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By J. G. Richardson, J. W. Graham

The purpose of this paper is to present the results of theoretical and experimental studies of water imbibition. The imbibition processes are involved in recovery of oil from stratified and fractured-matrix formations in natural water drives and water flooding. An understanding of the role of inhibition in implementing the recovery of oil from such formations is deemed essential to proper control of these reservoirs to achieve maximum recovery. The theoretical studies involved development of the differential equations which describe the spontaneous imbibition of water by an oil-saturated rock. The dependence of the rate of water intake by the rock on the permeability, interfacial tension, contact angles, fluid viscosities and fluid saturatiorls is discussed. A few experiments were performed using core samples to determine the effects of core length and presence of a free gas suturation. The role of water imbibition in recovery of oil from a fractured-matrix reservoir by water flooding was investigated by use of a laboratory model. This model was scaled to represent one element of a frac-tured-matrix formation. Water floods were made at various rates with several fracture widths. Interpretations were made of the behavior expected in a system containing many matrix blocks. The presence of a free gas sntu.ration was found to reduce the rate of water imbibition. In the reservoir prototype of the fractured-matrix model, water imbibition rather than direct displacement by water was the dominant mechanism in the recovery of oil at low rates. INTRODUCTION Imbibition may be defined as the spontaneous taking up of a liquid by a porous solid. The spontaneous process of imbibition occurs when the fuid-filled solid is immersed or brought in contact with another fluid which preferentially wets the solid. In the process of wetting and flowing into the solid, the imbibing fluid displaces the non-wetting resident fluid. Common examples of this phenomenon are dry bricks soaking up water and expelling air, a blotter soaking up ink and expelling air and reservoir rock soaking up water and expelling oil. As increasingly better lithological descriptions have been made of the characteristics of petroleum-bearing formations, it has become obvious that imbibition phenomena which were once considered laboratory curiosities are of practical importance. For instance, in reservoirs composed of water-wet sand strata of different permeability in intimate contact, the tendency of water to channel through the more permeable stratum is offset by the tendency for water to imbibe into the tight sand and expel oil into the coarse sand. Also, in fractured-matrix formations the tendency of water to channel through the fractures is offset by water-wet matrix blocks. As some imbibition of the water into the of the largest fields in the world are fractured-matrix reservoirs, it has become increasingly important to understand all the factors involved in the imbibition process. Examples of fractured-matrix reservoirs are the Spraberry field in West Texas which produces from a fractured sandstone', the giant Kirkuk field in Iran', the Dukhan field in Qatar, Persian Gulf2, and the Masjid-I-Sula-main and the Haft-Kel fields in Southwestern Iran, which produce from fissured limestone3. Research into recovery of oil from fractured-matrix formations was stimulated by the rapid decline of oil productivity of wells in the Spraberry formation. One result of this research was the water imbibition process developed by the Atlantic Refining Co.4 Another idea was that much of the Spraberry oil could be recovered by conventional water-flooding procedures5. Subsequently, pilot floods were conducted in this field to test the feasibility of these ideas. It was felt that an understanding of the role played by imbibition processes in displacement of oil from a fractured-matrix reservoir could not be obtained from field data alone because of the many complicating factors and uncertainties involved. Therefore, theoretical and laboratory studies were undertaken to provide this understanding. Study of the equations which describe the linear, countercurrent imbibition process provided an insight into the role of various factors in the process, such as the permeability of rock and inter-facial tension. In addition to the theoretical studies, imbibition experiments were conducted with core samples to determine the effect on the rate of imbibition of such variables as core length and free gas saturation. The principal experimental studies were conducted by water flooding a scaled model of an clement of a frac-tu red-matrix reservoir to evaluate

By C. S. Brooks

Several of the commonly used methods for deter-mination of effective grain densities of minerals have been evaluated for accuracy and precision, and their applicability to various types of petroleum reservoir minerals tested. Consideration was given to the effect of consolidation, swelling clay, and wettability on these grain-density determinations. Procedures with helium displacement, a modified low-pressure Boyle's law porosimeter, and liquid displacement (water, n-decane, and n-octanol) were tested. The helium-displacement procedure was used primarily as an independent reference for which the effects of fluid-solid interaction, e.g., adsorption and wettability, were negligible. The procedures with the Boyle's law porosimeter and with liquid displacement are the methods in common use. Minerals investigated included calcite, West Texas limestone core samples, California sandstone core samples, and Wyoming bentonite. The number of samples tested, while not large, was adequate to permit application of statistical methods. Grain-density measurements on calcite (Iceland spar) provided a basis of comparison for the accuracy and precision of the procedures under investigation for a material with a high degree of chemical and physical homogeneity and for which fluid-solid interactions were at a minimum. Liquid-displacement procedures (water or n-decane) provided the greatest reliability and precision for measurement of the grain densities of the limestone tested. Gas-displacement procedures (helium or Boyle's law porosimeter) provided the greatest reliability and experimental convenience, if not the greatest precision, for the measurement of the grain densities of the swell- ing clay, Wyoming bentonite, and the shah sandstones which -contained appreciable amounts of swelling, in-terstitial clays. These experiments were designed to evaluate the applicability of the grain-density procedures in common we to some of the less tractable minerals—tight limestones, shaly sandstones, and swelling clays. The results should provide a useful basis for a selection of the method most suitable for application to the various petroleum reservoir minerals encountered and especially for use in the estimation of formation porosity. INTRODUCTION These experiments were undertaken to evaluate the accuracy and precision of several commonly used methods for determination of effective grain densities of minerals, and to test their applicability to various types of petroleum reservoir minerals. Consideration was also given to the effect of consolidation, clay, and wettability on these grain-density determinations. Procedures used included measurements with helium displacement, with a modified low-pressure Boyle's law porosimeter, and with liquid displacement (water, n-decane, and n-octanol). The minerals investigated included calcite, West Texas limestone core samples, California sandstone core samples, and Wyoming bentonite. The accurate determination of the porosity of petroleum reservoir formations, in particular those of low porosity, has received much attention. Both bulk-volume and grain-volume measurements necessary in the evaluation of porosity have been examined by many investigators with the objectives of improving both the accuracy and the precision of these determinations. In this investigation statistical criteria for selection of the most reliable and suitable procedure for use with a given type of mineral are demonstrated. The grain-volume measurement procedures appear to be the most susceptible to improvement.

Jan 1, 1958

By P. E. Baker

An experimental study of heat flow in steam flooding was carried out with steam displacing water in a plane-radial fluid-flow model. Temperature distributions in the model reservoir, overburden, and substratum were measured at frequent intervals during flooding by means of a fixed array of thermocouples. Heat losses to overburden and substratum and the division of heat between the steam and hot-water zones in the reservoir were calculated from the temperature data by performing numerical integrations over the appropriate volumes. Temperature profiles and heat distribution results are presented. Some results were (1) the fraction of injected heat that is lost to overburden and substratum, when expressed as a function of time, did not depend on injection rate; (2) a significant portion of the injected heat was contained in the hot-water zone ahead of and underlying the steam front in the reservoir. The division of heat between the steam and hot-water zones did depend on injection rate; and (3) considerable gravity override was observed. The experimental results are compared with those of several published theoretical studies of heat transfer in hot-fluid injection; approximate agreement on heat losses is found with some of the most idealized theories. A valid comparison cannot be made in regard to steam us hot-water zone; basic interpretation of the hot-water zone, in terms of a vertical steam front, is clouded by gravity override in the experiments. INTRODUCTION The experimental study reported here primarily concerns two problems of heat transfer in steam flooding that have economic and engineering importance. These are (1) the fraction of injected heat lost to the overburden and substratum, and (2) the division of the remaining heat between a zone containing steam and a zone of hot water at lower-than-boiling temperature. The volume of the steamed region, which has a direct effect on oil recovery, can be calculated readily from these results. Several valuable theoretical studies have been published on this problem, but no experimental tests of the theories have previously appeared. Tests were needed because the effects of the necessary simplifications in the theories could not be determined otherwise. The basic, untested theoretical problem is the effect of "decoupling" heat and mass transfer, and the resulting inability to include the effects of heat transfer and condensation on fluid flow. The theoretical problem solved so far has been a heat transfer problem in which fluid is present as a heat carrier and flows in a stipulated manner, either linear or plane-radial. The simplifications that have sometimes been made in regard to thermal conductivity by, for example, Marx and Langenheim,l Lauwerier2 and Thomas3 apparently do not have an important effect on heat loss calculations, although the results of the different methods disagree to a certain extent, as shown by Spillette4 and by Ramey.5 A recent paper by the author6 compared the results of experiments in hot-water flooding with several theoretical results and showed a rather poor agreement. It will be seen that the experimental heat loss results for steam flooding, however, show acceptable agreement. These comparisons are discussed in a later section. EXPERIMENTAL APPARATUS AND PROCEDURE The system studied was one in which a sand pack, initially water saturated and at a uniform temperature, was flooded with steam at a constant rate and temperature. The temperature distribution in the "reservoir" and adjacent "strata" was measured with an array of thermocouples, and heat losses were determined from these measurements. The sand pack contained no oil because the only concern of the experiments was heat flow, and it was important to have the flow pattern as simple

Jan 1, 1970

By N. L. Hancox

Previous researchers have used the cutting and rejoining model to represent a porous body when attempting to place Archie's law on a theoretical basis and calculate rock permeabilities (F = Ø-m, where F = formation factor, = - porosity and m = a constant). In using the model, a porous solid is assumed to consist of sections with parallel sides containing straight. continuous pores of total porosity 4 (pore to total volume or, for a homogeneous body, pore to total area). The joining of pores at an interface between two sections is assumed to be random which results in a common pore area Ø2. This reduced area Ø2 is then taken as that area which limits electrical conduction or fluid flow in the body. Agreement between experimental results and results calculated using a model is not always good Because of this, various modifications of the model are presented — for instance, the random nature of the joining is questioned. Formation factor measurements on specially constructed porous specimens are described in this paper. it is concluded that the common pore area limits flow completely only if sections containing straight pores are very thin, and that, in most natural materials, the common pore area at an interface must be greater than that given by random joining. However, the model is still useful in permeability calculations, provided that the fractional area limiting flow is determined from formation factor measurements. INTRODUCTION Two important properties of porous media are permeability k, which describes the ease with which a fluid can flow through a porous medium, and formation factor F, defined as the ratio of the resistivity of a saturated specimen to the resistivity of the saturating liquid which describes the electrical properties of the saturated material. k is defined by Darcy's law,l k= µqL/P Ae . . . . . . . . . . . . . . . [1] where µ = liquid viscosity, q = flow rate, L = length of specimen, p = pressure across the specimen and = - the effective area over which flow occurs, F is given experimentally by Archie's law,2 F = Ø-m ,..............(2) where 4 is porosity defined as the pore to total volume or, for a homogeneous body, the pore to total area; and m lies between 1.3 for loose sand and 2 for cemented sand. Consider a porous body of total cross-sectional area A and length L, saturated with a liquid of resistivity R and with an effective area for conduction of Ae. Then, from the definition of F given previously, F=A/Ae , ................(3) or, from Eqs. 2 and 3, Ae = A Øm ..............(4) Thus, in any attempt to calculate k or F, some assumptions must be made concerning the pore structure of the body and a value derived for A,. To determine Ae, Childs and George3 and Wyllie and Gardner4 suggested an elementary statistical representation often referred to as the cutting and rejoining model of a porous body. The body is assumed to be composed of two or more parallel-sided sections of porosity 4, containing straight, continuous pores with a common pore area of Ø2 at an interface between two sections. The value is given by a random joining of the pores in either section. Two such sections,

Jan 1, 1969

By E. F. Johnson, V. O. Naumann, D. P. Bossler

A method is presented for calculating individual gas and oil or water and oil relative permeabilities from data obtained during a gas drive or a waterflood experiment performed on a linear porous body. The method has been tested and found both rapid and reliable for normal-sized core samples. INTRODUCTION Individual oil and gas or oil and water relative permeabilities are required for a number of reservoir engineering applications. Chief among these is the evaluation of oil displacement under conditions where gravitational effects are significant, such as a water drive or crestal gas injection in a steeply dipping oil reservoir. Numerous proposed methods of obtaining relative permeability data on reservoir core samples have been too tedious and time consuming for practical use, or have yielded questionable and sometimes inconsistent results. A method has been developed by which the individual relative permeability curves can be calculated from data collected during a displacement test. The method is based on sound theoretical considerations. Using this method, with a properly designed experimental procedure, relative permeability curves can be obtained using core samples of normal size (i.e., 2 to 3 in. in length and 1 to 2 in. in diameter) within a few days after receipt of the core. In a recent publication D. A. Efros1 describes an approach to the calculation of individual relative permeabilities that is based on the same theoretical considerations. We believe the approach described in the present paper is more adaptable to practical application than the method implied by Efros. In addition, comparisons with independently determined relative permeabilities are furnished to substantiate the reliability of the new method. DERIVATION Previously the theory of Buckley and Leverett as extended by Welge2 as been used to calculate the ratio of relative permeabilities. In the derivation which follows, this theory is further extended to permit calculation of the individual relative permeabilities. The theory assumes two conditions which must be achieved before the method is applicable. They are that the flow velocity be high enough to achieve what has been termed stabilized displacement,3 and that the flow velocity is constant at all cross sections of the linear porous body. In stabilized displacement the flowing pressure gradient is high compared with the capillary pressure difference between the flowing phases. The high pressure drop insures that the portion of the core in which capillary effects predominate will be compressed to a negligibly small fraction of the total pore space. The assumption of constant flow velocity at all

By R. E. Gilchrist, R. F. Nielsen

When a gas is displaced by mother in a porous medium, and there is a relative immobile liquid present. there is a transition zone in which the gas composition varies from essentially that of the original gas to that of the injected gas. The change in extent of this zone with time may be governed by a mixing process in the direction of flow and by the rate of establishment of gas-liquid equilibrium, if the gases are appreciably soluble in the liquid. The relative importance of these two factors in deterri~ining the growth of the transition zone is discussed, with appropriate equations and experimental data. INTRODUCTION The various solution methods of oil recovery involve a transfer of the components of a mixture, whether within a single phase or between a liquid and gas. The similarity of a natural porous medium to systems used in separation and chromatographic processes makes it seem reasonable that certain concepts used by chemical engineers in "mass transfer" might be applicable to petroleum reservoirs. One approach, the HETP (height equivalent to a theoretical plate), involves a distance or length of time over which equilibrium might be con. sidered as established. Another approach involves direct use of a rate of mixing or rate of establishment of equilibrium. The latter approach has been preferred in relation to production problems. DISCUSSION AND PROCEDURE When a gas or liquid in a porous medium is displaced by one of different composition but miscible with it there will be set up a transition zone in which the composition varies from cssentially that of the original fluid to essentially that of the injected fluid. If the fluid being displaced is a gas, and there is a relatively immobile liquid present, the transition zone may be

By R. I. Lahring, C. G. Inks

A surfactant waterflood, using a block copolymer of ethylene and propylene oxide (PLURONIC L64*), wa., started in 1956 in Kansas to determine the commercial feasibility of such a process. The production formation was the Bartlesville sand at a depth of 1,400 ft. Leares in the test area had attained both primary and secondary production peaks, and were on a normal decline when the tests were initiated. The injection wells were taking water at a reasonable pressure and none of the operating conditions were particularly unusual. The leases were selected primarily because of normal operating conditions, and not because of any serious problems demanding remedie>,. Oil production and water injection data were recorded during the 10-year test period, and their compilation indicates that the surfactant resulted in an increase of about 9 percent in secondary oil production. Other benefits derived from the surfactant injection were lower injection pressure that resulted in reduced power consumption and reduced erosion of the pumps; and lower calcium scale buildup. These fringe benefits more than offset the cost of the surfactant used. Introduction During the past several decades, considerable research has been carried out on secondary recovery of crude oil that remains in the producing formation despite the most efficient, current primary production practices. Methods are constantly sought of increasing recovery of oil, while at the same time improving the economic efficiency of operations. One method that has received intensive study over the years is the use of a surfactant as a waterflood additive. Laboratory tests show conclusively that dilute solutions of surfactants remove considerably more oil from uncon-solidated sand cores than does water alone.'- The loss of surfactant by adsorption at solid-liquid interfaces has caused serious doubt that surfactants can be used economically as waterflooding additives. This is especially true with anionic and cationic surfactants, and to a lesser extent with nonionic surfactants. This work was undertaken to determine the effectiveness in increasing oil production by injecting water containing small quantities of a nonionic surfactant. The surfactant selected was PLURONIC L64*, a block copolymer based on ethylene and propylene oxide, having the following structure The selection of the test site, a typical shoe-string sandstone type of field located at Batesville pool, Toronto District of Woodson County, Kans., was made because of its normal primary and secondary recovery operating conditions, and not because of any serious operating problems. When the tests were begun, the field was experiencing the customary post-peak production decline of a waterflood. Using for the tests a field that was on a normal production decline, while maintaining constant all variables other than the addition of the surfactant. was of varamount importance if success or failure was to be attributed to a particular cause. It was reasoned that if a surfactant was added to the injection waters and nothing else was done to influence the oil production or injection of water, the effects of the surfactant on the flood could be determined precisely. Production History The first well in the tested field was drilled in 1935. Gas repressurization began immediately, but it was not until Aug., 1938, that the major portion of the field was developed. During the period of repressurization, 1,700 MMcf of gas was injected. As can be seen from the curves of Fig. 1, the production peak from gas injection was attained in 1937 and a normal production decline followed. Early in 1942, water injection wells were drilled and injection of water Was initiated. The amount of water injected increased as more injection wells were drilled until a peak injection rate of 2,700 BID was reached in

Jan 1, 1969

By O. C. Holbrook, George G. Bernard

A new theoretical treatment has been obtained for the behavior of pattern waterflood injection wells when closed in. Two cases are treated: Case I where oil and water are assumed to have the same properties, and Case 2 where they arc different. In applying the method, one plots log (p — p,) vs closed-in time, where p is well-bore pressure at any tims and p, is static pressure. The value of p. is determined by trial and error as that value which makes the plot linear at large time. A value for the permeability-thickness product can be determined from the intercept of this linear part, and a value of the skin factor from the injection pressure at time of closing in. Application of the method to data from water floods at three fields seems to give reasonable results. For the case of unit mobility ratio, it is proved that this new method should give the same value for permeability-thickness product as the conventional pressure build-up method. In addition, the new method gives correct values for static pressure, whereas the conventional method does not, often indicating negative static pressures. The new method may be used in cases where the surface pressure persists after closing in as well as in cases where it does not. INTRODUCTION It is of considerable interest and importance to be able to determine the characteristics of the reservoir in an area surrounding a water injection well. Thus, if we can determine early in the life of an injection well that there is a considerable "skin effect", remedial measures can be started before a full-scale pattern flood begins. Similarly, if it can be shown that a gradual buildup of skin effect is occurring with time, measures to free the water of plugging material can be taken. Determination of static pressure in the water-injection well may show that the water is entering a thief zone and not the desired reservoir. Finally, determination of the permeability of the sand around the injection well will allow estimation of the future relation between injection pressure and rate. It should be possible to determine average reservoir permeability, skin effect and static pressure from pressure fall-off data. However, at the time we began work on this subject, it was thought that no adequate theory on which to base such determinations' was available. According to the conventional method which considers the reservoir to be filled with one fluid of small compressibility (see Van Everdingen, Joers2, and Nowak2), shut-in pressure is plotted vs log where is injection time, and At is closed-in time. The physical significance of injection time, may well be questioned in this case, since in a reservoir completely filled with a single fluid (as required by this theory) and with input and output rates equal, the pressure behavior after an initial transient is independent of t,. Attempts by our Tulsa area to use this theory led to negative values of static pressure in most cases. Because of these limitations of the method discussed above, it was decided to attempt to develop a new theory of pressure behavior in water injection wells, one which would apply when there is a gas saturation, as is so often the case in water floods. In the following treatment the assumptions and basic equations are given first, then the method of application of the equations. A complete example is given to clarify details of application. All difficult mathematics has been placed in the appendices so that the reader can follow the text without difficulty. However, if he wishes only to apply the results without knowing the basis for them, he can learn how to do this from reading only the sections entitled "Plotting of Experimental Results" and "Example." ASSUMPTIONS AND BASIC EQUATIONS Statement of Problem It will be assumed that a horizontal layer of constant thickness contains in its pore system a mixture of oil, gas and water. While water is being injected into this pore- system through a well at constant rate, an oil bank is built up, gas being expelled from the space taken by the oil as shown in Fig. 1. The saturations within each

By S. C. Jones, J. A. Davis

This paper describes displacement mechanisms of micel-lur solution slugs, displuced by a thickened water "mobility buffer", in a glass micromodel and in consolidated Berea sandstone cores. Colored motion pictures, taken through a microscope, recorded interactions on a pore-to-pore bash in the micromodel. Production and differential pressure data from consolidated cores previously waterflooded to residua1 oil saturation revealed displacement mechani.sm.s macroscopically. Oil is displaced miscibly, while interstitia1 water is displaced immiscibly by the injected micellar. solution slug. Part of this water is dispersed and carried near the lead ing edge of the .slug, and some may remain as a residual saturation. Thickened water displaces any residual water. It also emulsifies with the. slug material, thereby displacing it. The displacement sequence forms banks of oil and water with relatively constant fractional flows, saturations, and mobilities. The mobility of oil and water flowing ahead of the .slug defines .slug viscosity required for stable displacement. Equations are presented that describe. several aspecis of the displacement flow behavior: oil breakthrough, oil cut during production of the stabilized bank, breakthrough of banked water and the resulting decreased fractional flow of oil, fluid mobilities, etc. Breakthroughs, calculated from the equations, are compared with experimental data. Slugs of micellar sollution, ranging from 1 to 5 percent PV, produced unusually high oil recoveries (up to 100 percent) in previously waterflooded consolidated cores. Investigation of the effect of mobility buffer size on oil recovery indicated that recovery is reduced in 3-in. X 4-ft cores for mobility buffer sizes .smaller than 50 percent of the pore volume. Introduction A new oil recovery process,* conceived in 1962, utilizes a slug of micellar solution' that has sufficient viscosity to avoid an unfavorable mobility ratio with respect to the oil and water it displaces. These micellar solutions are composed of hydrocarbons, water, and surfactants, and may contain electrolytes and co-surfactants. The micellar solution is displaced by a "mobility buffer". The mobility buffer is a bank of reduced-mobility fluid, preferably an aqueous polymer solution. The entire system is displaced by unthickened water. This paper describes some of the microscopic and macroscopic displacement mechanisms observed during laboratory development of the fluids for a potential field test. Microscopic phenomena were observed photographically in a specially constructed glass flow model and macroscopic data were taken from constant rate, tertiary displacements conducted in Berea sandstone cores. Qualitative Description of Micellar Solution Flooding This section qualitatively describes displacement mechanisms that occur when a slug of micellar solution is injected into a previously waterflooded core. The slug completely removes oil from the portion of core it contacts, and banks oil in the process. The oil saturation in this bank and the flowing WOR reach definite, fixed values that are determined by water and oil mobilities and slug characteristics. This zone of constant WOR and oil saturation will be called "stabilized oil bank" in the following discussion. When this stabilized bank reaches the outlet end of the core, the oil cut jumps from zero to a constant value equal to the fractional flow of oil. Saline water has little or no solubility in the slug. Because of this. the slug displaces in-place brine immiscibly. During slug injection, most of the brine is displaced ahead of the slug, some is carried along in the leading portion of the slug, and part of the brine may become immobile. The immobile, residual brine is later miscibly displaced by thickened water. Displacement of all the oil and only part of the water during slug injection results in a higher fractional flow of oil ahead of the slug than occurs when all the water is displaced after injection of thickened water is begun. The slug spontaneously emulsifies fresh (low-salt content) water. By emulsification, the slug can become infinitely diluted, and hence totally displaced, by thickened water. Fluid saturations in the core are shown schematically in Fig. 1. Parts A and B show conditions before and after the start of the thickened-water injection, respectively. Since thickened water displaces any residual brine by-

Jan 1, 1969

By A. R. Gregory, A. L. Podio, K. E. Gray

Dynamic elastic properties of dry and water-saturated Green River shale samples were computed from compressional- and shear-wave velocity measurements. P- and 5-wave velocity measurements were made in three mutually perpendicular directions with respect to the bedding planes. Measurements were also made in several different directions by varying the angle between the bedding planes and the direction of propagation of the wave for angles of 0, 30, 45, 60 and 909 The oriented samples were subjected to both confining pressure and axial loads, in excess of the confining stress, in the direction of propagation. In general, P- and S-wave velocities increased with increasing stress levels, with a corresponding increase in Young's modulus. Water saturation caused the P-wave velocity to increase and the S-wave velocity to decrease. Elastic moduli decreased upon saturation, except for Pois son's ratio, which increased, indicating some degree of weakening of the material. The samples showed a moderate degree of anisotropy; this was to be expected from the laminated nature and shallow occurrence of Green River shale. INTRODUCTION This paper presents some results of an experimental determination of the elastic coefficients of anisotropic materials (in particular, finely layered rocks and minerals such as Green River shale) from measurements of dilatational and shear-ultrasonic-wave velocities. Ultrasonic techniques have been used extensively in nondestructive testing. Several methods have been proposed by Mcskimmin, 1 and some of these have been used successfully to measure ultrasonic velocities in rocks. Hughes and Cross, 2 Wyllie et al., 3 and Birch 4,5 developed pulse first-arrival techniques for the measurement of dilatational and shear velocities. Williams and Lamb 6 used the method of cancellation of a traveling wave, which was later modified by Myers et al.' and perfected by McSkimmin. 8 Although this method is highly accurate, it has not been used as widely as the pulse-transmission methods recently reported by Jamieson and Hoskins 9 King,10 and Mattaboni and Schreiber. 11 It has been common practice to use some form of crystal transducers, either quartz or ceramic, that has been cut or polarized in different directions in order to generate either compressional or shear waves. However, accurate determination of shear wave velocities has been difficult due to problems that arise in obtaining a pure shear wave from cross-polarized crystals, which usually also generate a small amount of compressional energy. As reported by Gregory, 12 this energy can be seen as a long precursor preceding the sharp break of the shear first arrival. The need for generating pure shear waves led to interest in mode-conversion techniques, which are based upon conversion of the mode of vibration through wave reflection or refraction at a discontinuity. Arenberg13 showed that for certain materials and for certain angles of incidence it is possible to generate pure shear modes by reflection at a boundary. Jamieson and Hoskins 9 used a pyrex glass-air interface for generating pure shear waves, and King 10 used this method successfully for measuring shear-wave velocities in rocks. Gregory l4 arrived at a similar result by refraction of a wave at an aluminum-oil interface. A plane compressional wave, traveling in the oil phase, is incident on the aluminum at an angle larger than the critical angle for compressional waves, and thereby generates a purely transverse, plane-polarized wave in the aluminum. During the last few years methods have been developed that allow the simultaneous determination of shear and compressional velocities in solids.

Jan 1, 1969

By D. O. Wyble

Five water-driver fields, for which pressure predictions had been made within a few years after discovery, were re-studied when four to five years of additional pressure and production data became available. It was fortnd that forecasts of the pressure in water-dri~fe reservoirs may be subject to considerable unccrtainty. Predictions car be erroneous when there is insufficient presswe history to define the cllaracteristics of the aquifer, and when there are errors in the pressure, production or geologic data. INTRODUCTION The efficient operation of an oil or gas reservoir requires that the important characteristics of the reservoir be determined soon after its discovery. It is especially desirable to evaluate the pressure-production relationship quickly, for this relationship provides a basis for deducing the type of production mechanism that is operative, and for making pressure predictions. The pressure predictions, in turn, provide a basis for calculating efficient rates of production, determining whether water or gas should be injected, etc. The importance of pressure prediction is such that forecasts should be made as accurate as possible, and the limits of error well defined. Historically, pressure forecasts for some fields have proved very accurate, while for others the forecasts were so erroneous that they resulted in the adoption of wrong, or at best, unnecessary production practices. Pressure predictions made for water-drive fields are especially subject to error. Accurate forecasting for these fields requires that much be known not only about the reservoir proper but about the adjacent water-filled rock as well. The water-containing rock, or aquifer, may extend for many miles, and often there is only meager data regarding its thickness, permeability and porosity. Much of the information must be inferred from the previous pressure behavior of the reservoir. A brief pressure history may provide little information about the characteristics of the aquifer. Hence, pressure predictions made early in the life of water-drive fields can be incorrect. The work described in this paper was done to determine the degree of reliability which can be expected of pressure predictions made within a few years after water-drive fields are discovered. Five fields, for which the original pressure predictions had been based on less than seven years' history, were re-examined with the aid of four to five years of additional pressure data. Where the predictions were inaccurate, the reasons for failure were ascertained. The studies were accomplished by use of the electric reservoir analyzer.' The analyzer was first adjusted to repeat the original pressure-production match using the same values for the various parameters as were used in the original study. Then, the actual production rates experienced during the time elapsed since the initial study were set into the analyzer, and the pressure behavior indicated by the machine for this period was compared with that which actually OCcurred. Where the pressure profile traced out by the analyzer deviated from the observed field behavior, the analyzer was readjusted to obtain a better over-all match. The old and the new solutions were compared to determine the reasons for any differences. COMPARISON OF PREDICTED RESERVOIR PRESSURE BEHAVIOR WITH ACTUAL PERFORMANCE Field A The original pressure-production match is entirely adequate as evidenced by the pressure match of the field's history subsequent to Jan., 1952 (see Fig. 1)'.

By C. E. Kemp, A. B. Dyes

Results of a field research project on the thermal recovery of oil by movement of a combustion front are presented. This field test was conducted in the South Belridge field, This The war paitern was a Southsing1e 2.5-acre five-spot pattern in the 700-ft deep Tulare sand. Results of this project indicate the technical feasibility of producing high viscosity, low-grallity crude oils by thermal means. It was demonstrated that heavy oils could be moved rapidly over considerable distances by thermal drive, and that high percentage recoveries of oil could be effected. The bulk of the oil recovery came after arrival of the burning front at a production well. However, rapid corrosion and high-temperature failure of conventional pipe and equipment both below and above surface occurred. Stainless steels gave more satisfectory performance under these conditions than did conventional steels. Oxygen was seldom observed in gas from wells ahead of the burning front, and it appears that the front was continuous throughout the life of the experiment. INTRODUCTION During late 1953 and early 1954, industry-wide attention was focused on the thermal recovery of oil by movement of a combustion front through an oil zone. Two separate Oklahoma field tests were described in publications by Kuhn and Koch, and Grant and Szasz. To obtain basic information needed for making engineering and economic appraisals of commercial recovery of heavy oil by this thermal recovery method, a new field research projecta3,4 was started May 31, 1955. Twelve oil companies contributed to this project with General Petroleum Corp. as operator. Since combustion recovery experiments* with heavy oils in other areas had been conducted in small patterns,"' an important purpose of this test was to determine whether the process could be operated in a pattern of near-commercial well spacing. The south Belridge field, located 30 miles north of Taft, Calif., was chosen because its oil sands are similar to many heavy oil sands which typically have low primary recovery and leave a large amount of oil in place. such fields are not particularly suited for common secondary recovery methods. DESCRIPTION OF TEST SITE The test interval was a Tulare sand at a depth of about 700 ft. Nine test wells were drilled, cored and completed from Oct., 1955, through Jan., 1956. The initial pattern shown on the map, Fig. 1, consisted of an injection well (II)', four producing wells (1P-4P), and four observation wells (IT-4T). An old well (81) was also recompleted for use as an observation well. The area enclosed by the original four production wells was 2.5 acres; the distance from the injection well to each production well was 233 ft. During the combustion period, Well 5P was drilled to replace Well 2P, and Well 6P was drilled to replace Well 3P. The new wells were 25 ft beyond the old production wells along a diagonal from the injection well, and resulted in an increase in pattern area to 2.75 acres. Isopachs are shown in Fig. 1. The average thickness of the oil sand in the test pattern was 30 ft. Fig. 2 presents two cross sections through the test pattern. Oil sand is represented by white and claystone by gray. The dashed lines indicate layers of conglomerate grading to claystone. The sand thickness was fairly uniform throughout the test pattern except in the 1P area. The sand body dips gently from the northwest to southeast, about 3 degrees from horizontal. The sand is unconsoli-dated, and grades from very fine material to pebbles as large as 2 in. in diameter. Table 1 presents the average initial properties of the formation. The in-place porosity of 37 per cent was estimated from the porosity of coke-consolidated cores taken after combustion. Original core analyses data were adjusted to 37 per cent, giving pore volume saturations of 60 per cent oil, 37 per cent water, and 3 per

By L. Yarborough, L. R. Smith

This paper presents results of a laboratory study of retrograde condensate recovery by revaporization into dry injection gas. Flow tests were performed in 10.6-ft long sand packs at 100F and 1,500 psi. In three runs methane revaporized the liquid from a n-pentane-methane mixture in the presence of immobile water. TWO of these tests were water-wet, and the third was totally oil-wet. In the three runs n-pentane recovery was complete after 2.5 hydrocarbon PV of injection. There was no significant performance difference between the two wettability extremes. In a fourth experiment, a methane-hydrogen sulfide mixture revaporized a synthetic light, sour condensate. No water saturation was present. Equilibrium compositions and volumetric data were obtained for the four-component condensate. The heavy component, n-beptane, was removed after 6 PV production. Comparison of the effluent fluid compositions with known equilibrium data shows that the flowing fluid was equilibrium vapor and that the mixing zone between equilibrium vapor and dry injection gas was short. Data indicated that complete recovery of retrograde liquid occurred after it was contacted by a sufficient quantity of dry gas. INTRODUCTION When pressure declines below the fluid dew point in a gas condensate reservoir, a liquid phase forms. In this process, referred to as retrograde condensation, the quantity of liquid formed is frequently small enough that the liquid is not a flowing phase. To prevent loss of valuable retrograde liquids, the process of dry gas cycling has been employed for several years as a more or less standard practice. In this procedure the reservoir pressure is maintained above the fluid dew point so that the liquid components may be produced as vapor and then separated at the surface. Although full pressure maintenance by gas cycling seems ideal in terms of preventing liquids loss, several factors can reduce the atuactiveness of such an operation. From a study of a condensate reservoir in Alberta, Canada, Havlena et a1.1 con-cluded that cycling under conditions of declining pressure leads to economic advantages and to a high recovery of hydrocarbon liquids. This study considered effects of volumetric sweep efficiency, retrograde behavior of the original wet gas and revaporization characteristics of the retrograde liquid when contacted by dry gas. The first major work concerning revaporization of liquid in a gas condensate system is that of Standing et a1.2 Calculations based upon the PVT behavior of a recombined gas condensate fluid indicated that all retrograde liquid can be recovered if it is contacted by a sufficient quantity of dry gas. The paper considered the effect of variable permeability upon the recovery of retrograde liquid. Standing et al. concluded that recovery of heavier components in the retrograde liquid is greatest if reservoir pressure is allowed to decline below the dew point prior to dry gas injection. Since the work of Standing et al., several laboratory studies have been reported which show that recovery of hydrocarbon liquids by vaporization into dry injected gas can contribute to increased recovery above that obtained by ordinary production practices. Vaporization from retrograde condensate,3 conventional oil4-7 and volatile oil8 reservoirs has been considered. There is little work that deals with revaporization recovery from condensate reservoirs. This study demonstrates that vapor-liquid equilibrium exists when a dry gas is injected into a porous medium containing retrograde condensate; investigates the effect of wettability on revaporization of retrograde condensate; and provides an indication of the amount of dry gas necessary to

Jan 1, 1969

By W. A. Roper

Since 1950, several papers have been published which have described various methods for studying mobility ratio effects. The methods which have been described for studying mobility ratio effects include X-ray shadowgraph techniques, stepwise potentiometric model procedures, stepwise numerical procedures, gelatin model studies and fluid mapper model studies. The purpose of this paper is to describe a procedure for using a resistance network for studying mobility effects. DESCRIPTION OF APPARATUS The basic unit of the experimental model consisted of a network of 840 resistors fastened to a sheet of black Lucite which was mounted on a wooden frame, Fig. 1. The dimensions of the Lucite sheet were 36 X 36 in. The resistors were fastened to the Lucite sheet at mesh points by means of snaps. The mesh points were arranged in a square pattern, 30 X 30 in., spaced at intervals of l1/2 in. The snap fasteners permitted several resistors to be "stacked" in parallel. By adding additional resistors to the basic unit, the specific resistance on each side of the flood front could be varied. The square network of resistors represented one of four symmetrical elements of a five-spot well pattern. The current input was analogous to an injection well and the current output was analogous to a production well." The power supply was essentially a full wave transformer-rectifier system for converting alternating-current into direct-current. The DC voltage could be varied from 41.43 to 290 v in steps of 41.43 v by selecting the proper outlet jacks. The major components of the power supply were: a Triro power transformer, R 11-A; a dual 5Y3 vacuum tube; two gas filled OD-3 tubes; and a Triplett voltmeter, model 227-1, range 0 to 500 DC v. The voltages at the mesh points were determined with a Keithley electrometer, model 210, No. 185, manufactured by Keithley Instruments, Cleveland, Ohio. A Heathkit vacuum tube volt-

By C. D. Stahl, S. M. Farouq Ali, W. E. Culham

One- and two-dimensional mathematical models have been developed that simulate transient, two-phase flow of hydrocarbon mixtures in porous media in a manner that accounts for interphase mass transfer. Numerical simulations of one-dimensional depletion-drive experiments using a two-component hydrucarbon fluid were used to establish the validity of the mathematical models. In addition, the experimental and numerical data were used to demonstrate that production rate had a relatively insignificant effect on the recovery of individual hydrocarbon components from the experimental system, and that attainment of equilibrium between phases is possible for a wide range of liquid and vapor velocities in reservoirs containing light hydrocarbon fluids. Results of some two-dimensional numerical simulations are also presented. INTRODUCTION This study was undertaken to develop a mathematical model that would simulate transient, two-phase flow of hydrocarbon mixtures in porous media under conditions that result in interphase mass transfer and to test the validity of the assumptions used to set up the model. In addition, the study was designed to determine if production rate influences the recovery of individual hydrocarbon components from reservoirs producing by depletion drive. Two-phase flow in porous media, with interchange of components between the two phases, is important in many petroleum recovery processes. Studiesl-5 conducted within the last 3 years have outlined methods of solving multiphase flow problems incorporating mass transfer. Some of these studies have also indicated the importance of accounting for mass transfer under various producing conditions. An earlier work6 first demonstrated the importance of combining relative permeability data with equilibrium ratios in compositional balance methods. The mathematical model presented in this paper is formulated so that a phase behavior package, as described in previous papers,5,17 is not required as an integral part of the routine employed to solve for the primary dependent variables. The finite difference formulation is designed so that all the primary variables can be solved for simultaneously. This is accomplished by utilizing one basic set of equations. These innovations, which are in contrast to other models2,3,5 but are similar in some respects to the approach used by Taylor,l render the total problem computationally simpler than any of the previously referenced formulations. The mathematical model was developed by combining Darcy's law with a continuity equation for each hydrocarbon component. The principal assumptions invoked in the formulation were (1) that capillary forces and diffusional effects are negligible, and (2) that thermodynamic equilibrium exists in the reservoir at all times. No assumption as to the type of vaporization process was made in formulating this model. Experimental data were required to complete this study. These were generated by conducting several depletion drive experiments. The experimental apparatus consisted of a sandstone core enclosed in a pressurized casing. The apparatus was designed in such a manner that the core could be charged with a liquid hydrocarbon mixture and depleted at different production rates. The experimental tests were designed to determine the effect of production rate on component recovery. In addition, direct comparison of experimental and mathematically predicted

Jan 1, 1970

By J. L. Huitt, IM. L. Slusser, E. E. Glenn, M. L. Slusser

This paper is concerned with: (1) an analysis and interpretation of the filtration characteristics of drilling muds on filter paper, and (2) an interpretation of early stage "filtration" on consolidated synthetic porous media, herein referred to as the "surge loss" period. It is shown that water-loss data can be plotted so as to disclose the existence of a surge period, a non-uni-form cake thickness period, and a constant pressure filtration period. The same periods occur during exposure of consolidated porous media to muds. The data on consolidated porous media indicate that: (1) mud particles bridge pores inside porous media during the surge period, and (2) "filter cake" build-up inside porous media occurs during subsequent filtration. The data also show that the surge loss increases with sample permeability and with pressure differential. INTRODUCTION The phenomenon of mud filtrate invasion into porous sands of a drilling well has been recognized for many years.1,2,3,4 Part of API Code 29 is concerned with the filtration testing of a drilling mud to yield data on this subject. In a number of investigations "plastering properties" of muds were studied,5,6,7,8 "plastering" apparently meaning cake forming properties. In such studies the characteristics (thickness, permeability, and toughness) of the cake and the rate at which the liquid phase of the mud was separated from the solids (filtration rate through the cake, commonly referred to as water loss) were measured or at least considered. From the standpoint of its effect on well productivity, filtrate invasion into "clean" water-wet pay sands (i.e., those containing no clay minerals) may not necessarily be a serious problem. An increase in water saturation in the zone immediately around the wellbore would cause loss of oil permeability. However, in such a system, much of this water may be removed by oil production. If filtration occurs into "dirty" sands (i.e., those containing appreciable clay minerals), clay swelling or other mechanisms may cause losses in oil permeability which cannot be overcome by oil production.'.' The phenomenon of mud particle invasion into the pay sand of a drilling well was considered to be a possible serious factor in the early days of rotary drilling. Opinions varied as to whether drilling fluids "mudded up" and "sealed" oil sands. In 1932 Rubel,9 Gill,10 and Parsons," in separate papers, presented differences of opinion together with some laboratory and field data concerning loss of oil production caused by drilling fluid invasion into pay zones. These papers, and the discussions, furnish a large background on this subject and indicate the very clear thinking of many men at a time when petroleum production technology was in its infancy. Jones and Babson,6 and Silent,' experimentally found that the depth to which mud particles invaded water-saturated sand packs was small (no greater than 1/16 in). Arn-quist12 in 1937 observed loss of water permeability due to invasion of mud particles into water-saturated, consolidated, sandstone samples. Williams and Cannon5 studied the filtra-tion properties of drilling muds and mentioned "sludge" which penetrated a short distance into the water-saturated filtering medium (sand pack). Byck5 concluded that formation permeability had no effect on the plas-

Jan 1, 1958

By J. H. Henderson, L. A. Wilson, R. L. Gergins, R. J. Wygal, D. W. Reed

This paper presents a method of predicting the production history of an underground combustion recovery process. A rigorous solution of the thermodynamics and hydrodynamics involved is beyond the scope of this report. However, a practical scheme to appraise combustion recovery performance has been Worked out. By prudent assumptions, regarding the attainment of steady-state conditions, a trial-and-error solution, which satisfies both material balance and three-phase relative permeability requirements has been evolved and tested. As a combustion zone moves through a formation the interstitial water, the water of combustion and a portion of the oil wil1 be vaporized. These vapors will move downstream to a cooler region of the formation where they will condense. Part of the remaining oil will be displaced by the imposed gas drive and what remains behind will be consumed as fuel. The fluids will distribute themselves downstrean? in a manner which will satisfy the three-phase relative permeability characteristics of the formation. Three distinct zones will exist ahead of the combustion zone: (I) a zone termed a water bank which contains three mobile phases, oil, water and gas; (2) a region containing two mobile phases, oil and gas, called the oil bank; and (3) a zone having the fluid saturution distribution which existed prior to combustion. The mobile water will impose a water flood on the oil in the region containing three flowing phases. The process can be visualized as a simultaneous water .flood and combustion-supported gas drive. It is possible to estimate the saturation distributions in the water bank and oil bank as well as the production history for any combustion temperuture if the porosity, three-phase relative permeability characteristics, oil viscosity and initial saturations are known. This has been done for injection pressures ranging from I to 100 atm, oil viscosities of 10 to 1,000 cp and porosities from 20 to 40 per cent. Several of these calculations have been checked in the laboratory, with the result indicating rather good agreement between the predicted and observed production histories. The analytical and experimental techniques are described and the comparison of predicted and observed performance presented. INTRODUCTION Considerable research by the oil industry during the past 10 years has been directed toward the thermal recovery of crude oil. Of all schemes considered, the one receiving the most attention is in situ combustion. This is a process in which part of the oil is actually burned within the reservoir rock. The uniqueness of the process lies in its potential. Theoretically, this potential generates from the following: (1) it may open the door to vast reserves previously not economically producible, (2) it may induce increased production rates where low rates are now realized and (3) it may give high over-all recovery efficiency. The general idea of the process has received considerable attention in the literature '-' and the following brief description is intended for orientation only. In situ combustion can be visualized by considering a linear system, one end of which represents the injection well and the other the producing well. The formation at the inlet end is raised to ignition temperature by placing a gas or electric heater at the face of the formation or by the ignition of thermite, charcoal or similar material. While heating, the water and part of the oil in the heated region is vaporized and moved out into the reservoir by the injection of gas containing oxygen. In addition, part of the oil is displaced as liquid by the imposed gas drive. The oil not displaced is modified to an immobile, coke-like material (hereafter referred to as coke) by the high temperatures existing immediately ahead of the combustion zone. This coke is used as fuel, and the combustion front advances through the formation at a temperature which is probably in excess of 800°F. As the combustion front advancek it will drive ahead all interstitial oil not rendered immobile and all interstitial water plus the water formed by the combustion of hydrogen. As a result the sand behind the combustion zone is freed of all oil and water. Although the general description of the process has been outlined in the literature, there has been no consideration of the behavior of the fluids moving downstream from the combustion front. It is the purpose of this paper to present a technique

By J. C. Savins

The practical analysis of the hydrodynamics of the wellbore has long been a subject of interest to engineers. This paper presents a simplified solution to the problem of computing the pressure drop for the flow of drilling mud in the annulus of the wellbore. This solution is, however, an exact and rigorous solution under the assumptions which have been imposed. These assumptions are that the drilling fluid is a Bingham plastic fluid* and that the annulus is formed by two concentric, stationary, cylindrical pipes. It is further assumed that the fluid is incompressible and that its motion is isothermal and in a steady state. This problem under the same assumptions has been attacked by previous authors. Beck, Nuss and Dunn' proposed that the equation for the flow of a Bingham plastic fluid in a cylindrical pipe could be applied to an annulus if the pipe radius were replaced in the equation by the hydraulic radius. This equation, known as the Buckingham-Reiner equation' (see Appendix 1), was also used in an approximate form. Van Olphen pointed out that even for a simple or Newtonian- fluid the pipe equation (Poiseuille's law) could not be converted to the Lamb equation escriptive of flow in an annulus (see Appendix 1) by using the hydraulic radius. Van Olphen further attempted to give a solution for the annular flow of a Bingham plastic fluid by introducing approximations similar to those which have been used in the case of the Buckingham-Reiner equation. Other attempts to provide approximate or exact solutions have been made by Grodde' and by Mori and Ototakeq. The present authors some years ago in unpublished work derived the correct expressions relating the pressure drop and flow rate for this problem. It was found that the solution consisted of two simultaneous equations, one of which contained a logarithmic term. Thus, obtaining numerical results for any particular case of interest involves very tedious trial-and-error computations. Very recently Laird presented the correct derivation of the two equations which are given in full detail in Appendix 1. In order to reduce the amount of calculation time which would be involved in providing a complete tabular or graphical solution to the problem, a high-speed electronic digital computer has been utilized. For this purpose the two simultaneous equations were transformed into more compact expressions by introducing reduced variables. These expressions are given in the following theoretical section. A similar procedure in this problem has been developed by Fredrickson and Bird1" Their tabular results, however, are very incomplete in the range of practical interest for problems of wellbore hydrodynamics. We have furthermore been able to express our graphical results in terms of convenient and familiar dimensionless groups. THEORETICAL DEVELOPMENT Use of Reduced Variables In terms of reduced variables the two simultaneous equations just discussed take the following form, The reduced variables q, x, a and z are defined in terms of the various measured quantities, where Q is volumetric flow rate, AP/L is pressure gradient, Dl is OD of inner pipe, D, is ID of outer pipe, is plastic viscosity, and is yield point. Thus, we have a dimensionless volume flux, a dimensionless reciprocal pressure gradient, and the ratio of the pipe diameters Before introducing the fourth reduced variable, z, it is of interest to consider the physical significance of the parameter x. As may 'be seen from the velocity profile of Fig. 1 the Bingham plastic fluid has the interesting property that a portion of the stream flows at a uniform velocity without shearing action. This section of the stream is situated approximately in the center of the conduit and is known as the "plug flow" region. Its existence is due to the fact that the shearing stresses within the region do not exceed the yield point, which is one of the two flow properties characterizing the fluid. The parameter x then turns out to be the ratio of the

By C. S. Matthews, H. C. Lefkovits

The performance of depletion-type reservoirs at the stage in which the gas is at such low pressure that gravity is essentially the sole driving force has been investigated both theoretically and by use of scaled models. A simple theory is developed which completely explains the scaled model experiments. It also agrees with some field data. When the range of production rates to be considered is less than 25, the theory predicts that the production rate decline curve for homogeneous reservoirs in which there is a free surface will be of the hyperbolic type with n = 2: The theory gives a physical significance to the parameter y. INTRODUCTION This investigation was undertaken to determine a method for prediction of performance of reservoirs where gravity is the sole driving agent. Depletion-type reservoirs fall into this category after the pressure has fallen to some low value. Some examples of this type are found at Healdton ('Oklahoma), Oklahoma City (Oklahoma), Midway-Sunset (California), and East Coalinga (California). Generally the production rate per unit thickness is quite low in such reservoirs, with the conse- quence that this stage is sometimes called the stripper stage. However, per well production rates may be of the order of 50 B/D because of the considerable sand thickness open in some of the reservoirs. In the past, predictions of behavior of such reservoirs have been based on empirical extrapolation of the rate of production to the economic limit. Difficulties arise when new wells are drilled in the field or when production is artificially curtailed, since in such cases the decline rate of individual wells or leases changes and no clear trend can be detected. It was hoped that the present study would lead to improved methods for evaluation of such reservoirs. The study was initiated by building a scaled model for a particular field and by running a number of experiments on this model. Later a simple theory was developed which explained the laboratory results completely. The theory led, as shown below, to a hyperbolic-type decline equation which had been used previously' in a completely empirical manner. THEORY The General Case The problem to be treated here is the following: We are given a homogeneous cylindrical porous reservoir with exterior radius r, wellbore radius r., and height h,, containing, from the bottom up to a height hit oil, some gas, and interstitial water (see Fig. 1 A). The portion of the reservoir between K, and h, is filled with gas at low pressure, with residual oil, and with interstitial water. If at a time t = 0, the level of the oil in the wellbore is lowered to a height h,, what is the production rate from the reservoir at a time to The configuration within the reservoir at some instant

Jan 1, 1957

By C. W. Volek, G. Mandl

Steam-injection tests in the field have shoum that heat transport into the oil/water region, ahead of the steam zone, may have a significant effect on the production process. Earlier theoretical work on reservoir heating by hot-fluid injection can be used to describe the growth of the steam zone, but only if heat transport from steam zone into oil/water region is neglected. It therefore became necessary to re-examine and extend the earlier theory. It was found that this theory ceases to be consistent with the physical model of the process at a certain critical time, tc, which depends on reservoir thickness, temperature, and quality of the steam. The critical time marks an important change in the heat flow across the condensation frovt: the heat flow, which is purely conductive during 0 < t < tc&apos; becomes predominantly convective at t = tc. Accordingly, at t = tc, the equation which governs expansion of the steam zone changes. On the basis of the new equation, an approximate description for the steam-zone growth during t > tc is developed by making use of upper and lower bounds for the exact solution of the problem. The boundary conditions for heat and mass flow inside the oil/water region are established, and a simple method of detennining the saturation at the downstream side of the condensation front is presented. Experimental results show that the theory gives an accurate description of steam-zone expansion before the critical time. At times near critical time, and also after critical time, experimental data show theoretical steam-zone volumes to be high. Even so, the accuracy of the calculated steam-zone volumes is sufficient for practical purposes. THEORY OF STEAM-ZONE GROWTH INTRODUCTION The engineering evaluation of steam-drive processes is based mainly on a mathematical description of reservoir heating by hot fluid injection presented several years ago by Marx and Langenheim.l This theory can be used to determine the growth of the steam zone,2 provided the flow of heat from the steam zone into the liquid zone ahead of the condensation front (Fig. 1) is neglected. Since we must expect, however, that heat transport into and inside the liquid region (the region between condensation front and production wells) will affect both the water/oil flow and the growth of the steam zone, it is necessary to re-examine and to generalize the Marx and Langenheim theory as applied to steam drive, which will hereafter be termed the "old" theory. Observations made in several field tests emphasize the need to study the heat flow across the condensation front (CF). On several occasions it was discovered that heat was transported in the liquid zone far ahead of the advancing CF, resulting in an early breakthrough of warm water. This seems to occur most often when high injection pressures are applied, when "wet" kONINKLIJKE/SHELL EXPLORATIE EN PRODUKTIE LABORATORIUM RIJSWIJK, THE NETHERLANDS SHELL DEVELOPMENT CO. HOUSTON, TEX.

Jan 1, 1970