N 0 tCMC C — _ o o\00 N HLc O\O\ H o N o CM Cl Mo 4- CM ON o o\ II 0I - Ln\_= OOr-IOJ r —O 0\ O\ GO C o\_O\ Cm r- 0000 7\ 0\ o\.tr- L0 ro-It r -O0 ONCMIO rnON4 0 O NH4O H 0 C-\ \ C-LCmH OON rHON H CCO ONONO O\ C-:OJ- N\ KHONHI0ON -II O,, IHONONONON- fO rlOO OOO c ONO Cr MHO O O\\ONONC- LrCN CM OO X ^ ^..........^^............ CH rI _ N cO-_ 00\o, iH-IODO N cOc L\ I\ — o \ O C r-0 N CoMo C o0 M O r-I - HOJ CO -OCC _ONON 0 - \ KN CM M O ['- O \N O _: C o M H\ rIO r -I ON Z- ON \ \OO CoLUN L C- MH 0 OII 0CMHHOO N\O Hc CUN\JO 000\ (7C.-\[ L\OC — OiONOCMH0 00 C\ H HO Lr\ -roj C rr-C O00 IC 0 - l CM ON LrK4 00CM1 0 C- _O J K4\ O 0 ON- HN -O1- t- 0 ONC rO Co o \ \f -J c\ O -J 00 -J-C O oUC 0 LcKN, - 0o0 o 0 H- ro O NHC rq O C HO ON ICO \ON LrLCN\C — CM KNO CC ONCO ~fO~ON-CNMCMH ONONC- f~~0\ LC\-CMCMHOOO NCo Lr\K\Cm H 0 0 L\ O — HCM O HC M C —' OH C'-M L CO LC rN OO OO O0 -o,O -- Lf, -t Cy D 0 O N 0 VN KM\ Lr\ 0 CH- \ 0 r O O O\D 0 Co \ N r-IOJ-iO-l' -Or-I~f~~ON ~ ~ M~OJr-tOOJ

-505H- 0O\C -OMON6r- K\00\OOKN OrO0 NON \ 0C CcO cMoO-l-t h-\ o H H II0 N 0\ ON O \N 00 0\ a\ tc\\ CO 0 M O- O \ H 0 0(\ ON O - \ 0 0 0 Cl C m -\CCU \D rl-oK 8 -Z o 1o\ C=o N *4 \- 0 C\otCMrJ.UNuN 0 0\ CO \O r 0 a0Q 0Q HNo K\e 0 -t- c 0 II 0 ON <\O N\ C\J 000 CU\N o OC\Or-N 0N00C% ON c \ K\ O N0 o 110R 0\00\O r< CM 00 c 0\ r <y\^0 FO^ OO: s Cs t\O KO 0 00 o O\ N QJ a, \ 0 1- 0 r ON 1:0 0\ \0 h-,\ C> 0 i,^........... 0 o *0 0\ ONa- 0QCONOO C-H AUNH C4^ C-CO UN\UON4 0 Hl H lr NO r-\C8 o aONH ii ONONON ONONON -U\ \O a\ K L- \ 0 0 El- H UL \ QO? 0 \ LOJI\ _ -l O ^-\ J\(7 \ r -1\OlA K ^ 00 I0ONU a C U 0 0N H 8 0 00 0 0 C^ 1 C 0 0 0 H C- C..'4 *. 00 U..... UN ON UNUN'0' IT~i-l K^^CVJCHiUILNHi-l'^^rcO O 4^-OJOOOJOJ-4-r-lON Oi V00^flN0 0C80 ~ r9 9?. — I 11 O SCrO CUH 00... 0 0-U CU.. \................... b...._... m U S 1-* r.C Orl C r \COC\rlP O C p 0 0 * co U i\ m cOH U- N HaH 0 UNO Hr\ 0 -ON co C C).-Q LA coOU\ n W W HON.M',, ~, C -, O, r,,, ~ ~ ~ ~ ~ ".. II ON u\ONCc vO64UcNHHO S0 0 O C00 u\ -0 H \ CO cFU-\ "CO-H000 ~II jON -O4H-HOO0 C\- r H COO\ ON' NO C-U N.-N h 0O cOO 40 0 O H CO 0 a\ \ H - i4l rL 0 CUCUCOC-L t i H C — O HO C0C-CO NH H O 0N 0U H — DOn- O H r- H \ ONCa C -- Or- t 0 c O CU \ b-L.. CcL \ II co CHO l Hr\ 00 CINU \ El- H, CC-Lo-t No CY\000NO CU \ Q O CUH\ OOO ~,.......... 0 KINOKr _: 0 H tU\N\lr-lHOLCNC-OH-HC Or\ r-lQC\tCO OO H\ H UN\Cc) -:t r-H 40H - 0 H 0 0o8 0 ON MII5 \D _t O M H rO - 0 0 CO -~ — \"- -: CON-l 0 0 0N r-\O CO " CU m r- N 0 0,_II -U N CH AH 0 Hr 0 00 \C UC..-. H H 0 cO 0 t — 0.0 \\ HO H ~ ~-O OS ~ \ C ~N HH ~ ~ ~ 0 ~H 6O 0 II0 ON-\O ~CMHHOO C-O- ^CUHOOOf O4UNCUCUHOO o -- \IO _0 t _:t C\I r~ 14 P 0 co \,O \o L \ (m ~lr 8 o -0 O \r-l - 0 lr M.I - 0. — EH cU~~~-.o o..o.C.. O.-.oo H.... - 0-000 H0 OJ.-4 \,-{ H

APPENDIX K DIMENSIONLESS PRESSURE DISTRIBUTION, P(ry,~), FOR TWO-LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE Tables valid for all rb/H > 10o0, for all values of y at any r, Q Calculated from: P(r,y,Q) = [ l Cne Ymnm F(b)U(br) (V-188) m=l n=l Eigenvalues b from: Appendix B Eigenvalues a from: Appendix E Truncation Test: ITerml < I|0l for m series Only one term used in n series Estimated Accuracy:: 0.001 Definition of Dimensionless Variables r - a (V-135) rb R re (V-156) rb Y (v-137) klta lta (for liquid flow ) 01crb - ~kl ~~~~~ta (v-14o) = klta ( for heat conduction ) PlCplrb2 * k2 All values values presented are for K2 = K2 = -306

C) K INOJO-C - O-n O0\ C Lfd KN CM -I\IC% H0 Lu\Ii C u CU 1 u OJ 0 O i [ iO CO uLn 0 0 C -cOO -o --- iu Cl N l — CH HI 0 0 0 0 0 O'\ OC\ cO - -— N CM rH 0 00 k ^''' o o g ^S^ P'o oo c o' oo'. ^................................. II u 0 0 \' 01' Ll CO C)\0 OI\ OL\ \J O t- N' QJ0 CM1 0t OKr tcK' o HHOHCOC- L \K-Lr\C-LN - 0.......0 0.......................0 OJ II SOM O I7 u~ (00 O u~ M1 O u CO L 0 t!C a- 00 OJ 1 1 0OJ I O 0Q KO -O l —OOOJOOr\OJH uL - ONH t — H\ 0C\ U' 1 I0 t-) K cu I' C- 0 p tf- \O Lr\^^^ U-\:I. _:t cr O q O O O O O P; k t- O Lnr\ LM -1- 00 0 H ~C 0i 0 0 0 0 0 ~~~~~H CMOJ~~~ tCK'\^~~ L( C->-CO HOOOI Ki l< I\\Ot-c.....CD.............0. 00000000 C H; CMq N-:4- u\ C -Oo H CM JH 1O o 0 N 0 ~ 00 ~ C 0 ~ ~ H H ~ ~ ~ O i 00 ~ ~ C \ \ -\.L K COj OJ - 00 Oj N — C\ C —L LcocO C-r-4 NL' 0 ~-On LL- 00 II CM -LC"CO O)\ C1O\ oCO- H 000 -0 00 0 HO II KUNO HU 0 OC —CT\K'OO H0\L\000II 0OO MO MOO OOHOO CM K'P - Lro tCc-00 00 000 0 0 C H O K' K\t L\D N C-CO....... 0 0 00.....C......:........ LC O\C-\-O 0 C t0- Lro\ CP\ MOJ H0 H0 O' L —OIO b\i-1CO ON co \ l-q 00 0C0\CO HCOHCMCMCM MK' -C —K H0 -0 0\ \\0 —\-K H00000C 00O [- O OL 0-O O \000O II K'\ C- \ t-C?\^ I 0 -CO Hr\ H O_ 0 00 0 0 00 0 4 CC)ID 00LC'LC co'0CO IC'_: C-C —C0 0 t\)COl NT r- C HC Mr\00K'- ICC O r-COOH 0 \ O H 0 co \ 0Lr\ c 0 UN - c- o-o Kc H o o 0 C m- 0K\ 0'\HO 0\ODL \ Ul\V iu -=I- oj 00000 p 0,C\0\(7\OIN0\ u\t- < C\c\OLC' I-\ C0t HO000000 CM II C —LO\ HUDHC-0C-PC- K-Hooo'i DC\HO n 0- — J- I NC\ 0 0 00 t-O'\ r-I O C, OH r- O,\ co \ O' -OC 4-' — \ co U o II C H I NO\CMCCOH C\NOCH C\O M- 0 tC tO\ O0 CM HO co H 01\ CC) L\ L"- L " - L'- t.. VI\'~ C r l r-'l O O O O L "-'kD U'x..~-' h~ C- Jl r-I O1 O O,1 -.". C-C O,\o - OD r-'l Co O'O D t- t O O O C- Cr LO-D O C-0-C > 0 d 0 0 d H 00 oj 0K t-L \LC'\C -oCO OS CS o * r" io tCol o CD 0000HCmK'CHlrkcM~\0 K D C H;,. c - o u~~x~O b-CO 0 0 0 0,-I (~l I~- t~~~~~~~~~~~~rxk. O b-OD ~ ~ ~ r

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-316o t- 2SN~ ~~~~~~~~~~~~~~~~K ti —!;'^^t-^0!^0^0? 0'-00 g lN O UN \o or-f mo —ioo.o a)o o r - -i C-^ ~ ~ ~ ~ 0-~ P1 c O\CL O, ^ - HO if-O 7 I, IN'Nc CO nCm 888 00............................................C- L H1 UHO CsO UcN H cI C-O C( \C-KHC L M \ C ON 0 i oo o C-\C 000 I S I \ 0 RI g 0oO r -t O-O^ CMH 3 o o II I I II C II II- Ou t 0- 0C 00L\ H - C MC C 80 I l 00 0 0 II II <7-C\OCM\O \ H -j O\ 0 0 000 0 0 0 - \o \- H H HH 0 0-C OH C\.. H. 0 H. 0000. o o I o o C H o.o o.0.... II I II ICC M C M C M 00 00 11 IIOfOVc- MHC0 Mr-t-d-h-r KN H 0 0 0 0 0 0' IO \O \0 1L^0\ AC M CR - -t ho C- c\ H 0 0 0 0 HOJ 0r\-:t HlHH\. LI-00 O O O Q O OQ OO O i HHHt l^^O t 0 0 98 (9 000 0 0 0 O 00 ol\- 0, o -o O, o -H i,0 _ ~ Co o8 000000000r-4 oj o, 0 C- 000(\HCJ-CMO- I rIt MCMHC\OC OOOO\. —0HCO 0 C>- 0 H 0 cMo K_40 Vo0 UL\O OOHO M M-IC - HO rH 0 O CO * OO (O\ M,-O LC-MO H rC 4CO\0CM 0OCM S OO 8 HR COOO CMc0OH C-CM-\t- 0 0 0 11 II O0 0\ \ t 00000~ CO 0 rC-C-MO-C 1M HD tO 000 t II II 0000h10CCO MCO OOCr (T Or l r O OOO M. ~.o... I. I \,O L. C........ j...... 0 4U\ 0 gCMM-OJOH\Ot rcOcH0OCMN-OJ OC CMC CHOO u 0 t- I I \\ oCMC \MOHO C -HO O HOO CZ- IC t C HOo H MOIO MOM O tCM OMCMQ 00 0 II II -CMIOC MHHOJOOOOO ^ 1 11 oI 00 o\ on\ co oo b- - ^ —0 o LI Lr\ K^ \ 1- o o o o 8C)1o oCC o t <y o o- \o ot o o o o ot o o 0 *^ ~tL\O CrY\ p \O Ot'O i -i rc\ _ I -ON t- -4- NQN r-* " UN 0 t -1 -\ — C\N C~- UN0 o "lo-:d. o S ON \0, II 00 C0 M -CO L- - CO CM O M000 0 H\ CMl (C 0O r-i 0 0 0 0 0 CD HCM CDlk C or-l0 LtCM 0-Co0 0 00 0- o0 C-ClM H -C-M to0 l r O C- 0 I CO OM l\- 00 M L C1o -MO Lf\CO C\0 0M 0 0 H 0 CM. l00 -0r0l-00Oir\ rl CMH 0M H U HMO irMO Oc oH oi CMO I 0(| CO cCo frO 0\CM- UO\rl-tOO C H\ H 0800 II II 0 CM0C MC MO 4C0C C\M lHHn _t-_i- r\OMUr-OO C-OOO II II \OO L \ 0 C-MOCM-CM MCM -l'O(M l - co00 0 HH ~ ~ ~ ~ ITO OH ro 0J C-d- C C \H 0 C -MCO 0 0 00000M000 0- C- C -- M CM 0CHO 0 t- LCH\ CJ O H C- 0 CO C\cO tC- C - OJ rH COJ -:d UN 0 t. 0 C-H\ H\ O UJ t-'O U CMN-\ 0 L I " rI 0 OJ o0 M^ CO \0 K<' rH) 0M\0 HH<O \Lr\ \O0 0iLr\ r-q^00OJ \l \ I 0 M O M OOOi 00 t- i0Q Uo 0 M\ M M\ Mo\oo _o l 0 o t- M C0 o\ OC c\ a\ -\ o- O\Coo tCHo 0n\ E rnCM\ 0' C o 00 00 C- M0CM 400- ( 0 c o t O\_O C-UCM 0 1000 10 C"\0 t1N- O LO0CMOK-\O - \ HO CM 0 C 0C HOr-IO C- *- Or-tHOCOMOO0 0 0 -t- - -r-4 O Lr\Or O K U _o rN lr\ 0 Cm Ln cM —C\j OH 0 c0 o II Mc - 00 t — \ MC j r- rH 0 0 0 II C \ c \ cC-O- o0 C 0 rtC\ 0 0 0 0 0 CM I >.........CM......OJ t C-..................... II II II C o- \ CM0 0O-t HO co -1-L 0\H 0 oIC-Mc OCM H0 0Kr\ H \ CMM- CM OC -Lr _t H' 0Mt _t -\ N0C O H cM K% MO 0\_t 0 CQ H 0 OII II 4r- \I 0 1 CMHHHco00-t00\00-4 \ 0 0, c II II 00 t 0M aO t- 1 0_ - \ MCMCMOOO LC O\ O Ot CO H r 0 0 O 0 00 CO t- O-0 0 0 - J r- r-0 0 0 0000 CH 11 oc 0 C 00 9 0 0 C.M CM OO - 0CM 0 0 00 _-. or\ r- 0 H\ CM M4 c MO C-CoM 0\ U 0 H CM CM O _I- 1 "K nOO c. -......................................

-317~~~~~~o\~~~~~~ ~ I.o\ o l 0-4 C\oOA-\ A-\ 00\0 0A o- 0-C Ir\ 4 K r N C\O. H - H ICo\H c L o OCrA100H.................................... 0., C0 * 0 -st- ^D ^0 0 t\0 t- 000\ (^ QJ ^0 M-4 0 8-0 co 0 * i U`\ 8^ L- 0 Ca r-l 0 II II OS OSCO CO C O-CO fr —0 t IO NN% CMHH II IC 00IC-A- N0 co E- N co P K CH A Cm H-\\ 0 0 0 0 00 AC............... o.. ~........... C... A CAI 911,,~ _ C o o 40 C H C.,-' 0... II II CO tt OR LnCUO r, rk l 0 0 II t - ISLI\- C b 0 \O N U.\ c\ C r-1 0 6 0 0\ 0\ 0. M -0 t- \0 0N HH -0 00 t- 0 QONA-CAC-Q O N ON 0-4 HCA\0-C- -r\ Q H Ct 0 r-1 OJ9 0 cO ClA0 SOO \ 0 E - NO Qt Q lllltl l j t — -\ 0 -0O L CN 0 00 ONJ H 0 H..J. It 0 OiOv lON 0-0 vCO ON0r\ AC N; 0 —OrN 0cy OO CON -A clr f A t 11-.t ~-t No o b f o88880 o o\ ~ ~oo ( o o O o88...................................... Cd II 1 ON R ON 0A.*Q A-0. 0. -. O C00\ 0 C...CO H....HA CO.00 A O N.. 0. f 0 ll t ONLAr\A 0cA r\ 0 U fr-L0ICY\0 111 - N1 4t L -ON \O0CND 0-U 000 UN (PUN - Hr- QQO? t- CM UN -1O`~~~~~~~~~~~~~~~~~LN A-CO ONr ONO Ol 0 OOO \00 A H - C 0 A 0 1 1000 Q O < H> HCOH C -HOO o0- 0 H r CA 0 II ^- O N r1 ON ON-0 00 t- lC HO QQ OJ 0 1................... c......h.................. Ln0 R., r 4 K^ QJ O n IA r4 OJ O -- r 0 0 n I~ Q\ 00 \ KC 0 U` -- I\ CO O 0 J0 H t ON -- ON0S8 8 00 0 rd CNLr\D t4000 og0 0 go 8 8 8 8 8 8 0I r4 0tOH 00CA0800- CA HOOOuN8rr-O 0 0 0. 00H OC 0 nOV OVOHC 0 KCO C CO -A t-OO$OLNAO HOOOOO CA II II C0\ 0a rl ( +COC L\-\O AC A -....... II I 0O I 00 -CAO ON A-N MTN HO 0 H rA- 0 0 0 0 0 OJ II 100 0r4CA 0CA0L irOALAN1 000 0000 II \O cO u\ Mf m 0UC~fr100M~nu\C MO L. ON Ur OX. ~ -o0 o0 000 \ CM 0 r- K^ = gH " C\CO 00r 00 0 0 0 00 t< O 0 0\- il O 1 CC)OM \- CmO 0 0 - 0 It 7 ON WN8o8 Cr-IN CP P t"-,o \0'IO U l\'LIN ~Of; 0 0 aNH\0Cvl\ H CA' 0-40 A-C OS a, U`\ 00 ~ 8 ~ | 8~\ ~ 8< CC a O - c\ N Le ~ ~ ~ 8 1 r-HCAI " H \9C t c 7 L \ ON 0 N-I I C U7 M0" Nt- U\"I o,,, C- %O,DCoM O - O \ 0 0 O\9 _O 0 O -0 o ~8 rll 0v CM o U. C oNo~,~o U \ 0 o o, o Ho0 0 0 14 ~ ~~~~~~ ~ ~ ~ o............

APPENDIX L INDIVIDUAL INSTANTANEOUS DIMENSIONLESS FLUXES, ql AND q2 FOR TWO-LAYER LINEAR FLOW, CONSTANT TERMINAL PRESSURE Calculated from: 200 -a. ___ (Va2Q l = m n Cmne [sin 7] (V-114) _ a 2 ne -ae [cos yc sin (1-c) V-115) L m=l n=l mne L cos P(l-c) Eigenvalues b from: b = ( ) - m = 1,2,5, (v-45) 2 L Eigenvalues a from: Appendix D Truncation Test (Both Series): |Terml < ISum Estimated Accuracy for ql and q2 + 1 part in 104 Definition of Dimensionless Variables: x -a (v-3) H y Ya (v-4) H.La L = - (V-5) H Q= - — a (for liquid flow) ktlCH2 klta (v-15) -= PC p~- (for heat conduction) * k2 All values presented are for K2 = K2 = k -318

-319L 5; c = 0.5 K2 = 2.0 K2 = 5.0 K2 10.0 CL1 _ 1q _ 0 Q q2 q, l2 9lkl ql q2 4l.2 1.1021.9955.52543.5423.4421.55086.3910.2951.56989.3.8832.8223.51786.4202.3665.53411.2927.2441.54520.4.7564.7166 51352.3530.3195.52497.2412.2122.53203.5.6717.6433.51080.3098.2866.51946.2093.1898.52443.6.6102.5887.50897.2792.2620.51588.1871.1729.51974.8.5251.5112.50669.2380.2272.51156.1573.1483.51470 1.0.4679.4580.50533.2107.2032.50914.1370.1303.51239 1.5.3800.3747.50353.1687.1645.50635.1023.0980.51059 2.3281.3246.50266.1420.1389.50542.0780.0749.51030 3.2653.2633.50190.1046.1025.50500.0457.0439.51024 4.2242.2227.50166.0779.0763.50495.0268.0257.51024 5.1921.1909.50158.0580.0569.50495.0157.0151.51024 6.1654.1643.50155.0432.0424.50495.0092.0088.51024 8.1229.1222.50154.0240.0235.50495.0032.0030.51024 10.0914.0909.50154.0133.0131.50495.0011.0010.51024 15.0436.0434.50154.0031.0030.50495.0001.001.51024 20.0208.0207.50154.0007.0007.50495.51024 30.0047.0047.50154 40.0011.0011.50154 50.0002.0002.50154 60.ooo001.001.50154 K2= 20 K2 =50 K2 = 100 q1 ql Ql 0 ql1 92 q! 1 q q1+sq q1 ql+q2.2.2975.2057.59118.2144.1251.63146.1640;0728 bo9269.3.2129.1690.55746.1372.0953.59020.0884;o473:6513k.4.1706.1451.54039.0987.0743.57054.0532,0314.62914.5.1445.1274.53153.0750.0587.56129.0338.0209.61763.6.1262.1133.52679.0585.0465.55690.0221.0140.61169.8.1001.0914.52273.0366.0295.55375.0098.0063.60700 1.0.0811.0745.52143.0232.0187.55297.0044.0029.60569 1.5 9b4go.0451.52082.0075 o0060.55272.0007.0004.60517 2.0298.0274.52078.0024.0020.55271.0001.0001.60514 3.0110.0101.52077.0003.0002.55271 4.0040.0037.52077 5.0015.0014.52077 6.0006.0005.52077 8.0001.0001.52077 10.52077 15 20 30 40 5o0 60

-320L = 10.0; c = 0.5 K2 = 2.0 K2 = 5.0 K2 = 10 q ql ql 2 ql 0 ~l 2 ql+~ q1 q2 q+q zl q2 qz+a.3.2925.2441.54510.4.3530.3195.52496.2413.2123.53198.5.6717.6433.51080.3098.2866.51946.2094.1900.52428.6.6102.5887.50857.2792.2620.51587.1875.1735.51941.8.5251.5112.50669.2380.2273.51154.1586.1501.51375 1.0.4679.4580.50533.2109.2034.50905.1400.1342.51062 1.5.3801.3747.50353.1701.1661.50587.1124.1094.50678 2.3283.3248.50264.1464.1439.50434.0966.0947.50500 3.2673.2655.50175.1189.1175.50286.0779.0768.50343 4.2312.2300.50131.1026.1017.50214.o661.0653.50287 5.2067.2058.50105.0914.0908.50175.0571.0565.50266 6.1885.1879.50087.0829.0824.50153.0497.0492.50259 8.1631.1627.00o66.0700.0697.50133.0379.0375.50255 10.1455.1452.50054.0600.0597.50126.0289.0286.50255 15.1163.1161.50042.0414.0412.50124.0147.0146.50255 20.0958.0956.50039.0286.0284.50123.0075.0074.50255 30.0660.0659.50039.0137.0136.50123.0019.0019.50255 40.0456.0455.50039.0065.0065.50123.0005.0005.50255 50.0315 0314.50039.0031.0031.50123.0001.0001.50255 60.0217.0217.50039.OC 5.0015.50123 80.0104.0104.50039.0003.0003.50123 100.004.0049.50039.0001.0001.50123 150.0008.0008.50039 200.0001.0001.50039 K2 20 K2 = 50 K2 =100 e 21 q2 ql+2 q2 qlq2 q q2 ql+q2.2.62300.2156.1305.65317.1744.0926.65317.3.2131.1697.55670.1445.1071.57440.1091.0751.59241.4.1717.1471.53862.1l19.0923.54797.0806.0634.55965.5.1472.1314.52837.0939.0821.53373.0652.0548.54321.6.1308.1197.52212.0824.0743.52578.0551.0479.53502.8.1099.1034.51523.0678.0630.51828.0418.0373.52871 1.0967.0923.51163.0581.0547.51532.0326.0293.52693 1.5.0771.0748.50759.0418.0397.51331.0180.0162.52618 2.0652.0637.50611.0307.0291.51301.0100.0090.52614 3.0494.0484.50530.0166.0158.51296.0031.0027.52613 4.0381.0373.50517.0090.0085.51296.0009.0008.52613 5.0295.0289.50515.0049.0046.51296.0003.0003.52613 6.0228.0223.50515.0026.0025.51296 7.0008.0007.51296 8.0136.0134.50515.0002.0002.51296 10.0082.0080.50515 15.0023.0022.50515 20.0006.ooo6.50515 30 4o 50

-321ICU r\ Cu - -INH r CVM ~\ O\ f' H \D ~ f I Ls | OJ Ks-ri < \ 0 K\D KM - O 0 s\O \D \D\O \ iH HCu C — oCNo\CO^ o o o \-o NVo 0to oD o o H o o o' Mo o- rN E-~-Co\0 HO Lf- o O D o- o\ O o \o 04 0Uo-noi\^NU-ncucUH-iHO0OOc >0'- LNU'K -CuHHrHo 0ooo,O,0 C — C o -- O H _-O cO O- O t 0 - Cu 0 000 0 rHHr-OOO 00000 0000000 11 0000000000000 g.................................... o \D 0 _t \ r ou> i H Kcu\ KMn <^K ^'"lq" 00 n CO t\ o \D o- (7\ o r-i-=t- LI \ i Cu -) O Cxl Y MX > O.cuNX()O OCu H H H O H H H HC cO cO H 0o C c -1 cO\ CuC O O dH fOOJ 0 0001 0>0000C CJ- 0 r 0 H0 0 0 c\ Ui -\O CU \J r OOO~ rH 0 0 0 000 0 00 00000000 0 O ooO.......................... d MCO \D X4 O a X Fn M M n 1 M M M d c 0 w N \) \0 Oa\ CUJCM C CUJ N CM r + Ckl Cl r-I H HO O O OOOOOOOOO+ C< X= \0 OJ tt 1N n 1... KN. n 00000000000000000 CU OOOOOOO 0000000 0o 0 n n ON L\O. H. n c o NH00 L Cj co o\ o cu n N \o Z O \ o oooo cu oi OJ K ^ -r-lr-IK^(M UAQ(KIO 0\q H " OH O CM 1l\ _ 0 _ \- 0 CM t — r" 0\ -t 0 -o CM 0 0 II 0 - r1 0 N j i f O\ C \ CM O -0-4 1 - ONN 0C C 1 o 0 — \o0 r u f — N O.\ 0- O.4 Cu 0 0000000000000000 00000000000000................................ Cu c_ L \ H n o - Ll \ C n tl- n 0 a H 0 0 0 0 0 0 0 o 0 c - H \, \ \ \ n C\ o o o o CUf ^ - i K ^ L ro o~o o o o o or o. o\ ooooooo- o H C o; o o o-O oN o HCO Cu H Crl CuM NCNA O\ O0. O IH C 0 t. COO- H O CMU0 CMC CQ O 04 C ~ r 0 0 WH0 C 0)U 0I C) tt 0 N 0 O O OuCu OH 0O 4 tO ) C CU 1NHH000 o000000000oo000000000000 HHOOOOOOOOOOOOO < i\ ip~ IS ipi ir n in r^ 10 ir LP mn irc Lr \ i\ if i\ ir in n no n n n IP n m ir in in n n in nr in i mn i CM O\n 0 COO \N0O m 0HH \O r O - lCO 0 -Ct 0 C o - CMu CUM O0 0O\0co t — C \0 r-O HO HM CO C, —r HOO MO\ nU \o 0 -OCr)i O OC HOM HD H _I - CM Hn Cn -C t- C\-t c C- uM 1-N 0o 04 OCM\\ OCOO-c H0% -- t K, \C.MH000 I 0 40 0u \0 u _t "CM r-IH00000 KCCCHHHHHOOOOOOOOO000O HOOOOOOOOOO00000................................... K^ KN CM t —\D CMC Hi-m ri H - li\ r-IOC CO = 0 0 t- CM n 0 _ Om C —:- _ CC)- M" D 00 0 m \0 0 r-1 H " r- 00 -- H \Q CO KM\n S M CM d- co t- O\ in n \0 CM ri- 0HU\ C\0 t.- \> Kl \0 r-q t-CM tO- C: 4- _CM rN- 0 CO CM\O \ OCO \D _=t r-q I t- \DlO \ - Cr-I 04 0 ( CO \ t-W \D I t 1 — 1 " CMM OOO 0 0 HKOCMCM0 0Ul00000r 0000HfO000000000000 000000 Co ir\n- i-\oo co io r CM ooo ooooo ooooooooW r0 00 0 C ooonoooooO o....... C. K......r\...Oooiroo0ooo r....H H Cu......O 1- 1 1 CM~ H D- 0

-322L =50; c = 0.5 K2 = 2 K2 = 5 K2 = 10 9 Cql ql qlQ2 3 ^l q2 ql2 ql q2 ql ll q2 q 4.06754.06690.50241 5.06028.05982.50192 6.08358.08311.50141.05495.05460.50159 8.07228.07198.50105.04750.04728.50119 10.6460.06438.50Q84.04244.04228.50095 15.05268.05257.50056.03461.03452.50063 20.1031.1030.50026.o4560.04552.50042.02995.02989.50047 30.0841.0841.50017.03721.03717.50028.02444.02440.50031 40.0729.0728.50013.03222 03219.50021.02115 02113.50024 50.0652.0651.50010.02881.02879.50017.01891 01890.50019 60.0595.0595.50009.02630.02628.50014.01725.01724.50016 80.0515.0515.50007.02277.02276.50010.01485.01484.50013 100.0461.0461.50005.02036.02035.50008.01309.01308.50011 150.0376.0376.50004.01650.01649.50006.00986.00986.50010 200.0326.0326.50003.01395.01395.50005.00751.00751.50010 500.0264.0264.50002.01029.01029.50005.00436.00436.50010 400.0223.0223.50002.00765.00765.50005.00254.00253.50010 500.0191.0191.50002.00569.00569.50005.00147.00147.50010 600.0165.0165.50002.00423.00423.50005.00086.00086.50010 800.0122.0122.50002.00234.00234.50005.00029.00029.50010 1000.0091.0091.50002.00129.00129.50005.00010.00010.50010 1500.00029.00029.50005.00001.00001.50010 2000.0043.0043.50002.00007.00007.50005 3000.0021.0021.50005 4oo000.0005.0005.50005 5000.0001.0001.50005 K2 = 20 K2 = 50 K2 = 100 2,.,. 2.22 - - - l q2,ql+q2 q2 ql+q2 q1+ 2. 1.5.07738.07518.50724.04829.04685.50760.03400.03297.50771 2.06643.06503.50531.o414o.04050.50552.02915.02850.50560 3.05377.05303 50345.03349.03301.50358.02357.02323.50362 4.04637.04590.50256.02887.02857.50265.02031.02010.50268 5.04138.04104.50203.02576.02554.50210.01812.01796.50213 6.03771.03746.50169.02347.02331.50174.01650.01639.50178 8.03259.03243.50126.02028.02018.50130.01421.01413.50138 10.02912.02900.50100.01811.01804.50104.01256.01250.50119 15.02374.02367.50067.01472.01468.50071.00963.00959.50105 20.02054.02050.50050.01259.01256.50059.00749.00746.50103 30.01674.01672.50034.00963.00961.50053.00455.00453.50103 40.01443.01441.50026.00748.00746.50052.00277.00276.50103 50.01275.01274.50023.00581.00580.50051.00168.00168.50103 60.01139.01138.50022.00452.00451.50051.00102.00102.50103 80.00921.00920.50021.00273.00273.50051.00038.00038.50103 100.00748.00747.50021.00165.00165.50051.00014.00014.50103 150.00445 oo445.50021.00047.00047.50051 200.00265.00265.50021.00013.00013.50051.00001.00001.50103 300.00094.00094.50021.00001.00001.50051 400.00033.00033.50021 500.00012.00012.50021 600.oooo4.0oooo4.50021

-323| H HH OO O OO OO-t CO O 10 \ N N'IO KN K, K^ N^N KN KIN KD KH\ fr N K0 [ C r \ cm co 0> H 0 H - n 0 0 t lC CU C 0 CU \0 \0 Cu \u II 0' KQ a,, re 11H 00^^0o 08 8J 8 8 OJO O r-! r-Q O O O O O O OO O OO S000000000000000000 HII L \000 N00000000 00000...................... I^-d- i-o I-ir t-Qj Ircoo^ o Krc \r\ - o.d t'- f i>-ojoji)<i> r\ o\ i \ i'< o DO -=t - ct j O- ri CO U\v ~O IPD\O -o. - H o o uCrc I \ o H g -\ H\0\Koj M " coj - l o- Icoju o0o OJ K^ - t *cO N- 0 -c u O ~ * K-Chu 0000r- LH- alr-0 D H Co -- D n 0 0 0 0 0 UtCoN Cu U ~0\4 H HH H~ ~ Q 11 i1 ji> ^< st -H O<^ I 0000000 i\ C\uCU HHHHHQ O 008 0 OOooo O0000000 0 0 000000000000 000 080088888888 00 CU " oOooo888888 -1 C)H- C U 00Cu0 -C UC 0 C- H o c K CHo - n0 0. o0 -0 N 0o C Hj o. —... - o Hr-qH Hr-iiH r-i-r. rQiirACr-I H I- 0 0 Cu -00 H00 U0 0 0000000000 1 0 0 K0 K0C U -I O OJ-\ OH 0 000000000000000000 0000000000000000ooo N' C-H0 Cu0\'^30\HC."" ^ X3"" " HO H0 C " H 0~c- t- t —8o0 03 u3 ou\ u cu o o H Cuj t~ 0 -D H OCOU uH0 H\ \D n -cm rq 0 0 C- OH 0o00 O NN-HHHHOOO-OOOOOOOO -0000 0 O r- HHHHOOO OO-OO-r 000000000000000000 00000000000000000000 H 0 L — t — " 0 LC\ Co CO Lr,\ e- cO'N \D O p H W N t- \\o t — CC) g CC) Cm \oD -01 CC)t coco coU M t- Ci i NI OO- MVD -.. f- H 0 C (Ri 0\ M N\O " rq 0 cO oE — _: CQ trI E- -_ CH H 00 0 00000000000 HH- ~l c uOCO 00 00C H H0 Cu r"-I U 0 0L 00 X)4XXCSCSCSCxX XXXXXXX H LC\ Cu N- t L(\o co Cf\ Cu "-4~- LN\\O O 0 000 000000000000000000 C\u U-\""0NHMCHHH000000 0 000000000000000000000 0000000000000000000 d- o0\O -0 0 H O K\H ONo - OH 0 0 t-0 "ON \NDWN\oC\ O C- L-CO Cu Cu- O\ \o - Cu 4 C~0 \ Lr\ \'O 0 Cuj U'D n LOW\H 78 H 0 0 Cuj 0 \Dn t\ — Oco Cuj H~ 0 CC)- Cu rl-CO 0 - 00000000000000000000 o 0000000000000000000 ON H C- CmOD 0 rn 0 H CO KN H C-Or\- I M C_- ONCO M H c - 4 O H t —4\ tC-o-C C — Cu K \ m O \D LC\ L\ LC OC\I W\ O\ C O CuWNC L CnL\MiH H 0 _:I 00 H C-w S - \C)N- u Cu Ho "C~co C-cO CuO \ t- 0i H oHHH0 00 H 0)O-CuH0-COL4tXNCUCUH000 00000000 K\ 0 r 0 0 0 0 0 0 000 0 c ooq o2 oooooooo oO 9oo HOOOOOOOOOOO 00000000000000000 ~o~0~ r o kO ~ ~f LO Co~ O \CuO C 0O WN0u Ho 0 o co H Hu:~~r oo o riooooooo

-524OOOOOOOOOOOOROOOOOOO o Hf O00000000O00000000 00000000000000000000 0 \ \\-HCT C l' t- ) -- H00 0 0\C- O" c - \i lOO 0 K'rM CMH 0 0 R~ io I~ ii I o I H ii. If I t I~ I~ If)............. i........i..i....i..n h-^ ^ < \ o..... co"CU 00 H, U- 0 L0 L0 a-K o'0 - 0 o ooo o o \ -oo ooo oo oo _o:oo -:d- K\NO OOOIooo o oo HO OOOOOOOO 00000000000O H 000 OOOOOO00000000 00000000000000000000 000000000 0000000000 so\ oo 4Nm oj o, oo \ oj o o o oj r4Ns ip-fol oo if oj t -t Ir <o co H H H H H H H NH O O H - \ \ \ Jn { \D \ CM\o co 04 0000000000000000000 aj Q %040400 H- H- - H H0H ~ 008 88888 co?oo8-3 o ^ c0 0 88 iU M r<^ ~ IP cu0N - oUN irH CM KU Ni=t ir'^ oI\ o CM f n i u oo ir \ CM In io oo IM CH H r-1 H H t O 8 oo 88 ~ 88S ~ 8 88 888 g H..ooo. ooo.o...................... co N0Ia \ t- u -\ 04 -N' H co- 8" aN O N L 0 0* n-H 004 o' co \0 Kr\ 00 n r_-l MCo I 88888 8888 1ooo 0 00,Om00000000000000Ca 0 0 0ll000000cm00o 0 00t 0n N N N 4 ~ N 8 N N U\ N 0 C8NS O NO CO N 0 N N _: CO UN 0 tN u UN0 CO 0080800800 HHHHHH 00 000000000$00 000000000000 co 00n U\ Uo U NHH 0\O 0 0\' o U-\- OC 0t H 0 co n C-t\0 co + 0000000000000000000 0000000000000000000 t- \ t- o CH CC) \b - a\ nz\-s~ tl a\co H co \,O CY\ C\\8 M a'\0 \ t M\C aC-C 04 m cm' 0! \!D-1.......04H 008 CU \H HH- C0 \D H OZ)\\DO 0 0 HCU 0 8 040404 HH HH \ HOOO N - 00000 t 0 O8 ""NN000000000000 C 000000 000000000000 0000000000000000000 00 1 00000000000000000 000000000000000000 000000000000000 U\o 04\ \V00\ UD CO Un04 KN Ln I 0 CnO (N f04 CN\0 $ Co Uo00 to CC) o o o =- CO o \D tH- C\ H H c \ \\ -C IV8~88 8888800000000000 0000000000000088 00o 0000000000000000 000000000000 00000 11"1 "uu w"" o NAo. ood I ooooooooow ww

-325L = 20; c = 0.1 K2 = 2 K2 = 5.0 K2 = 10 _ _ _ _ _ _ _ q2 q_ _ _ _ _ _ _2 ql__ _ _ _ q2+ q2.6.02694.24149.10043.8.02332.20913.10032 1.03301.29595.10033.02084.18704.10026 1.5.02691.24164.10022.01700.15272.10017 2.05801.52093.10020.02329.20927.10017.01472.13225.10013 3.04733.42535.10013.01901.17086.10011.01201.10798.10008 4.04097.36837.10010.01646.14797.10008.01040.09351.10006 5.03664.32948.10008.01472.13235.10007.00930.08362.10005 6.03344.30077.10007.01343.12082.10005.oo848.07625.10004 8.02896.26048.10005.01163.10463.10004.00729.06558.10003 10.02590.23298.ooo10004.01040.09355.10003.00641.05768.10003 15.02114.19023.10003.oo844.07595.10002.00479.04314.10003 20.01831.16473.10002.00716.06446.10002.00362.03255.10003 30.01492.13427.10001.00534.04808.10002.00206.01856.10003 40.01281.11528.10001.00402.03616.10002.00118.01059.10003 50.01123.10110.10001.00303.02723.10002.00067.00604.10003 60.00994.08942.10001.00228.02050.10002.00038.00345.10003 80.00784.07052.10001.00129 01162.10002.00012.00112.10003 loo00.00620.05576.10001.00073.00659.10002.oooo4.00036.10003 150.00345.03103.10001.00018.00159.10002 200.00192.01727.10001 oooo4.00039.10002 300.00059.00535.10001 400.00018.00166.10001 500.oooo6.00051.10001 600.00002.00016.10000 K2 = 20 K2 = 50 K2 = 100 q q q q q q q q l 91 q2 91 22 1 2.08.02177.19127.10221.1.02770.24421.10186.01937.17103.10173.15.02245.19933.10121.01571.13960.10113.2.01937.17260.10090.01356.12088.10084.3.02569.22945.10070.01576.14090.10059.01103.09868.10055.4.02221.19869.10052.01362.12202.10044.00954.08545.10041.5.01984.17771.10042.01217.10913.10035.00852.07641.10033.6.01809.16222.10035.01110.09962.10029.00777.06968.10028.8.01565.14048.10026.00961.08627.10022.00668.05994.10022 1.01499.12565.10021.00859.07714.10018.00587.05276.10020 1.5.01142.10259.10014.00697.06266.10012.00440.03954.10018 2.00988.08884.10010.00592.05326.10011.00333.02992.10018 3.00806.07244.10007.00444.03991.10010.00191.01716.10018 4.00693.06232.10006.00336.03019.10010.00110.00985.10018 5.00610.05483.10005.00254.02285.10010.00063.00565.10018 6.00542.04871.10005.00192.01730.10010.00036.00324.10018 8.00431.03881.10005.00110.00992.10010.00012.00107.10018 10.00345.03102.10005.00063.00569.10010.0000oooo4.00035.10018 15.00197.01775.10005.00016.00142.10010 20.00113.01016.10005.00004.00035.10010 30.00037.00333.10005 40.00012.00109.10005 50.00004.00036.10005 60.00001.00012.10005

-326L = 20; c = 0.3 K2 = 2 K2 = 5 K2 = 10 l 92 qaql ql + 1, q 2 q1'2 1 1 q2 ql9+q__2 ql+q2 ql+q2 1.0709.1626.30354 1.5.0896.2068.30234.0575.1327.30232 2.0774.1791.30174.0497.1149.30172 3.1505.349Q.30094.0630.1462.30115.0404.0938.30114 4.1302.3028.30071.0545.1266.30086.0350.0812.30085 5.1164.2708.30057 0487.1133.30069.0312.0727.30068 6.1062.2473.30047.0444.1034.30057.0285.0663.30057 8.0919.2141.30035.0385.0895.30043.0246.0573.30043 10.0822.1915.30028.0344.0801.30034.0219.0509.30037 15.0671.1564.30019.0280.0653.30023.0171.0398.30031 20.0581.1354.30014.0240.0560.30019.0136.0316.30030 30.0474.1105.30010.0186.0435.30016.00oo86.0201.30030 40.o408.0952.30008,0147.0343.30016.0055.0128.30030 50.0361.0841.30007.0116.0271.30016.0035.0082.30030 60.0322.0751.30006.0092.0214.30016.0022.0052.30030 80.0260.0606.30006.0009.0021.30030.0009.0021.30030 100.0210.0491 30006.0036.0084.30016.ooo4.0009.30030 150.0124.0290.30006.0011.0026.30016 200.0074.0172.30006.0003.0008.30016 300.0026.oo60.30006 400.0009.0021.30006 500.ooc3.0007.30006 K2 = 20 K2. 50 K2 = 100o 1 q g2r 1 q2 ql q, q2 q1q.5.0693.1561.30740.0427.o64.30728.0300.676.30724.6.0628.1424.30602.0387.0879.30592.0272.0616.30589.8.0539.1232.30439.0333.0760.30431.0233.0532.30437 1.0480.1101.30346.0296.o680.30340.0206.0473.30361 1.5.0389.0899.30226.0240.0554.30225.0161.0371.30294 2.0336.0778.30168.0206.0477.30177.0129.0297.30281 3.0274.0635.30111.o161.0374.30147.oo84.0193.30278 4.0236.0549.30085.0129.0299.30141.00oo54.0125.30278 5.0210.488.30072.0104.0241.30140.0035.oo0081.30278 6.0189.0440.30065.0084.0194.30140.0023.0053.30278 8.0157.0365.30060.0054.0125.30140.0010.0022.30278 10.0131.0305.30058.0035.0081.30140.oo0004.0009.30278 15.oo84.0196.30058.0012.0027.30140 20.0054.0126.30058.0004.ooo9.30140 30.0022.0052.30058 40.ooo0009.0022.30058 50.0004.ooo009.30058 60.0002 oo0004.30058

-327L=20;; =0.7 K2 = 2 K2 = 5 K2 = 10 9 q, q2 qi ql q2 r ql q2, ql~z 91 2 ql42 P 92 ql+q 1.5.1953.0799.70963 2.1671.0692.70705 3.1949.0822.70347.1348.0565.70458 4.3474.1481.70120.1682.0712.70259.1160.0489.70339 5.3106.1325.70096.1501.0637.70206.1034.0438.70269 6.2834.1210.70080.1368.0582.70171.0942.0399.70223 8.2453.1048.70060.1183.0504.70128.0813.0346.70166 10.2193.0938.70048.1057.0451.70102.0726.0309.70133 15.1790.0766.70032.0862.0368.70068.0591.0252.70090 20.1550.o664.70024.0746.0319.70051.0507.0217.70072 30.1265.0542.70016.0606.0259.70036.0395.0169.70062 40.1095.0469.70012.0516.0221.70030.0313.0134.70060 50.0976.0418.70010.0446.0191.70029.0249.0106.70060 60.0884.0379.70009.0389.0166.70028.0198.0085.70060 80.0742.0318.70008.0296.0127.70028.0126.0054.70060 100.0629.0269.70008.0225.0097.70028.oo80.0034.70060 150.0421.0180.70008.0114.0049.70028.0026.0011.70060 200.0282.0121.70008.0058.0025.70028.0008.0003.70060 300.0126.0054.70008.0015.0006.70028 400.0057.0024.70008.0004.0002.70028 500.0025.0011.70008 600.0011.0005.70008 800.0002.0001.70008 K2 = 20 K2 = 50 K2 = 100 9 q 92 ql +q qq2 + q 2 q2l 1.5.1399.0567.71166.0901.0362.71356.0642.0256.71468 2.1190.o4go.70838.0762.0312.70950.0539.0220.71038 3.0956.0399.70536.0608.0253.70601.0420.0174.70741 4.0821.0345.70394.0520.0218.70455.0342.0142.70660 5.0731.0309.70312.0458.0193.70384.0283.0118.70637 6.0665.0282.70258.0410.0173 70348.0234.0097.70630 8.0573.0243.70195.0334.0141.70322.0162.0067.70627 10.0509.0217.70162.0275.0116.70315.0112.0046.70627 15.0401.0171.70131.0170.0072.70313.0044.0018.70627 20.0324.0138.70125.0105.0044.70313.0017.0007.70627 30.0215.0091.70123.oo40.0017.70313.0003.0001.70627 40.0142.0061.70123.0015.0007.70313 50.0094.0040.70123.0006.0003.70313 60.0062.0027.70123.0002.0001.70313 80.0027.0012.70123 100.0012.0005.70123 200.0002.0001.70123

-328L = 202 c = 0.9 K2 = 2 K2 = 5 K2 10 ql q2 l q2 q q1l 2 q ql+q2 ql+q2 ql+q2 3.5596.0619 90039 31162.03417 90118.24004.02609.90196 4.4845.0537.90029.26947.02965.90o88.20708.02263.90147 5.4333.0580.90023.24080.02655.90071.18479.02026.90117 6.3955.0438.90019.21969.02425.90059.16842.01851.90098 8.3425.0380.90015.19011.02102.90044.14557.01604.90073 10.3063.0340.90012.16996.01881.90035.13005.01436.90058 15.2501.0278.90008.13868.01537.90024.10602.01173.90039 20.2165.0240.90006.12006.01331.90018.09174.01016.90029 30.1768.0196.90004.09799.01087.90012.07471.00828.90020 40.1531.0170.90003.08472.00940.90009.06411.00711.90016 50.1368.0152.90002.07539.00837.90008.05621.00624.90015 60.1244.0138.90002.06809.00756.90007.04971.00552.90014 80.1060.0118.90002.05662.00629.90006.03921.00435.90014 100.0917.0102.90002.04729.00527.90006.03101.00344.90014 150.0651.0072.90002.03081.00342.90006.01726.00191.90014 200.0463.0051.90002.02001.00222.90006.00.00.107.90014 300.0235.0026.90002.00844 oo.00094.90006.00298.00033.90014 400.0119.0013.90002.00356.00040.90006.00092.00010.90014 500.oo61.0007.90002.00150.00017.90006.00029.00003.90014 600.0031.0003.90002.00063.00007.90006.00009.00001.9014 800.ooo8.0001.90002.00011.00001.90006 K2 = 20 K2 = 50 K2 = 100 9 ql q2 ql3 ql q2 ql ql q2 -ql ql+q2 ql+q2 ql.q2 2.12141.01245.90699 3.18668.02010.90280.12946.01379.90375.09542.01010.90429 4.16046.01742.90208.11063.01192.90274.08118.00871.90310 5.14288.01559.90165.09819.01065.90216.07176.00776.90247 6.13004.01423.90137.08918.00971.90178.o6484.00704.90209 8.11220.01233.90102.07670.oo840.90133.05480.00597.90171 10.10013.01103.90081.06820.00749.90108.04732.00517.90157 15.08149.00900.90054.05430.00598.90082.03369.00368.90148 20.07033.00778.90042.04477.00493.90076.02416.00264.90147 30.05629.00623.90032.03106.00342.90073.01244.00136.90147 40.04656.00516.90029.02163.00238.90073.00641.00070.90147 50.03884.00430.90029.01506.00166.90073.00330.00036.90147 60.03247.00360.9002.0109 49.00116.90073.00170.00019.90147 80.02271.00252.90029.00509.00056.90073.00045.00005.90147 100.01589.00176.90029.00247.00027.90073.00012.00001.90147 150.00651.00072.90029.00040.oooo4.90073 200.00267.00030.90029.00007.00001.90073 300.00045.00005.90029 400.00007.00001.90029

APPENDIX M INDIVIDUAL INSTANTANEOUS DIMENSIONLESS FLUXES, ql AND q2 FOR TWO-LAYER RADIAL FLOW, CONSTANT TERMINAL PRESSURE Tables valid for all rb/H > 10.0 Calculated from: q 7 Cmne -a2 r 2b2V2(b) sin )7c (v-189) m=l n=l ] (b) 189) 2 - -a C b 2b2V2(b) [cos yc sin P(l-c) ] (V q2 Z Cre [: (v-194 m=l n=:l b2V2(b) - C cos P(1-c) Eigenvalues b from: Appendix B Eigenvalues a from: Appendix E Truncation Test: JTerm| < ISu for m series 10fre Only one term used in n series Estimated Accuracy for ql and q2: + 1 part in 104 Definition of Dimensionless Variables: r a (V-135) rb R = re (V-136) rb y = a (v-137) H Q =klta or liquid flow ) [l1lCrb (v-14o) = klta (for heat conduction ) PlCplrb2 *. k2 All values presented are for K2 = K2 = -329

-3300g Hayo KIN a0 O\ H H ^ -~ I 0as st- O ITMu iH o o o H -tt MMnKClO CJ H 00 0 0 00 0 0. 0............... ~ 0 0 0000000 H HC KT* ^ i-l 00 H U HCU KlP -UiV6Dd H Q J, t'- nH H mI a A R C~ O\ t n C H U4 C4 U4 r ra.n........~.. o......... w ^ t"- S H^0\.st- ITN1- <SK\O\ K^ t- - rl 0 o\ h"c Po\ N \ irX) 1 O O KO ir N H o 0 Q C.................~............ CUI CU N I r-H CN HHt HHD CO 0 U) I H I Q U C HHH 0000o W m ~ ~ O H < I 0 1 U C C U HH H 0 0 0 0 00 co a t- 0 \n 0 H 0CUN 0U a1 H i 0 O O 000 - tN- N C\ N N H - C O \O O O 000 O,~........................ 110 0 0 KOoQ0 H H cm nt U\ c Ho CUU H CU rcy 1 C 1 0 QC r o.......................... o.............. OCU... 0 0. C 1 HCU M..t CU00 O O0,\ t( NH MF. 1N o U-\ _ ~. H, t C 4 on* t UC - H — 2N 88N o' o\0. C... O. O N i a O S oo Ho CU j U 2............................. oo 0 0 H 0 O 5 R 1,1N -t 1n 0 n\ od01o I CM0 H H C K l 00_ IT\nsO g (O c l\28 00H CU 0.*O..........0,..0.r.c o., o o oH 0 0 I,.. ~ Q. K Ht H o-t'o -.oO I 0 0' n \0 Qf~- K\, N:t V S cO0 n N.~ rI c tw \H \00 c0l 0 01'O 0C\\00000 t00 O1 Oj:t- ~C1Cq > -^ f N 0 t~H 0 H CU KM LO H 0 O N NN N NOO H CU 1 LIM O OC 0 OI\ 0 0.y o<...................- H............ i-lH Ni ns Ur\VO co ir\ \1 o lrl\........ r 4 ^ d CY n Do H, co $ \ CR) H * N -F-t irvo Co o \o O o o o O o H > nx K0\ 0 i 00 0 n 1 nol o H - 0 o \ \ 0 0 - \ o 0 o r-l _lt n n n o H 0~ o0 n n nC o04 0 0 I O g o< 2 ~ ~~.............................,~,~~~~~~~~~~~~~~. W n,^~r-.H N CM CM I 0 0 \l 0 H CC) r-| \n0 t — OJ 01 n t -I O K-I \J O\0 N On 01 O\ a Oh - i U\0 CU C n 1 Cho \ iU5 D o \- _Odt t- i-l _~._ O:0 0 0rlN 0 8- r- O) Od 8 \ ( \0 f 0 0D 0\ 0 0 0 t0 0 O O H| E \ n WN 0 VO KO T N N N8 H 0 0 O O 0 O O \ t H 0 \ n G\ ^ OOOO O - 0 O 0 _.H _It I~" KIA O Od ( r H, - O 0 O00 0 J rl,-I 0 0 0 0 0 0......................... io co n c,0 ri *N *roco \ o o oo o _ 0 o o o o,i, rl N K_^

-331H | T 5-Q 8 i g'IO S c IN c o UN o o 0 88k foJ 0 |H O | J sb O ^ c c -~^ OD co lS " O R v p-t m Q 1 H II HJ NCU It 1 H H C CUQ IOOo U' HiON^ r>~\C\ H HOO \ 0J H 0O Ku O, CU 0C 0U H II L|n CMO N. 0 8O > O U X r O ( Q H Q 2 ~ I > t~0 H D u 0\ I~ ~r | -U rI 5 0 \ 2c\ 0s ~~ ~ ~ 9 0 ~' ~ 2~ ~I ~d't,~ ~, ~ 2 CH H ~ 0 H ~ C ~ H CU ~ ~ H~ 2 0o 0 C CU o o a o, C N Ch 00 i CU HH CU KO u"-D\4O> O O O' O-0 U-' HH CUH Z 0 MLO O i c^ cu S II.s o O^ <b ^ ffi < Oir o~ o o o 8 88 ^ P -< V fj ^ ^ -oi r o 0,\ o o o 8 8 8 0 H H CU i H H CU Hr O 0 4........................................ II II 00 oH 00000000Q HHHHH O HHOOOOHOOOOOOOOOOOOO,- o, I.- OJ r -co oi t o o o o 2o O O o o0 0~ i rcI._- I k0 2ou'O OOOooOooo ^ tO H~ r' s~ n N^ i - N1 0r N 0 D C t 0 I0 0 00 0 t O r10C)UN000 0 0 0 0 0 000 r CUCU~IU n tt 1C8J M Cl=100 0 U) H CU z F~ O'O HH0CUt10K tU UJ 0U U' O C OJ O O O 0 O 0 O O O O O o O' O O OJ O O O 0- O O O O O O O0 O K-0 _D 0U U,I CU U\- 0'0 UO Lr O o O o Q o rr\S 60 00w o u O O O O QO O 0 01 r \0 0 ||I C0ONt-'.0U'.4t 04 0r\ -Ur r CU 000 0II l ri,1,I CU 0 H0 U O l O H 0 0 0 0 04.44C r lR,1... HC HH H w... 2 d o.................... o o....................o ir \u o o o o oJ o0 o\ o o o> oU o o0 O ) \ cOO - of o o- o - to- o- oo Lr NDO0o00 000000000000 L00000000000000000000 H3,H(p1 Hj X- X CUXO XMO U O'O O 000OXX OJ CU COJ -- 0 CU O0 O\ 0C- 0 o 00 CU C- 0 C-CCO'0.O 0\ 0 0 0 0 0 0 0 r 4t — 0 K OJ CU 0\T C - H 0 CC) 0H 0L O O C CU O OC\lf C - O 0 CV U0 U0 UN00 H C\CUC'IO UHHHHH- coH\HHOO 0 00 000o \co H\ CC\CY \-t0 0~ o rO F- -o II a rl C3hO c a-'O LI \ OJ \D l"8; r\ ONr \ 0 0II \'0 ON "C r 0rC J OJ U t0 uCO C- N O 0 0 r o OO rr-li-IOO- 000 D:,1 H -00000 0 000000000000000 ooo000000000000000000 H4 H DH.Ot< U\ 0CXXXXXXX XXXXXXX U' OJ rCU -\tO 00 CC) U' CU \- t U\"0 CO r -I

-332R = 100; c = 0.5 K2 = 2 K2 = 5 K2 = 10 l 9l'q2 9 ql q2 8 ql'q2 5 x 102.1300 2 x 102.1337 102.1352 6 x 102.1271 3 x 102.1271 1.5 x 102.1285 8 x 102.1228 4 x 102.1228 2 x 102.1240 103.1195 5 x 102.1195 3 x 102.1182 1.5 x 103.1137 6 x 102.1170 4 x 102.1140 2 x 13.1091 8 x 102.1127 5 x 102.1106 3 x 103.1009 103.1091 6 x 102.1074 4 x 103.0934 1.5 x 103.1009 8 x 102.1014 5 x 103.0865 2 x 103.0934 103.0958 6 x 103.0801 3 x 103.0801 1.5 x 103.0832 8 x 103.0686 4 x 103.0686 2 x 103.0722 104.0588 5 x 103.0588 3 x lo3.0545 1.5x 104.0400 6 x.05044 x.0411 2 x.0272 8 x 103.0371 5 x 103.0310 3 x 104.0126 10.0272 6 x 103.0233 4 x 10.0058 1.5 x 104.0126 8 x 103.0133 5x 104.00272 x 104.0058 104.0075 6 x 14.0012 3 x 104.0012 1.5 x 104.0018 8 x 104.0003 4 x 104.0003 2 x 104.0004 105.0001 5 x 104.0001 K2 = 20 K2 = 50 K2 = 100 8 qllq2 ql'q 2 8 qlq2 50.1360 20.1366 10.1367 60.1329 30.1297 15.1298 80.1282 40.1252 20.1253 102.1247 50.1219 30.1194 1.5 x 102.1189 6o.1193 40.1153 2 x 102.1147 80.1152 50.1119 3 x 102.1082 102.1118 60.1089 4 x 102.1025 1.5 x 102.1045 80.1033 5 x 102.0971 2 x 102.0978 102.0981 6 x 102.0920 3 x 102.0858 1.5 x 102.0861 8 x o12.0826 4 x 102.0753 2 x 102.0757 103.0741 5 x 102.0660 3 x 102.0584 1.5 x 103.0566 6 x 102.0579 4 x 102.0450 2 x 103.0432 8 x 102.o446 5 x 102.0347 3 x 103.0252 103.0343 6 x 102.0268 4 x 105.0147 1.5 x 103.0178 8 x 102.0160 5 x 103.0086 2 x 103.0093 103.0095 6 x 10.0050 3 x 103.0025 1.5 x o13.0026 8 x 103.0017 4 x 103.0007 2 x o03.0007 104 oo0006 5 x 103.0002 3 x 103.0001

-333CV 4H 00 C 0O H\'0Huf N0000000000000 0 000 0000 0 b-b- OJ0000000000000000000 -00 0cu0 o 0 o0 LOr Ch r L\ 0 -'- cu O0 - 0 r-o~ CI'DCO0r.~ C8 cu 0 C CuUJNc L \ \, \.tLrC\JC UCS- CHM N\~0 \Oruro^(o.~'?O Cu C M N0 \ C-uC- U Lr\- c Ho 0jo 0m d1 0 000 0\ CTC\7CNCCt 8~ t-^ OldCM r- g CSQO 000 b- --- bo 0 Dc - -C CO0 O c o ~ 0-.4 N HO-C0 H. V —CuO 9 l OCM K ^~~~~50000 S i0 ^ Cy-t OJ^- 0004 i-]7-i2 H- C-^ ON KQ'c Ds oj OOO zrCH tOKNIL u.OO Oc Q r ir\ I-j lC\ 00,O0C-CO C. H CM Mf -:d-H \f CO lS OO l' -+ ~ \-0 -to - K K.o - uH- t O CO KLr\ UN QCo o o o o ooH H Ho;8o HH HHH IHr- HqHH H H r- H Hi-,Hr-l - HHH H Hr-H IHHHH -H HHHHHHHHH H H H -l -4 11 m m m x -' x x omm''''k 0x 0H - x'k OO U'x x x x'..-x x x xO H x x m LiV.0- Ur\ u H,.O U-\MO0 LCt Cu \,' OD Lr\ N " L n S OV Cu N LV-00 LACI\O CD Lr\ C-j \O Lf Cu CLt UN HD O0_-5 0 H M o c rC 0,H H Ht- - 0 CC - - q O o 0 g r i0 - 0 C00.C- O C 00 H -_ t wo 0 K o00C - u 0 0L 999;000000000000 000000' t c ~ 8 8oO O\ 0 O-J - 0 - H <\ OLKH Ir\o 0M7I'A 0 COJ 04 H-^ O - 0H 0 t-oN _CO ^\ O OJ l^ 0 t-,( 0 OJ mO 0U 01 0\0\ 0\03000000d~t~- [~-VO'sulAd-.d-OJHOQOOOOM 01N O t O O 1 Cl O H ooo oo ooooooooo ooooooooooooo8888 0HOH20 H 0000 r4 r r r1 r i r' OJ IA <r - CC) \\CC t\0 t - 0 C \ H' ^-8 - O > 00I O OO su, US - - O O O OH H C!.............................................. Cu QI CMKNQ^C- LCr~i CO 00tOO~l -OCOj-[ -00\\ H\ Cu r4H0 C t0 O -HiAO n -\OlO\ Or\ CU Uo 0 00\O d ""a~oooo 0, C 0 Koooooooo~ HH00o o O 0 0o 0 0 00 0000 0000000UO0000000 0 ooos oS~Il ifciltS'SoI'i^-liS~looSHSooooo O~nsk~ze10CTBt- tL- CHO 0 0 0 0 00r, C o t-\ C OJC —- ~ n M^K\CUHHO\ 0 000000 0000 0000000000 0000..........-.-.-................................................M........................ CuOJ ~ 4 LrCM AO-^ 00_j Ctr\r-CC)uN(\ D- ui -lt\ 0CM0 LCKC \ \X4 LI H 0 CO Cu D I LA_: 0 M CC K 4 V0 0 L-l M Cu0 N'N 0 Cu\ M M M-O-O^-OMuf 00Mtrlf 0 Cg\~1H C-H Q 0 000 M \ CM r-l HlH0 Cr\ " " H N0 0 0u O,0 oLr, C. Ut O uHHOOOOOCe C-cdU0.0 t 0000 00000 0 00000000000000000000 00 0000000000000 XX X X X H XX X X XXXXXXOX 0 r4 Cu =CuX X 0HXX -4XX X4 CuC —K1 L C~CU 00 \0 L( 8- 5: m - 4uu\0 \ EHI cr\ C-\-H HO SH O 00 0 4C0 H.HO OO H \ L C"P\\.O M — a- 4 LC-0 O u O "O. ONI0 Cu - X XXXXXXX XXXXXXXO L (, 0 m \XX C X X X X X 0\u0co Lr C-C-C —\otc N\ \CuHH u 00 C-0-C-t \,O,\ \ N\CD) 000 o 00000000000 00000 0 0 0'0 0'888 08 000 40'0\0 000 0H0000 0 4 O H00000 000000000000000000 000000000000 Lc U o t\ \0 \C N Nco - r- u - 0o 0 8 C \,D 0,_K - t4 fO N 0"-4\_0 " oo H o b-i b-m b -O 0 - - - H H H k 0 0 0H0 c0 o t'o~0oo~ ~. ~0 ~"~ 0 O0, an a Ot_ kO a O Oh O [- \ -._I\ v\ t\ h\ c i H H 0 - \ 0\o 0nn\8 60H 00000000o0000000000000 0 - - b- 00 000 mN00COU\(UV\ C1 0r, t \ Cy, 0 H, 8 00000000000000000-40000000000000000000 Lr\ Cm,\-:t Lr\\CO CC) n cCU rr\ -:t Lr\ \0 cD LO \ cr\ \,O 00 Ln\ \ C m " (y Mt Lr,\\OO OD b - by \ —t L\\O CO UN ooooooo;8 o o o o o o o ~ 800000;

-334R = 20'j c 0.1 K2 2 K2 = 5 K2 = 10 a 4ll<29 q q42 9 q, L, 10.04659.41932 5.04485.40361 2.04700.4230 15.04303.38723 6.04329.38958 3.04338.3904 20.04077.36696 8 o4lo01.36910 4.o4110o.3698 30.03790.34107 10.03938.35442 5.03946.3551 40.03593.32340 15.03660.32935 6.03819.3437 50.03433.30897 20.03457.31110 8.03623.3260 60.03289.29602 30.03120.28081 10.03465.3118 80.03027.27242 40.02824.25415 15.03131.2817 100.02788.25089 50.02557.23008 20.02836.2663 150.02270.20429 60.02314.20829 30.02330.2096 200.01848.16634 80.01897.17070 40.01913.1722 300.01225.11029 100.01555.13990 50.01572.1414 400.00812.07312 150.00945.08507 60.01291.1162 500.00539.04848 200.00575.05173 80.00871.0784 6o00.00357.03214 300.00213.01913 100.00587.0529 800.00157.01413 400.00079.00707 150.00220.0198 1000.00069.00oo62 500.00029.00261 200.00082.0074 1500o.00009.00ooo080o 600.00011.00097 300.00011.0010 8oo.00001.00013 400.00002.0001 K2 =20 K2 = 50 K2 = 100 9 %1 32 e q% a2 e' 1.0o4705.4234.5.04502.4052.2 04709.4238 1.5.04343.3908.6.04345.3910o.3.04346.3911 2.04114.3702.8.04116.3704.4.04117.3705 3.03823.3440 1.03952.3557.5.03953.3558 4.03627.3264 1.5.03673.3305.6.03827.3443 5.03469.3122 2.03471.3124.8.03630.3267 6.03329.2996 3.03139.2825 1.03472.3125 8.03075.2767 4.02846.2561 1.5.03140.2826 10.20843.2558 5.02582.2323 2.02847.2563 15.02337.2103 6.02342.2107 3.02343.2109 20.01922.1729 8.01927.1734 4.01928.1735 30.01299.1169 10.01585.1426 5.01587.1428 4o.00878.0790 15.00973.0876 6.01306.1175 50.00594.0534 20.00598.0538 8.00884.0796 60o.00401.0361 30.00225.0203 10.00599.0539 80.00183.0165 40.ooo85.0076 15.00226.0203 100.ooo84.0075 50.00032.0029 20.00085.0077 150.00012.0011 60.00012.0011 30.00012.0011 200.00002.0002 80.00002.0002 40.00002.0002

-335R = 20; c =0.3 K2 = 2 K2 =5 K2 = 10 9 q, q2 q, q q2 9 91 92 10.1430.3336 5.1398.3261 2.1476.3443 15.1319.3077 6.1348.3145 3.1359.3170 20.1249.2913 8.1275.2957 4.1285.2998 30.116o.2706 10.1223.2854 5.1232.2875 40.1101.2569 15.1137.2653 6.1192.2781 50.1054.2460 20.1078.2515 8.1132.2642 60.1014.2365 30.0987.2302 10.1086.2535 80.0940.2194 4o0.0908.2119 15.0997.2325 100.0873.2038 50.0836.1951 20.0920.2146 150.0727.1695 60.0770.1797 30.0785.1833 200.o6o5.1411 8o.0654.1525 40.0671.1565 300.0419.0977 100.0554.1234 50.0573.1336 400.0290.0676 150.0368.0858 6o.0489.1141 500.0201.0468 200.0244.0569 80.0357.0832 600.0139.0324 300.0107.0250 100.0260.0607 800.0067.0155 400.0047.0110 150.0118.0276 1000.0032.0074 500.0021.0048 200.0054.0125 1500.0005.0012 600.0009.0021 300.0011.0026 800.0002.0004 400.0002.0005 K2 = 20 K2 = 50 K2 = 100 1.:1482.3458.5.1419.3311.2.1488.3471 1.5.1364.3183.6.1368.3191.3.1369.3194 2.1290.3010.8.1293.3017.4.3294.3019 3.1196.2792 1.1240.2893.5.1241.2895 4.1136.2651 1.5.1152.2688.6.1200.2800 5.1091.2545 2.1093.2551.8.1140.2659 6.1052.2455 3.1005.2344 1.1094.2553 8.0986.2300 4.0930.2169 1.5.1006.2347 10.0926.2161 5.0o861.2009 2.0931.2172 15.0793.1851 6.0798.1861 3.0799.1865 20.0679.1585 8.0685.1598 4.0687.1602 30.0499.1164 10.0588.1372 5.0590.1376 40.0366.0854 15.o4o1.0936 6.0507.1182 50.0269.0627 20.0274.0639 8.0374.0872 6o.0197.0460 30.0128.0298 10.0276.0643 80.0106.0248 40.0059.0139 15.0129.0301 100.0057.0134 50.0028.0065 20.00o6o.0141 150.0012.0028 60.0013.0030 30.0013.0031 200.0003.0006 80.0003.0007 4O.0003.0007 100.0001.0001 50.0001.0001

-336R = 20; c = 0.7 K2 =2 K2 =5 K2 = 10 10.3529.1512 6.3518.1507 4.3434.1471 15.3244.1390 8.3312.1419 5.3279.1405 20.3065.1314 10.3167.1357 6.3161.1355 30.2841.1217 15.2930.1255 8.2990.1281 4o.2697.1156 20.2779.1191 10.2868.1229 50.2590.1110 30.2583.1107 15.2665.1142 60.2503.1072 40.2443.1047 20.2528.1083 8o.2354.1009 50.2323.0995 30.2318.0993 100.2223.0952 6o.2213.0948 40.2137.0916 150.1930.0827 80.2011.0862 50.1973.0845 200.1677.0719 100.1829.0784 6o0.1821.0780 300.1266.0543 150.1442.06l8 80.1552.0665 400.0956.0410o 200.1136.0487 100.1322.0567 500.0722.0309 300.0706.0303 150.0886.0380 6oo.0545.0233 400.0439.0188 200.0594.0255 800.0310.0133 500.0273.0117 300.0267.0114 1000.0177.0076 6oo.0169.0073 400.0120.0051 1500.0043.0019 800.00oo65.0028 500.0054.0023 1000.0025.0011 600.G024.0010 1500.0002.0001 800.0005.0002 K2 = 20 K2 = 50 K2 = 100 9 qI iq2 9 92 9 q,q2 2.3507.1503.8.3555.1524.5.34o8.146o 3.3225.1382 1.3392.1453.6.3282.1406 4.3047.1306 1.5.3125.1339.8.3099.1328 5 2921.1252 2.2957.1267 1.2970.1273 6.2825.1211 3.2745.1176 1.5.2756.1181 8.2682.1149 4.2607.1117 2.2617.1121 10.2576.1104 5.2500.1071 3.2419.1037 15.2373.1017 6.2408.1032 4.2258.0968 20.2203.0944 8.2244.0962 5.2112.0905 30.1906.0817 10.2096.0898 6.1976.0847 40.1648.0706 15.1769.0758 8.1730.0742 50.1426.o611 20.1492.0640 10.1515.0649 60.1234.0529 30.1063.0455 15.1087.0466 80.0923.0396 40.0757.0324 20.0780.0334 100.0691.0296 50.0539.0231 30.0402.0172 150.0335.0143 60.0384.0164 40.0207.089 200.0162.0070 80.0195.0083 50.0106.0046 300.0038.0016 100.0099.oo42 60.0055.0023 40o.0009.0004 150.0018.ooo8 80.0015.0006 500.0002.0001 200.0003.0001 100.0004.0002

-337R = 20; = 0.9 K2 =2 K2 =5 K2 = 10 15.4316.04794 10.4467,o4962 8.4390.04876 20.4072.04523 15.4110.04566 10..4193.04658 30.3767.04184 20.3885.04317 15.3872.04301 40.3573.03969 30.3603.04003 20.3670.04076 50.3433.03813 4o.3422.03801 30.3411.03788 60.3321.03689 50.3285.03650 40.3234.03592 80.3141.03488 6o.3171.03523 50.3090.03432 100.2987.03317 80.2973.03303 60.2960.03288 150.2648.02942 100.2796.03107 80.2724.03026 200.2351.02612 150.2403.02669 100.2509.02787 300.1853.02059 200.2065.02294 150.2043.02269 400.1461.01623 300.1526.01695 200.1664.01848 500.1152.01279 400.1127.01252 300.1103.01225 600.0908.01008 500.0833.00925 400.0731.00812 800.0564.00627 600.o615.00683 500.0485.00539 1000.0351.00389 8oo00.0336.00373 6oo.0322.00357 1500.0107.00118 1000.0183.00204 800.o141.00157 1500.0040.00045 1000.0062.00069 2000.0009.00010 1500.0008.00009 2000.0001.00001 K2 = 20 K2 = 50 K2 = 100 9 9liq2 eq C 2 8 q1 5.4434.04925 3.4254.04726 1.5.4324.o48o4 6.4269.04742 4.4016.04462 2.4079.04532 8.4030.04476 5.3847.04274 3.3773.04192 10.3859.04287 6.3718.04131 4.3579.03976 15.3580.03977 8.3528.03920 5.3438.03820 _0.3400 ~.03777 10.3390.03766 6.3327.03696 30.3148.03497 15.3137.03485 8.3147.03496 40.2947.03274 20.2934.03260 10.2993.03326 50.2766.03072 30.2581.02867 15.2657.02952 60.2598.02885 40.2271.02524 20.2362.02624 80.2291.02545 50.1999.02221 30.1866.02073 1oo.2021..02245 6o0.1760.01955 40.1474.01637 150.1477.01641 80.1363.01515 50.1164.01294 200.1080.01199 100.1056.01174 60.0920.01022 300.0577.00640 150.0558.00620 80.0574.00638 400.0308.00342 200.0295.00328 100.0358.00398 500.0164.00183 300.oo82.00091 150.01l0.00122 6oo.oo88.ooo98 400.0023.00026 200.0034.00038 800.0025.00028 500.0006.00007 300.0003.0000oooo4 1000.0007.ooo8 6oo.0002.00002

APPENDIX N TOTAL CUMULATIVE DIMENSIONLESS FLUX, Q(t) FOR BOTH LAYERS, FOR TWO-LAYER LINEAR FLOW, CONSTANT TERMINAL PRESSURE Calculated from: Q(t) - L(1-e) (V-126) 2 -a2Q 2 sin yc + 2 cos 7c sin P(l-c) Z Z'.mne,_2, e m=l nl ( yb2L2) 7 +1 E cos P(!-c) (v-4) e =ml n= l' (V-134) (c)] [C t + (l-c)] Eigenvalues b fromo Equation (V-43) Eigenvalues a from: Appendix D Truncation Test (Both Series): |Term < IS Estimated Accuracy: + 1 part in 104 Definition of Dimensionless Variables x x (V-3) y Y ^(v-4) y =-H La L La (V-5) H klta Q = (for liquid flow) 401cH2 ~ki~~~~~ta N/ ^ ~~(v-15) k= lta- (for heat conduction) P1CplH2 # ~k2 All values are for 01 - 2 K2 K2 = -338

-339K L\ t-_ r4 Nu N c- O ON N\ ) CO t- N cO co 00 LKC V0 C)O rc 0\ - C- 0 00 cO - -- H 0 Hr UL C LNL - CUO O\H NN \O CU N- O\ 0 iC' rN 1 H- r4 ON CU CO KN CO- \ CO4 O CU0 (\N \ C\ 0\ CN 0 O00 Hl K l,-! \ 0-co c co I~\ o ir \ -CO Or\ O L ON co 1 OJ' \ O i-IOQLN CO raMJ^^t^ H rzI r I O CC Qj C^ \C\ CC)\ 0 T\ 3O\rIO\Mc OH cO H 0\ O NHc CO 4 HcU \QNJClO 0 d * * *''' ^ ^....o o~~ ^ ~ o ~o ~o o^ o d N\ 1-l N - KIN\ C 0 3 t 0 co t- t- H -, CO c1 0 0 rl L\ Lr\ co O 0 1 \o 0 00 r)H:L\ 0 O NKO-d:tr LI", N N0 OCOcOCO H HU'N 4 S) CO O _Q 0\ 0 K; r O H l rC 0j C\0 OJ HCO\ \0 H coCloN 0\Ca 0 Lr0 - 1 M\ 1 M r'~ 0r \S ~ LO C~ ~) o CJ O -O O -I O d....'' r4 r4 r4 C; KI I L rO I I I~ 6~ d ~0.~ ~ ~.. No, - (? h"c- 1H c i0 c -SO O - CU H-t L^ - O\ KI rl \~o oO cO ro o N C. o 0 ~Dw~ ~ ~ ~ ~ ~ ~ O^ O.L — —..- oc O O N *\ O) 0 IC' HOJO~ O CU C K4 LQ t- s OONONONO O.O U0 \- -- CU- H L\ O\ O 0 Hl H CU KN4 L.N 0 CO 0 L 0o 0 0 0o H H o\CU r -q on k O C O O n u1nO rJ \ O -cO H — OC\U O-N CC 0 O 0 00\ O ~n o... c oo..... o o o o U OO 0 r 4 l,r,, N4 ( _ K) -~ =r N I ^. N DO t — 0 — o O c - O 0 KC OJ 0- CO- \H O Lr- \ 0 -4-\CO ~ C ONC OO O (n\ N"-O NO0 o\0 Lr\ <\ 0 0 0 rH H HHL \\,\ n\0 CT\ Nn cO (7\ 0\ 0 O-I CO O r \O O OO t- r- \ t- r- -l C -I 0 -O O OO 0 HN r OlNL \0. H CO-r-ON... ON 1- 0N.4- o o O o o0 -IOJ CH OU r-I r-I^ 0 NH HL O- -L O\HCO CO H ON - 0 NH O J \ L t —CO NO-H \ O C - \ - O - O 0O \ 0\ 0N O................ 0O... r1 r4 r1 rJ C } CN N; 1 -d J- L( - r: o Lr1 0 0t — L C O QJO i CM O\N O K P -l t\CD \ 0. \O -- \, ON H t — H CY\ rc\-:J- coH a\\ ^O^\ O L~...O.. r4 4 U) CM K l K n a O o o N r Lr\N H 0 - 0 o rO o- O\ o\ ~~. X uo uo FO H n F~L r-\0 On d r-I Lr\ O\ 0\ 0\\ C U l H- H HC U C U\O4 0 0C iO t3) N"COCOL' *' "1'C O H \OKNC O roCUCCNC> O O O. HCUC' t ( HH' O' bN4 Nf NO OtCO HO rNN- ONO O 0 H H H HHCUCUCU~~t44IC' H N N C = t

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-341o.............-. 0H. OO:\: 0s:OHOC: CN 0 H o u\ c -o ~ 0 O uo O o0 o~ o~w oa a o 0 4 0HH C-,r -,- O - (Ni (N 0I 0 Ci' K'-d H L 6 0 Cs r- r- H H H J C KN Lr\ o4 Lnt\O -0 \ OM O( D O\ 0\ H 0 HLr\ a\ Ctl^ o o D 0- a\ o C-o 0 o Co aO, 0 o H H - ri C H C H C OM O I \O K t- Co C0 C\ C0 0 0\ 0\ oN 0 I. 0 H 1 o 1 N \ \ H Co Ch 0O 0 d 0 0 0L\ C-O u C 0 C00~ 0 \\ CJC-O O LOO O ON I- 0 0 CU L HS t H N 0 1 C\ O H 0u o o UN H o\ ~O o o oN o 00 HCN N HO CU H 0 C \-00 O OU On CU \O CU rI O \O ~ LC OC L- ON ON O00 C \0 c\ l 0 0 0_ 0\ OS( O - K- l Lr\ u: Cc bc CO ONh h O 0 k o O00 ~'hO u~O colO oo 1-O O l- 0N- OkO unN 0OhO HJ Ck It t. - k, b-o, Oh Oh O NC X - O0 -C) 0NKD H C — \ 0 XL CN\ > N 0 NONmHO HKNHOnuC~ KN \u C- ON \ 0 0 0 0 cs\ o H 0uCHo * 0N0OHO0 0-N0 — H-I H H H H C Cu cN M N N) o CC O O O o O 0 OL O OL O CLrC \ O r\ 01 K-4- LU\10 0C H I C01 t\-:- Lri\O \ 0 H rC H COJ KN-=- ir\0 00 rCC H 01 \-=t Ln\\O CO CoHl 0H C _ \ tCM>I\ \ 0 t — Co;t-C\ -:d- r-1 t- KAO 01 n\ 4- LV 0 o O\ H 0 u CC) Lr\ ~Hm 00 1 > —-K OO C 0-D OON0 0 O Hr N cON O r- OO N O 4 _ O - C O - - H\CC) OONO H Hrs r0 0 CH H H H Cu Cu K0N- t 0N UN O \ - C O O C ON\ O 0 0 O rr \ H ~ - r-O D \ Co C1 C to 0000 r 003 C\ 0 H H H0NH CU C -- C L -rO\O Lr\O \ NC —O m ) O N\ 0 u\ O\ O \ CU)\~~~~~~~ H~ H H Cu0r<'O -OJIO irO irC O O t <7C.rO NOCCO\ ONO\NO C0 -0 OC 0 ON^uVtO O Kn C 0 C -- \,,. -..-. o 01 o... oo o..cO.. \II C —ON H 0 OL NpO\,-, — H 0 -ON CO0O CC) ONOO r-q H H C u _t _MI- \ 0M ODNON O Ck C- - \ C N 0 NOONNO o (~1r~K C'h_\ ~ Co o)\o r-I\ 0N\ 0 r-0- O OOOOO O O 0\ O 0 O\ O O O t0 CcCO COOH \kO ONOUCL) \ c\ \-DON O Oc 01 rKONON H8 O j C rC \- 0r ocu t Ln cO - O\ 00 H Cu CI L O O 0O C- \ O O O O O ON OO N ifN O H C 0 0 H O C — 0^ 0 COO O-N H 0 LF \PO 0 ON0 oCdU -— O — CoO1 No 0 o H H Cu Kn

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APPENDIX 0 TOTAL CUMULATIVE DIMENSIONLESS FLUX, Q(t) FOR BOTH LAYERS, FOR TWO-LAYER RADIAL FLOW, CONSTANT TERMINAL PRES>JURE Tables valid for all rb/H > lOO Calculated from: R2 Q(t) = (RKl-e) (V-193) 2 ) Ce -a2 [sin yc + 02 cos 7c sin P(l-c) ] [ 2V2(b) R2 - m- 7 P, cos P(l-c) ) 4 b2V2 (b) 02 [c + (l-c)] (V-194) Eigenvalues b from: Appendix B Eigenvalues a from: Appendix D Truncation Test: ITerml < 1I- 1 for m series Only one term used in n series Definition of Dimensionless Variables: ra r = a (v-135) rb R re (v-16) rb Ya (v-157) klta Q klta (for liquid flow) (v-140) kl1 ta lCrb (for heat conduction) PlCplrb2 Estimated Accuarcy: + 1 part in 104 l v f2' k2 All values are for p1 = p2p K2 = K2 11 - -344

-345R = 5; c = 0.5 R 10; c = 0.5 9 K 2 2 K2=5 K2=0 K10 =20. K =50 K =100 9 K =2 K=5 K2=10 K2=20 K2=50 K2=100.010 1.030.05 2.852.015 1.319.06 3.221.02 1.037 1.578.08 3.915.03 1.327 2.042.1 2.871 4.565.04 1.588 2.461.15 3.772 6.067.05 1.055 1.829 2.851.2 2.526 4.597 7.455.06 1.178 2.055 3.219.3 3.307 6.110 10.02.08 1.407 2.477 3.904.4 4.021 7.508 12.39.1 1.085 1.617 2.870 4.532.5 3.019 4.691 8.829 14.60.15 1.391 2.094 3.763 5.894.6 3.413 5.328 10.09 16.68.2 1.144 1.665 2.525 4.562 7.007.8 4.152 6.532 12.48 20.47.3 1.468 2.158 3.304 5.930 8.662 1 3.200 4.845 7.668 14.71 23.83.4 1.759 2.603 4.007 7.047 9.768 1.5 -4.216 6.448 10.31 19.71 30.62.5 1.311 2.029 3.018 4.650 7.958 10.51 2 3.200 5.149 7.930 12.75 23.98 35.61.6 1.468 2.283 3.409 5.241 8.702 11.00 3 4.215 6.862 10.67 17.16 30.79 41.98.8 1.760 2.757 4.134 6.283 9.804 11.55 4 5.147 8.448 13.19 21.03 35.78 45.43 1 2.029 3.198 4.795 7.164 10.54 11.80 5 6.024 9.946 15.54 24.45 39.43 47.30 1.5 2.642 4.197.6.214 8.818 11.47 11.97 6 6.861 11.38 17.74 27.45 42.12 48.31 2 3.198 5.077 7.353 9.906 11.81 12.00 8 8.447 14.07 21.72 32.42 45.53 49.15 3 4.197 6.550 9.003 11.09 11.97 10 9.945 16.56 25.20 36.27 47.36 49.40 4 5.077 7.709 10.07 11.61 12.00 15 13.4; 22.05 32.10 42.51 49.05 49.50 5 5.857 8.622 10.75 11.83 20 16.56 26.63 37.05 45.81 49.40 6 6.550 9.341 11.20 11.93 30 22.53 33.62 43:12 48.47 49.50 8 7.709 10.35 11.67 11.99 40 26.63 38.47 46.23 49.21 10 8.622 10.98 11.86 12.00 50 30.44 41.84 47.83 49.42 15 10.14 11.69 11.98 60 33.62 44.18 48.64 49.48 20 10.98 11.91 12.00 80 38.47 46.94 49.27 49.50 30 11.69 11.99 100 41.84 48.26 49.44 40 11.91 12.00 150 46.42 49.30 49.50 50 11.97 200 48.26 49.47 60 11.99 300 49.30 49.50 80 12.00 400 49.47 100 500 49.49 io R- 20;' c = 0.5 ~~R ~~~= ~201~ ~~c~ = 0.5 ~R = 50; J c =0.5 9 K2=2 K2-5 K2=10 K2=20 K2=50 K2=100 K22 K25 K2 0 K220 K250 K2 g K2 = 2 K2 = 5 K2 = 10 K2 = 20 K2 = 50 K2 = 100.2 7.463 2 43.47.3 10.03 3 60.31.4 12.42 4 43.85 76.30.5 8.837 14.69 5 52.47 91.71.6 10.10 16.88 6 60.83 1.067x102.8 12.51 21.05 8 76.96 1.356xi02 1 7.672 14.80 25.04 10 44.82 92.50 1.636x102 1.5 10.32 20.18 34.44 15 62.22 1.296x102 2.300x102 2 7.937 12.78 25.23 43.25 20 46.71 78.75 1.650x102 2.922x102 3 10.68 17.38 34.71 59.43 30 64.81 1.102xlO2 2.319xl02 4.054x102 4 13.24 21.69 43.59 73.92 40 50.05 82.00 1.401xlo2 2.946x102 5.053xl02 5 9.958 15.67 25.81 51.96 86.92 50 59.94 98.58 1.690xo102 3.534x102 5.933x102 6 11.4o 18.01 29.78 59.88 98.57 60 69.53 1.147x1o2 1.970x102 4.086x102 6.709x102 8 14.14 22.49 37.38 74.46 118.4 80 88.05 1.458xlo2 2.507x102 5.090x102 7.997xl02 10 16.75 26.77 44.60 87.52 134.3 100 59.89 1.059xl2 1.759x2 o22 5.974x102 8.998x102 15 22.89 36.85 61.23 114.5 161.7 150 83.45 1.46x10o2 2.471x1o2 4.149xlo2 7-749x102 1.063x103 20 28.66 46.27 76.08 135.0 177.6 200.059xl02 1.892x102 3.136x102 5.199x102 9.041x102 1.150xlO 3 30 39.48 63.46 101.2 162.3 192.2 300 1.485x102 2.657x102 4.334x102 6.879x102 1.067x103 1.221x103 40 49.55 78.72 121.1 178.1 197.0 400 1.892x102 3.366x102 5.380x102 8.173x102 1.153x103 1.24lx103 50 58.97 92.26 137.1 187.2 198.7 500 2.28x102 4.023x102 6.291x102 9.168x102 1.198x103 1.247x103 60 67.80 104.3 149.7 192.4 199.2 600 2.657x102 4.634xio2 7.0o86x10o2 9.934xo02 1.222x103 1.249x103 80 83.83 124.4 167.9 197.1 199.5 800 3.365x102 5.726x102 8.383x102 1.098x103 1.242x103 100 97.91 140.3 179.4 198.7 1000 4.023x102 6.666x102 9.369x102 1.160xlo03 1.247x10o3 150 126.1 166.9 193.1 199.5 1500 5.468xlo2 8.484xlo2 1.092x103 1.225x103 1.249x103 200 146.4 181.5 197.4 2000 6.666x10o2 9.735x12 1.170x1o 3 1.243x103 300 171.8 194.0 199.3 3000 8.484x102 1.119x103 1.229x103 1.249x103 400 185.0 197.8 199.5 4000 9.735x102 1.188x103 1.244x103 500 191.9 199.0 5000 1.o060x103 1.20x103 1.248x103 600 195.5 199.3 6000 1.ll9xl03 1.236x103 1.249x103 800 198.4 199.5 8000 1.188x10o3 1.246x103 1000 199.2 l1000 1.220x103 1.249x10 2 1500 199.5 15000 2000

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-347R = 20; c =0.3 R = 20; c = 0.1 0 K2=2 K2=5 K2=10 K2=20 K2=50 K2=100 K2=2 K2=5 K2=10 K —=20 KP2=50 K=100.2 9.494 11.41.3 12.83 15.48.4 15.94 19.28.5 11.22 18.91 13.51 22.91.6 12.86 21.77 15.50 26.42.8 15.99 27.24 19.31 33.12 1 9.612 18.96 32.47 11.45 22.95 39.51 1.5 12.99 25.98 44.75 15.53 31.52 54.39 2 9.763 16.15 32.57 56.09 11.50 19.35 39.57 67.87 3 13.20 22.05 44.90 76.32 15.59 26.51 54.45 91.17 4 16.41 27.60 56.27 93.69 19.43 33.25 67.93 110.4 5 11.86 19.47 32.91 66.80 108.6 13.70 23.09 39.66 80.16 126.1 6 13.60 22.41 38.01 76.55 121.4 15.72 26.62 45.81 91.25 139.1 8 16.91 28.06 47.72 93.96 141.9 19.60 33.39 57.39 110.4 158.6 10 10.93 20.07 33.45 56.83 108.9 157.0 11.86 23.29 39.83 68.10 126.2 171.8 15 14.81 27.52 46.10 77.27 137.7 179.6 16.10 32.00 54.82 91.46 154.5 189.1 20 18.44 34.53 57.75 94.79 157.3 190.2 20.07 4o.18 68.538 110.7 171.9 195.6 30 25.24 47.59 78.46 122.6 179.8 197.5 27.52 55.28 91.81 139.4 189.1 198.9 40 31.64 59.58 96.14 143.1 190.3 199.1 34.53 68.94 111.0 158.9 195.6 199.4 50 37.74 70.62 111.2 158.1 195.2 199.4 41.20 81.531 126.9 172.1 198.0 199.5 60 43.60 80.79 124.1 169.1 197.5 199.5 47.58 92.50 139.8 180.9 198.9 80 54.66 98.78 144.5 183.1 199.1 59.58 111.8 159.2 191.0 199.4 100 64.93 114.0 159.4 190.7 199.4 70.62 127.6 172.3 195.6 199.5 150 87.53 142.8 181.3 197.6 199.5 94.56 155.8 189.4 199.0 200 106.9 161.9 191.2 199.1 114.0 172.9 195.7 199.4 300 135.0 183.0 197.8 199.5 142.8 189.7 199.0 199.5 400 154.8 192.2 199.1 161.9 195.9 199.4 500 168.6 196.3 199.4 174.6 198.2 199.5 600 178.1 198.1 199.5 183.0 199.0 800 189.2 199.2 192.2 199.4 1000 194.6 199.4 196.3 199.5 1500 198.7 199.5 199.1 R =20; c =0.7 R = 20; c = 0.9 Q K2-2 K2=5 K2=10 K2=20 K2=50 K2=100 K2=2 K2=5 K2=10 K2=20 K2=50 K2=100.2 5.237 2.625.3 6.980 3.405.4 8.595 4.131.5 6.228 10.12 3.183 4.816.6 7.094 11.59 3.590 5.470.8 8.738 14.538 4.362 6.709 1 5.567 10.29 17.04 3.155 5.090 7.878 1.5 7.432 135.93 23.530 4.132 6.781 10.6o 2 5.974 9.160 17.534 29.18 3.772 5.037 8.346 13.14 3 7.988 12.37 235.71 40.20 4.977 6.705 11.25 17.88 4 9.857 15.537 29.70 50.45 6.o91 8.249 135.95 22.33 5 7.941 11.63 12.82 35.43 60.03 5.740 7.142 9.711 16.53 26.57 6 9.068 13.33 20.97 40.93 68.99 6.534 8.147 11.11 19.01 30.67 8 11.21 16.58 26.23 51.35 85.22 8.037 10.05 135.75 235.75 38.50 10 8.968 13.24 19.67 31.26 61.o8 99.43 7.938 9.459 11.86 16.32 28.28 45.93 15 12.10 18.02 26.96 435.08 62.70 127.7 10.68 12.78 16.11 22.29 38.95 635.01 20 15.02 22.49 33.81 54.02 100.9 148.0 13.24 15.88 20.08 27.90 48.89 78.19 30 20.49 30.90 46.60 735.65 129.3 173.0 18.01 21.69 27.53 38.43 66.94 103.7 40 25.62 38.78 58.536 90.63 149.5 185.8 22.49 27.14 34.53 48.24 82.82 123.8 50 30.53 46.27 69.22 105.3 163.9 192.5 26.77 32.36 41.20 57.44 96.80 139.7 60 35.25 535.39 79.24 118.0 174.2 195.9 30.89 37.38 47.59 66.07 109.1 152.3 80 44.27 66.66 97.02 138.5 186.7 198.5 38.78 46.93 59.58 81.80 129.5 170.0 100 52.76 78.72 112.2 153.9 193.0 199.2 46.27 55.90 70.62 95.68 145.2 181.1 150 72.01 104.3 141.o 177.4 198.3 199.5 63.46 76.07 94.55 123.6 170.8 193.8 200 88.73 124.4 16o.3 188.8 199.3 78.72 93.41 114.o 144.o- 184.4 197.8 300 115.9 152.9 181.9 197.0 199.5 104.3 121.1 142.8 169.9 195.3 199.53 400 136.4 170.5 191.6 198.9 124.5 141.6 161.9 183.7 198.3 199.5 500 151.8 181.5 195.9 199.4 140.3 156.7 174.6 191.0 199.2 600 163.5 188.3 197.9 199.5 152.9 167.9 183.0 195.0 199.4 800 179.0 195.2 199.2 170.5 182.3 192.2 198.2 199.5 1000 187.8 197.8 199.4 181.5 190.1 196.3 199.1 1500 196.6 199.3 199.5 194.0 197.4 199.1 199.5 2000 199.5 199.0 199.4