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By R. E. Cook

This paper presents results of a study to determine to what extent errors in estimated free gas saturation affect the results of static pressure calculations from build-up curves in two-phase systems. Use is made of the method of pressure build-up curve analysis developed by Miller, Dyes and Hutchinson1 , with modifications for two-phase flow advanced by Perrine2. Cornhined fluid compressibility of gas-oil mixtures is included, with special attention given to the discontinuity that occurs in fluid compressibility at the saturation pressure and its effect on static pressure calculations from build-up curves. The sensitivity of static pressure calculations to errors in gas saturation is considered over wide ranges of formation and fluid properties and reservoir pressure conditions. It is concluded from this study that for analyses of two-phase pressure build-up curves (I) the use of under-saturated crude compressibility data, even when producing GOR's are at or near solution values, will result in very significant negative errors in calculated static pressrrre, and (2) except for large values of buildup curve .slope and/or very low crude gravities, rather large errors in calculated gas saturation can he tolerated without .significant effect on calculated static pressures. The conc1usions arc qualified as referring only to the effects of gas saturation and excludc possible errors in the nuumerous other factors that enter into the ancllysis of pressure build-up curves. INTRODUCTION As part of the analysis of basic data for several major Eocene reservoirs in the Bolivar Coastal field (Western Venezuela), it was necessary to analyze a large number of pressure build-up curves for oil permeability and static reservoir pressure. From performance data it became apparent that gravity segregation is very active in these reservoirs. As a result of this performance it was recognized that the free gas saturation within the oil leg of the reservoirs would be small but would be subject to certain errors in calculation due to the inaccuracy of relative permeability data in the low gas saturation region. Since all of the various methods of build-up curve analysis were originally derived for sin- gle-phase, constant compressibility systems, and since the presence of a small but rather indeterminate gas saturation would greatly affect the reservoir fluid compressibility, a study was undertaken to investigate the effect of gas saturation on the results of two-phase pressure build-up analyses. TWO-PHASE BUILD-UP CURVE ANALYSIS The method of analysis used in this investigation is basically that originally developed by Miller, Dyes and Hutchinson1, with modifications for two-phase flow advanced by Perrine2. Miller, et al, showed that the original equation for single-phase permeability away from the well bore, held for two-phase flow except that the calculated permeability is the effective permeability to oil away from the wellbore. This finding was acknowledged by Per-rine, who also presented the equation for effective gas permeability* as Since Eqs. 1 and 2 contain no terms influenced by fluid compressibility, no distinct problem should develop in calculating permeability from two-phase build-up curves. In regard to the calculation of static reservoir pressure from build-up curves in single-phase systems, the modified Miller, et al, method as presented by Perrine is briefly as follows. 1. From build-up curve data, PVT data for the reservoir fluid and the well spacing and sand thickness, calculate the climensionless shut-in time corresponding to the highest value of actual shut-in time (t, hrs) on the straight-line portion of a semi-log build-up curve. Dimenrsionless shut-in time is related to actual shut-in time by the relation, t- 0.000264 kt 2. From a graph of dimension less shut-in time vs dimensionless pressure difference (Fig. 1 1 ) find the value of dimensionless pressure difference corresponding to the value of 7 calculated under (1) above. Dimen-

By I. Fatt

Study of a model which contains "dead-end" pore volume indicates that pressure transients are influenced by the amount of dead-end pore volume and by the resistance of the flow path between the dead-end pore volume and the main flow channels. A semi-empirical method is demonstrated that makes possible measurenment of the volume in the dead-end pores and the total pore volume. The difference between these two volumes is the volume of the flow channels. An indication of the flow resistance between the dead-end pore space and the lain flow channels is also obtained. INTRODUCTION In some engineering calculations the distribution of pore volume in a petroleum reservoir need not be known. For example, a material balance calculation treats a reservoir as an oil-filled tank. In other calculations the distribution of pore volume appears implicitly through use of the porosity and the relative permeability and capillary pressure characteristics. It is the purpose of this paper to point out that under certain conditions the exact distribution of pore volume must be known in order to interpret correctly transient flow behavior. Pressure transients during single-phase flow in porous media are usually said to obey differential equations of the form

By G. W. Nabor, M. Mortada

The effects of anisotropic or directional permeability on the areal sweep efficiency and the flow capacity are examined. The paper points out the importance of taking directional permeability into consideration in planning a flood. It analyzes the two-dimensional flow pattern associated with the skewed line drive for a unit mobility ratio. The direct and staggered line drives are treated as special cases of the skewed line drive. Analytical expressions are developed for the areal sweep efficiency at breakthrough and the flow capacity. They are related to the spacing between like wells, the distance between a row of injectors and the nearest row of producers, and the degree of skewness of the line drive. The latter quantity is defined such that it is equal to zero for the direct line drive and equals one-half for the staggered line drive. The areal sweep efficiency and the flow capacity depend also on the orientation of the flood pattern with respect to the principal axis of anisotropy. The paper provides a simple method for determining the areal sweep efficiency and the flow capacity for a formation in which the permeability in the bedding plane is anisotropic. INTRODUCTION Directional or anisotropic permeability is manifested by the ability of the formation to conduct fluids more readily along certain preferred directions. This situation occurs in many producing formationsl-4 and is usually attributed to depositional features in which the sand grains are oriented in a preferred direction. In some cases it results from the formation of a major and a minor fracture system. 4 Directional permeability should be taken into account in many phases of the production and exploration activities. Recognizing its existence in the formation of interest and planning accordingly can lead to increased recovery and substantial savings. For instance, the areal sweep efficiency in a water flood depends to a great extent on the orientation of the flood pattern with respect to the principal axis of permeability. Anisotropic5 permeability is specified by the directions of its three principal axes and the permeability along each axis. The principal axes of permeability are mutually perpendicular. This paper deals with the areal sweep efficiency at breakthrough and the flow capacity for formations with aniscltropic permeability. The flood pattern considered consists of alternate rows of injecting and producing wells. The rows of wells are parallel and form a developed, skewed line drive which is illustrated in Fig. 1. The staggered and direct line drives are treated as special cases of the skewed line drive. PRELLMINARY CONSIDERATIONS The formations considered in this paper are assumed to be of uniform thickness and porosity. The permeability is uniformly anisotropic, with each principal axis of permeability maintaining

By L. W. Emery

Undergrorlnd combustion operations were initiated in a 60-acre Bartlesville sand "shoe-string" reservoir in Allen Connty, Kans., in 1956. Tests in separate patterns were conducted using various co~nbinations of air and recycle gas to propagate combustion fronts from the injection toward the producing wells. These patterns were made up of 6 injection and 20 prodrrcing wells Gas and liquid prorluctiorz from each pattern was measured on an individual-well basis, and comparisons were made between the three patterns to ascertain the relative effects of injected gas composition on production behavior. Breakthrough of the combustion front at a well was characterized by an increase in water production from the well followed by an increase in bottomhole temperatrrre to approximately 250" F. After burning fronts had broken through at five producing wells, operations were terminated in 1960. From the total project approximately 79,000 bbl of oil were produced during thermal operations at a cumulative produced GOR of 23 Mcf/bbl. No appreciable change in the character of the produced crude was observed. Combustion in the reservoir was maintained with injected gas compositions ranging fronz 6 per cent oxygen in recycle gas to 100 per cent air. lnjectiotz of large quantities of recycle gas resulted in higher producing GOR's from offset wells than were measured from a pattern into ~vhich straight air ~vas injected. The air required to move the combustion front through I acre-ft of reservoir was computed to be 20 MMscf. This valrre was found to be relatively independent of the quantities of recycle gas injected. The recovery efficiency from the swept area was esti~izated to be about 59 per cent. Areas swept were similar in shape to tlzose obtained with a laboratory potentiometric model. Samples of sund taken from behind the burning front by coring indicated almost total oil removal from the sand. Petrographic analysis of the core samples indicated that the sand had been heated to peuk temperature of rlbout 1,200" F. No rignificant difference in peak temperature was forrnd in two areas where compositions of injected gas were quite different. Compression costs for thermal recovery were estimated to be $1,20/bhl of produced oil. INTRODUCTION The use of the "forward combustion" process as an oil recovery method has received a great deal of attention. This method involves ignition of the formation in an injection well, followed by propagation of a combustion front through the reservoir. Combustion is maintained by the injection of an oxygen-containing gas to react with reservoir hydrocarbons. As the flame front progresses through the reservoir, oil and formation water are vaporized, driven forward in the gaseous phase and recondensed in the cooler part of the formation. In turn, the condensed fluids push oil into the producing wellbores. Completed field tests of the process were first reported by Kuhn and Koch,' and by Grant and Szasz.' Results from other tests have since been reported by Walter,3 by Moss, White and McNeil,' and by Gates and Ramey." ach of these tests essentially utilized a single injection well surrounded by four or more producing wells. Sinclair Research, Inc., elected to do field experimental work using a number of test patterns in a single field in order that comparisons between various operating schemes could be made. The site selected and purchased in 1955 for this experimental work was a 60-acre Bartlesville sand reservoir located in Allen County, Kans. Combustion operations were initiated in mid-1956. Between that time and termination of the project in mid-1960, combustion fronts were propagated from injection wells to producers in three separate well patterns, using different mixtures of air and recycle gas. The test was terminated before sweep of the three patterns was complete so that information about the effect of combustion on the swept areas could be obtained by coring. Results from the test in the form of injection and producing well performance have been carefully recorded, and these form the general basis for this paper. DESCRIPTION OF RESERVOIR The reservoir in which the combustion tests were conducted is a Bartlesville sand "shoe-string", typical of a number of small reservoirs in Southeastern Kansas. Average reservoir characteristics are shown in Table 1. Fig. 1 is an isopachous map of the producing sand showing the reservoir to be approximately 400-ft wide and 2,500-ft long. Maximum net productive sand thickness is 21 ft. Fig. 2 shows a typical core analysis obtained by coring with water-base mud. The reservoir has no appreciable dip and is closed on the sides by degradation of sand into shale. The main body of sand is heavily laminated with shale stringers, which are not continuous between wells. The main reservoir is overlain by 30 to 40 ft of laminated low-permeability sand and shale streaks. No information is available on the original properties of

By B. Habermann

Artificially consolidated sand models, representing one-quarter of a five-spot, have been developed and used to study factors aflecting misciblt. displacrmenr. Sweep efficiency at breakthrough, size of the mixing zone between two miscible liquids and per cent area contacted by drive after breakthrough were determined tor the high mobility ratios encountered in actual resvrvoirs. Quantitative relationships between the degree of viscous fingering and mobility ratio were obtained by measuring the length of the fluid interface. Scaled miscible-slug experiments, supported by field evidence reported in the literature, have shown that when the slug is followed by dry gas the process is less efficient than expected. In addition to low areal sweep efficiencies encountered for high mobility ratio displacements, the effectiveness of a miscible slug is greatly reduced. This is caused by an accelerated growth of the mixing zone between the driving and the displacing fluids. INTRODUCTION Considerable interest in miscible displacement as a secondary recovery process has been shown by the oil industry in recent years. In addition to extensive laboratory investigations, numerous field operations have been initiated so that 39 miscible displacement projects had been reported' in the United States by early 1959. By far the most popular approach in these projects is the use of an LPG slug followed by dry gas. Laboratory tests have indicated that with a small miscible slug, amounting to several per cent of the hydrocarbon pore volume, practically 100 per cent recovery of the oil in place can be achieved and that all of the slug material can be recovered by a subsequellt gas drive. However, these experiments were conducted on small-diameter long cores that can be considered as essentially uni-dimensional reservoirs that do not represent actual field conditions. It soon became apparent that the inherently adverse mobility ratios existing in the reservoir cannot be ignored in any miscible displacement field project. When a less viscous fiuid such as an LPG slug drives a more viscous reservoir oil, fingers develop that reduce the areal sweep efficiency. The slze of the miscible slug to be used is a subject of some controversy. It has been proposed that the size of siug needed be calculated by setting lengths of the mixing zone proportional to the square root of the distance traveled. This proposal is based on experimental evidence' and theoretical considerations"' in which fingering was not a factor. A prerequisite for successful application of the miscible displacement process to field operations is a better understanding of the following factors. L What is the effect of mobility ratio on the rate of deterioration of a miscible slug? 2. How large a slug should be used for given reservoir conditions? 3. What is the sweep efficiency? The purpose of this investigation was to provide some additional laboratory data that should help to answer these questions. More specifically, it was intended to study more quantitatively the effect of viscous fiinger-ing, mobility ratio and slug size on the efficiency of this relatively new secondary recovery process by means of experiments on thin, horizontal, porous medium models.

By R. A. Salathiel, A. D. Kaiser, P. H. Monaghan, B. E. Morgan

The productivities of many hydrocarbon-bearing for-rnntions appear to be reduced during drilling, completing and repairing operations. The reason for these othcarwise unexplainable low productivities may be a drastic reduction of formation permeability immediately surrounding the borehole. A possible cause of the reduced permeability is swelling and dispersion of clays present in porous media when contacted by fresh water or mud filtrates. Hence, the invasion of fresh water or mud filtrate into hydrocarbon-bearing formations that contain clays may cause low productivities as a result of formation damage. Present laboratory investigations have yielded results which supplement existing literature on formation damage. An expanded-type clay, montmorillonite, way found in hydrocarbon-bearing formations not previously analyzed. When sand columns and formation cores containing montmorillonite were contacted by fresh water, their oil permeabilities were reduced. This reduction in oil permeability way restored partially but not completely by a repair procedure, i.e., flooding with kerosene containing organic agents or flooding with water containing electrolytes. Hence, where montmorillonite is present in formations, prevention of formation danzage was indicated to be the only complete solution to the problem. To prevernt formation damage, results suggested use of an approximately 1 per cent solution of calcium chloride or stronger solutions of sodium or potassium chloride. In conventional rotary drilling operations the wall of the hole is exposed to drilling mud for days, weeks, and in some instances for as long as several months. During this time, drilling mud filtrate seeps slowly into formations surrounding the hole. The permeability and stability of many formations, especially those containing clayey materials, are impaired by the mud filtrate. Often interaction between the mud filtrate and the formation causes drilling and production difficulties. Serious reduction in productivity of oil and gas formations during drilling, completing and repairing wells has been noted often in recent years. Cases are known of development wells which produced only small volumes of fluid upon completion. However, after a period had elapsed, they began to produce oil or gas in commercial quantities. In other cases, results of drill-stem tests taken during the drilling of deep wells have indicated good production should be obtained from shallow formations. Later difficulty has been experienced in obtaining production from these shallow formations which had remained in contact with drilling muds for an extended time. In yet other cases wells have failed to attain the productivities expected from previous performance after the use of drilling mud as a circulating fluid during workover operations. It is evident from the field behavior cited that the productivities of certain oil- and gas-bearing formations have been damaged by exposure to drilling muds. This reduction in well productivity, called "formation damage", causes serious consequences. In development wells, expensive remedial operations are incurred. In wildcat wells, good formations may appear to be marginal or even non-productive. The importance of formation damage has been recognized by a number of investigators1-s who have considered the effects of fresh water, saline solutions and aqueous mud filtrates on the permeabilities of formations. A general conclusion from such studies is that interaction between water and formation clays reduces permeability of reservoir rock and that the permeability decreases as salinity of the water decreases. Data in these references, however, concern the permeability change with only a single fluid present. A more pertinent and direct consideration in the production of oil and gas from wells is the permeability change when two fluids, oil and water, are present. Recognizing this, Nowak and Kreuger2 and Bertness1" considered several ways in which mud filtrates and saline solutions caused reductions in the oil permeability of formation cores. Investigations discussed here were designed to supplement information in the literature on changes in oil

By R. J. Wagner, F. F. Craig, H. G. Riley, J. D. Griffith

Peripheral water injection has been underway in the Sholem Alechem Fault Block "A" Unit, Stephens County, Okla., since 1955. In the engineering planning of the flood, it was recognized that maintenance of pressure in the large, dry gas caps was vitally necessary to prevent the loss of an estimated 27 million bbl of oil through saturation of the gas caps by the oil banks moving updip on the peripheral flood pattern. The development of an abundant, low-cost water supply and the rising cost of makeup-gas purchases indicated that gas-cap control by water injection was desirable from an economic standpoint, provided an eflective water barrier could be placed without premature water breakthrough into downdip oil producers. A laboratory model scaled to the existing reservoir conditions was comtructed to evaluate the feasibility of gas-cap water injection. Model studies indicated that (I) water injected at the edge of the gar-oil contact would form a continuous barrier between the oil and gas zones, and (2) water would advance uniformly downdip without segregation due to gravity effects provided a specified, minimum water-injection rate was maintained. The model test results also indicated that it was not necessary to fill the gas cap completely with water. Water injection at the edge of one of the gas caps was initiated in March, 1958, to field test gas-cap water injection. Water breakthrough occurred at a downstruc-ture producing well in April, 1959, after approximately 2.5 million bbl of water had been injected into the gas cap. This was the volume calculated to be required to form the barrier between the oil zone and gas cap. This performance indicates that prevention of oil encroachment into dry gas caps in a peripheral water flood by water injection at the gas-oil contact appears practical and that the model studies provided a valid prediction of field performance. INTRODUCTION The Sholem Alechem Fault Block "A" Sims Sand Unit was one of the fist major waterflood projects in Oklahoma to utilize a peripheral injection pattern. Design of control measures to minimize oil encroachment into the large crestal gas caps was one of the greatest single problems during engineering planning of this flood. Engineering studies indicated that saturation of the gas caps with oil displaced by the peripheral water flood could result in a loss of about 27 million bbl of oil, or over one-half the estimated waterflood recovery. This estimate was based on a residual oil saturation after water flooding of 35 per cent pore volume, compared to the negligible oil saturation initially in the gas caps. The Sholem Alechem Springer field, located in Stephens County, Okla., is an elongated anticline with structural dips varying from 400 to 1,000 ft/mile. The structure contains numerous faults, some of which are sealing. The portion of the field considered in this paper is Fault Block "A", which is the largest of the five major fault segments in the field. A map of the unit is shown on Fig. 1. The unit is bounded on the east and west by major sealing faults, on the north by a water-oil contact and to the south by a deterioration of sand development. Two domes, each having gas caps, exist in the unit and are separated by a north-south fault as shown in Fig. 1. The sands being flooded are the Sims sand members of the Springer series of early Pennsylvanian age. The Sims sands average over 100 ft in thickness in Fault Block "A" and have a porosity of approximately 19 per cent. Core permeabilities average 160 md, with interstitial water saturation of about 23 per cent pore volume. Three different zones can be correlated over most of the area. However, they are commingled in most of the producing wells. The two upper zones, designated as First and Second Sims, are the most important. Both the First and Second Sims have gas caps in the east dome, while only the First Sims has a gas cap in the west dome. These gas caps, whose limits are also shown on Fig. 1, initially occupied nearly 11 per cent of the total reservoir volume. The major concern of this paper is the control of the two gas caps in the First Sims sand. Fig. 2 is a cross section, to scale, across the east dome of this formation. Notable features are the relatively shallow dip, the deterioration of the First Sims sand on the south side of the gas cap and the water-oil contact on the northern edge. The formation dip at the gas-oil contact in this cross section is approximately 5".

By G. Thodos, W. F. Stevens

The point-source function introduced by Horner' us a solution to the general unsteady-state equation for the flow of fluids through porous media has been utilized to calculate pressure profiles for adjacent wells as a function of time. By trial and error, the time at which the pressure waves meet to cause interference can then be calculated. For this calculation, the time of interference is arbitrarily defined as that time when both pressure waves exhibit a pressure decrease of 25 psi ar the same point within the formation. To illustrate the method, an example involving two adjacent wells producing at different rates has been worked out and is presented in detail. INTRODUCTION The initiation of production of a new well is usually accompanied by pressure drawdown measurements in order to establish the extent and characteristics of the formation. The interpretation of such pressure drawdown data is based on mathematical developments involving a single well in an infinite reservoir. If more than one well is present in a reservoir, the possibility of interaction between wells exists. Therefore, care must be taken in interpreting the pressure drawdown data. It becomes important to be able to determine when this interaction, or interference, becomes significant. This paper presents a method for the prediction of the approximate time of interference between two such adjacent wells. The basic differential equation describing the pressure distribution existing during unsteady-state flow of a slightly compressible fluid has been solved by Hor-ner.' for a constant production rate from a circular homogeneous reservoir of infinite extent. The solution utilizes the point-source function to define the pressure at various points within the formation as a function of time. The resulting expression is: q = production rate, STB/D ,;= formation volume factor, dimension-less µ = fluid viscosity, cp k = formation permeability. md H = formation thickness, ft C - fluid compressibility, psi' F = formation porosity, dimensionless R = distance from centerline of well, ft T = production time, hours Ei(—u) = Ei-function of U Eq. 1 may be used to calculate the pressure profiles generated during production from a single well in an infinite reservoir.

By Floyd C. Kelley

The literature of the last decade is rich with information relating to the cause and means of control of grain growth in pure metals, but is deficient concerning the role diffusion plays in grain growth, although during this period many new products that depend on diffusion for their properties and success have been developed. Among these are cementation products, produced by calorizing, chromizing, and carbonizing, or by mixing, pressing, and sintering powdered metals such as Syke's iron-molybdenum die material, Genelite, and Elcon Metal (copper, tungsten) etc. This is a fertile field for new developments, but before it can be utilized to the fullest extent there must be a thorough invest,igation of the processes by which diffusion and grain growth take place. Grain growth plays no small part in the process and certainly the physical properties of these products depend largely on the grain size and state of equilibtiuni produced. The field of investigation does not end here, but may be extended to the effect of diffusion on impurities, segregations and dendrites in the broad field of ferrous and non-ferrous metallurgy. A number of years ago the author investigated several of these cases and noted, with striking regularity, the pronounced effect of diffusion in promoting recrystallization and grain growth. An example of this phenomenon is given in his paper on chromizingl which contained photomicrographs of large radial or columnar crystals produced by diffusion of chromium in the solid state. Another manifestation of this effect was discussed by Austen,2 who showed the part hydrogen plays in the decarburization of steel and the effect of diffusion upon the grain structure. Diffusion of Metals in Iron It seems to be a general law, whenever two metals are brought into intimate contact at a temperature at which diffusion takes place, that a comparatively rapid, or abnormal, grain growth results. The grains

By G. W. Tracy, R. D. Carter

Numerical calculations were made to determine the behavior of reservoirs with high-pressure drawdown and wide well spacing where the initial productivity is low and the wells are completed by hydraulic fracturing. The two-phase flow equations were solved for the flow into a single well. This well was assumed to be producing from a reservoir with hydraulically created horizontal fractures (four different systems with fractures were studied). For comparison purposes, additional two-phase flow calculations were made assuming a reservoir with uniform rock properties. The two-phase flow results were also compared with the conventional calculation methods, which do not include the effect of saturation gradients resulting from a simultaneous flow of oil and gas which are normal to this type reservoir. It was found that the conventional methods predict (1) a high and too optimistic value of ultimate recovery, (2) a high producing rate and a high reservoir pressure at a given oil recovery and (3) a low trend of gas-oil ratio with oil recovery. Included in the two-phase flow calculations were provisions to control the oil production rate by an allowable rate and, also, by a gas-oil ratio penalty rule. For the systems with hydraulic fractures, the producing rate was controlled by the gas-oil ratio penalty rule for most of the life. This is in contrast to the system with uniform rock properties which went "on decline" almost immediately. An unexpected characteristic of the systems which included fractures was the early rise in producing gas-oil ratio from 730 cu ft/bbl to approximately 1,200 cu ft/bbl, followed by a "leveling off" before the normally expected gas-oil ratio rise began. Additional features which are a result of hydraulic fracturing are (I) greater ultimate recovery, (2) higher average producing rates and (3) a lower average reservoir pressure at a given oil recovery. INTRODUCTION Some oil fields discovered during the past few years are producing from certain volumetric ally controlled reservoirs (often referred to as solution or internal gas- drive reservoirs) which are characterized by high-pressure drawdown at the wells. Since the available pressure drawdown at a well is limited by the static reservoir pressure and the producing rate is controlled by the available drawdown, wells completed in this type of reservoir usually produce at a rate less than the allowable from the time of completion. Because of this, this type of reservoir is referred to as a low productivity reservoir. Economic considerations require the use of wide well spacing and well stimulation by hydraulic fracturing to make commercial wells in this type of reservoir. Performance predictions for volumetrically controlled reservoirs have been made using a combination of two standard equations. 1. The "Schilthuis" or "Muskat" type material balance equation is used to relate the average reservoir pressure and the cumulative oil recovery. 2. The results from the material balance equation and the productivity factor as described by Pirson' are used to relate the cumulative recovery with producing rate and time. The material balance equation assumed uniform pressure and liquid saturation conditions throughout a reservoir. The steady-state radial flow formula allows for a pressure gradient toward a well but assumes uniform liquid saturation. These calculation methods are adequate for application to reservoirs wherein the drawdown at the well to realize satisfactory producing rates is small compared to the total pressure. In low productivity, volumetrically controlled reservoirs, the pressure drawdown at the well is large cornpared to the total pressure. Although a precise number cannot be given for the magnitude of a large pressure drawdown, values in excess of 1,000 psi would definitely be included. For practical considerations, this usually occurs when the formation flow capacity is less than about 100 md-ft. However, this limit of formation flow capacity will vary with the well producing rate. The low pressure in the neighborhood of the well which results from a high drawdown causes evolution of large volumes of gas. This causes the gas saturation to be higher near the well than at a greater distance—-hence, a non-uniform gas saturation. Also, the relationship between the relative permeability to oil (K/K) and gas saturation is nonlinear but decreases approximately in an exponential way with increases in gas saturation. Because of this, the following chain reaction is established.

By P. B. Baxendell

The performance of a water-drive reservoir having a gas cap depends primarily on the movement of the gar-oil and oil-water contacts. The movement of the contacts during production depends in turn on fluid withdrawals and how the reservoir pressure changes as fluids are produced from the reservoir; that is, on how effectively the aquifer maintains pressure by replacing withdrawals. inasmuch as pressure changes, fluid withdrawals, contact positions, and produced gas-oil and oil-water ratios are interdependent, the analysis and prediction of the performance of a reservoir produced in a given way must take into account this interdependence throughout depletion. This paper presents an analysis which, within the limitations of the assumptions made, yields an engineering approach to predicting future performance based on reservoir pressure and production history. The most significant assumption is the method of extrapolation of future gas-oil and water-oil ratios. The extrapolation procedure smoothly increases both ratios to preselected values as the remaining oil column undergoes a specified decrease in thickness. This preselection is made on the busis of previous field experience in depletion of similar reservoirs under similar conditions. For computing future performance a vdumebic balance L combined with the differential equation defining pressure distribution in the aquifer to obtain positions of water-oil and gas-oil contacts. From these positions are extrapolated produced water-oil and gas-oil ratios. Reservoir performance can be investigated when oil production rates are dependent upon various factors including the performance of the reservoir itself. Examples of practical application of the procedure are included. INTRODUCTION To predict performance of water-drive reservoirs with gas caps and thin oil columns, it is necessary to describe the motions of the fluids within the reservoir during the entire production period to depletion. These motions depend on the pressure changes and on the withdrawal of oil, gas and water. Production of oil from the oil zone primarily causes water to move in to take its place; production of gas from the cap tends to cause oil to migrate into its place. In addition, the volumes of oil and gas remaining in the reservoir depend on changes in the pressure, since a decrease in pressure causes fluid expansion, gas liberation and oil shrinkage. A method of relating the future pressure to total withdrawals is used to describe the motions of fluids within the reservoir under conditions arising in possible modes of production. The analysis is based on two relationships and will reduce the problem to one amenable to digital computation. The first of these concerns the dependence of the pressure distribution in the entire aquifer furnishing the water drive upon the total reservoir withdrawals. This dependence is dictated by the permeability distribution and the extent of the aquifer; at present, such information is most readily obtained from the performance history by means of the resistancecapacitance reservoir analyzer. The second relation involves withdrawals, pressure in the reservoir, and movement of oil, gas, and water within the reservoir. The analysis is subject to certain simplifying assumptions that are necessary to permit solution of the problem. A comparison of the methods of this paper with those presently practiced is pertinent. One method of analysis is to use the reservoir analyzer to predict reservoir behavior based on aquifer characteristics determined from production history. Inasmuch as the total withdrawal rate depends upon the gas-oil and water-oil ratios, which depend in turn upon, among other things, the positions of the gas-oil and water-oil contacts, these ratios must be assumed in advance. In this paper these ratios are, instead, related to the computed positions of gas-oil and water-oil contacts. Thus, our method can be applied to problems in which the oil production rate is limited by produced gas-oil ratio, or to problems in which it is desired to determine the variable gas injection rate that will maintain the gas-oil or water-oil contact stationary. These problems cannot be worked satisfactorily on the analyzer. Another method presently used is based on a paper by Hurst.' Since his procedure is dependent upon using the solution of the heat flow equation, which requires constant permeability within the aquifer, and our procedure recognizes variations of permeability in

By A. A. Stripling

The Cretaceous limestone wells of the Mara/Maru-caibo Dist. of Venezuela are extremely prolific producers. To maintain production on cessation of natural flow, large scale gas-lift operations were commenced involving high production rates on caring Flow. For the design of these gas-lift installations it was essential to have some knowledge of the pressure gradients involved in casing flow, so that the required injection pressures, optimum gas injection raws, etc., could be forecast. This paper presents a method of calculating these annular pressure gradients. Basically the method makes use of an energy balance equation coupled with an empirical energy loss factor derived from field data. The calculations have been put in a form suitable for "punch card-type" calculating machines. A set of gradient curves for Mara/Maracaibo conditions covering flow rates up to 12,000 B/D, and GOR's of 500 to 2,000 cu fd/bbl is presented. From these curves the flowing BHP can be predicted from surface data and the vertical pow performance under varying conditions of gas injection can be estimated. The method has been applied to La Paz and Mam gas-lift operations for over three years and has given consistently accurate results. INTRODUCTION The Cretaceous limestone fields of the Mara/Maracaibo area in Venezuela are characterized by the presence of widely distributed fissure systems. Under these conditions individual wells frequently have extremely high potentials and such wells are generally flowed on the casing-tubing annulus in order to reduce the pressure loss in vertical flow. To maintain production on cessation of natural flow, an extensive gas-lift system was planned. As a first approach to this gas-lift design problem, an attempt was made to construct a set of empirical flowing gradient curves from the available subsurface pressure data. Such a method has been used successfully for tubing-flow conditions in California by Gilbert'. Unfortunately, the empirical construction of casing-flow gradient curves presents additional difficulties, as pres- sure bombs cannot bc run in the casing-tubing annulus against high rates of flow. Thus, the Full flowing gradients cannot be defined by actual measurement and the best that can be done is to obtain spot pressures at the tubing shoe. Under these conditions a vast accumulation of BHP data is required to construct a set of gradients. To the La Paz and Mara fields the majority of the available flowing pressure measurements fall into a relatively narrow GOR range, in the neighborhood of the solution ratio. While there appeared to be reasonable prospects of constructing gradient curves for this limited range, measurements at higher GOR's, which were of major interest for gas-lift operations, were relatively few and the prospects of constructing empirical curves cor~spondingly remote. As an alternative approach, the possibilities of a mathematical analysis of vertical flow conditions were investigated. May and Laird2fV n 1934 presented an excellent analysis of the vertical-flow conditions in the Anglo-Iranian Oil Co.'s fields. Their analysis was based on an energy-balance equation and calculated the losses due to kinetic energy, friction and slippage separately. By combining these losses with the total calculated available energy, the required depth-pressure traverse could be deduced. On applying the method to La Paz conditions, good results were obtained for tubing flow at high tubing-head pressures. However, for lower wellhead pressures (300 psi and under) the discrepancy between calculated and measured values increased appreciably. It was considered that this was probably due to energy losses caused by slippage or liquid hang-up in a continuous gas phase, for which it was difficult to visualize any analytical approach. In 1952 Poettmann and Carpenter4 presented a method of pressure gradient calculation which, while based on an energy-balance equation similar to that employed by May and Laird, made no attempt to evaluate the various components making up the total energy loss resulting from irreversibilities of flowing fluids. Instead, they proposed a method of analysis which assumed that all significant losses for multiphase flow could be correlated in a form similar to that of the Fanning equation for frictional losses in single-phase flow. They then derived an empirical relationship linking measurable field data with a factor which, when applied to the standard form of the Fanning equation, would enable the energy losses to be determined. The total available

By D. R. Wieland, A. N. Barron, A. R. Hendrickson

A definite relationship has been found between the reactivity of flowing hydrochloric acid and its shear rate in a carbonate fracture. Both flow velocity and fracture width affect the acid reaction rate. Laboratory studies were conducted on acid reactivity at different flow velocities through horizontal-linear fractures, using 15 per cent hydrochloric acid at 80' F and approximately 1,100 psi. Fracture width varied from 0.02 to 0.20 in. These data provide a new basis from which the spending time and penetration of the acid can be estimated. Equations were derived expressing the relationship between injection rate, fracture width, acid concentration, time and fracture height, for linear and radial fracture systems. Because the penetration of the acid before spending is closely related to the extent of productivity increase resulting from an acidizing treatment, these data provide a valuable insight into some of the controlling factors that must be taken into consideration during treatment preplanning. INTRODUCTION Acidizing of carbonate reservoirs to improve production characteristics has been widely practiced since 1932. Originally, it was assumed that the acid uniformly penetrated natural formation pores and flow channels, enlarging them and thereby increasing their flow capacity. Little consideration was given to the reaction rate of the acid, or how far it would penetrate away from the wellbore into the formation, before spending. It has been shown recently' that, unless fractures are present in the rock, very little penetration is attained before spending, and the benefits of the acidizing treatment are largely confined to the immediate vicinity of the well-bore. Therefore, acid treatments may be classified into two categories: (1) matrix acidizing, in which the acid flows through multiple: formation pores; and (2) fracture acidizing, in which the bulk of the acid travels through fractures in the rock, whether natural or induced. When acidizing treatments are conducted at pressures of sufficient magnitude to open and extend such fractures, it is often desirable to inject a propping agent to hold the fracture open after the treating pressure has been released, thus providing a highly conductive flow channel through the rock. In some cases where acid attack produces surface irregularities, the resulting flow passages can be sufficient to provide high conductivity without use of a propping agent. During injection, the acid dissolves the carbonate rock with which it comes in contact until it is spent. Deeper penetration of the spent acid into the formation produces no appreciable benefit (if no propping agent is used) because the unetched fracture faces will rejoin when treating pressures are released and the fracture will "heal", with negligible resultant conductivity. The spending time of the acid thus becomes important in determining how far from the wellbore the improved-conductivity zone extends. The spending time of acid after injection into carbonate rock depends on the rate at which the acid reacts with the rock. This in turn is controlled by a number of factors, as previously reported. 2-4 These include temperature, pressure, acid concentration, rock composition, injection rate and the area-volume relationship between the acid solution volume and the surface area of the formation flow channels through which it penetrates. Thus, in matrix acidizing, where an extremely large area is exposed to the acid, spending is rapid. In contrast, spending time is prolonged in open fractures, where the area-volume ratio is much lower. Thus, greatest penetration of the acid before spending will be achieved in acid-fracturing treatments where fractures are held open by hydraulic pressure. Perkins and Kern5 have presented equations for fracture-width determinations which, for hard formations such as limestone or dolomite, indicate that high injection rates and viscous fluids are required to create wide fractures. Increasing the injection rate will, in itself, produce faster reaction rates. Because this type of reaction can be considered to be first-order diffusion-controlled, '-' the rate of movement of fluid past the rock surface affects the thickness of the diffusion layer, in turn affecting the reaction rate. Staudt, et al,10 report the results of tests in which a rotating marble cylinder was immersed in acid, and the weight loss at different speeds determined. Such test conditions, however, are not truly indicative of the situation in a formation during an acid-fracturing treatment. This paper reports reaction-rate data obtained under

By R. E. Schilson, F. H. Poettmann

The direct determination of the stabilized performance behavior of low capacity, slowly stabilizing gas wells is extremely time-consuming and wasteful of gas. From both field experience and theoretical considerations, a test procedure has been evolved by which the stabilized hack pressure behavior of such gas wells can he predicted without having to revert to long time flow tests. The method consists of using the isochrona1 test procedure to establish the slope of the back pressure curve, "n", and the short time variation of the performance coeficient, "C", with time. From this short time transient flow data and theoretical considerations, the value of C at large times can he established. By assuming the radius of drainage of a well to be half the distance between wells, one can calculate the stabilization time for various well spacing patterns. Once the stabilization time for a given .spacing has been determined, the value of C can be calculated and the stabilized back-pressure curve can he. establbrhed. The calculated perfortnance coefficient as a function of tinze was cotnpared to the experintentally measured values for a number of gas wel1.e. The deviation of the calc~*lated from the cxperimental res1tlt.s vary depending on the set of short time experimental points used to evabrate the parcrttzeters of the equation. The longer the tirne far the flow test data user1 in the calculations, the better was the agreement with the experimental results. The time necessary to obtain this data from well tests varies consitlerably, depending on the physical natrcre oj the rt3scrvoir under consideratiot~. INTRODUCTION For many years, the U. S. Bureau of Mines Monograph 7' has served as a guide for testing and evaluating the performance of gas wells by means of the back-pressure method. The back-pressure performance of a gas well is expressed by the following equation: Q~ CW-W.........(I) where the characteristics of the back-pressure equation are determined by C, the performance coefficient, and n, the exponent which corresponds to the slope of the straight line when Q and {P* - PN2) are plotted on logarithmic paper. Q is gas flow rate at standard conditions, and P, and P, are equalized and flowing bottom-hole pressures, respectively. Prior to the development of the back-pressure tcst, the "open flow" capacity method of testing a well was common. By this method, the wells were flowed wide open to the air and the flow rate measured. Such procedure was wasteful of gas and did not provide information on the deliverability of the gas to the pipe line. Monograph 7 Procedure The back-pressure method of testing wells was developed to overcome these shortcomings. Although much has been learned regarding the laws of the flow of gas through porous formations, the original development of the back-pressure relationship was based entirely on empirical methods. The back-pressure behavior provides the engineer with information essential in predicting the future development of a field. It permits him to calculate the deliverability of gas into a pipe line at predetermined line pressures, to design and analyze gas gathering lines, to determine the spacing and number of wells to he drilled during the development of a field to meet gas purchasers' requirements, and to solve many other technical and economic problems. As described in Monograph 7, the flow-after-flow method of back-pressure testing, when applied to fast stabilizing and usually high capacity wells, correctly characterized the behavior of the well. However, as the value of the gas at the wellhead increased, small capacity gas wells having slow rates of stabilization became economically operable. The flow-after-flow method of testing could not be used to describe the behavior of these slowly stabilizing wells. The procedure of Rawlins and Shellhardtl for establishing the back-pressure behavior of a gas well was based on the rcquirement that the data be obtained under stabilized flow conditions; that is, that C is constant and does not vary with time. C depends on the physical properties of the reservoir, the location, extent and geometry of the drainage radius, and the properties of the flowing fluid. In a highly permeable formation, only a very short period of time is required for the well to reach a stabilized condition, and, consequently, the requirements for the test procedure described in Monograph 7 are met. For a given well, n is also constant

By J. N. Chew, C. A. Connally

A correlation is presented for predicting the viscosity of gas-saturated crude oils under reservoir conditions. It is based on the dead oil viscosity and the solution GOR. The correlation was developed from a study which showed that at a fixed solution GOR, the relation between the gas-saturated oil viscosities and the correrponding dead oil viscosities is a straight line on logarithmic coordinates. Data from 457 crude oil samples from the major producing areas of the U. S., Canada and South America were used. The best straight lines through the data were fitted by the method of least squares with a digital computer. The correlation is presented in the form of an equation and also in a convenient gra-phical form. INTRODUCTION The viscosity of gas-saturated crude oils under reservoir conditions is an important physical property used in reservoir engineering calculations. Many difficulties are inherent in the sampling of gas-saturated oils and in the laboratory measurement of their viscosities. Therefore, it is frequently neither possible nor convenient to obtain measured values, so the viscosity must be estimated from other, more readily available data. This paper presents a correlation for predicting the viscosity of gas-saturated crude oils. The correlation is based on the amount of gas in solution and the viscosity of the dead oil at the temperature of interest. A previous correlation of these variables was presented by Beal. In comparison, the correlation presented here extends to a higher GOR and to a lower dead oil viscosity. Over the range of variables covered by both, this correlation is based on approximately 11 times as many oil samples. DEVELOPMENT OF CORRELATION A study was made of the variables or parameters which could be used to relate the viscosity of a gas-saturated oil to its dead oil viscosity. Preference was given to those variables which are easily and commonly measured in the field or in the laboratory. It was found that either the differential liquid formation volume factor or the solution GOR of the gas-saturated oil could be used as a correlating parameter. The solution GOR was se- lected for the correlation because of its simplicity and availability. In an attempt to refine the correlation by introducing a second parameter, the following additional variables were tested: differential liquid formation volume factor at the reservoir bubble point, the formation volume factor at the saturation pressure for the GOR of interest, reservoir temperature, viscosity of C+ fraction at 100°F and atmospheric pressure, weighted average gravity of gas in solution, oil gravity, reservoir bubble point pressure, saturation pressure at the selected GOR and reservoir temperature, and gas mol fraction. None of these improved the correlation to any significant extent. It was found that at any fixed solution GOR, the relation between the gas-saturated oil viscosities and the corresponding dead oil viscosities is a straight line on logarithmic coordinates, for the range of variables investigated. This straight-line relationship was used for the correlation. Viscosity data on a total of 457 crude oil samples were used. The data were obtained with rolling ball-type viscometers. The samples came from all of the important producing areas of the U. S., as well as from Canada and South America. Their geographical distribution is shown in Table 1. The ranges of properties represented by these samples are as follows: For each crude oil sample the viscosity of the liquid phase from each step of a differential liberation was plotted against its solution GOR on semi-logarithmic paper. A smooth curve was drawn through the points and values of viscosity in centipoises were read from

By M. D. Arnold, P. B. Crawford, P. C. Hall

An experimental study has been made to determine the optimum flooding pressures for four different oils. The oil formation volume factors ranged from 1.08 to 2.13, and solution gas-oil ratios ranged from about 200 cu ft/bbl to 2,250 cu ft/bbl. Viscosities ranged from 0.38 to 0.95 cp at the respective bubble points of the fluids and from 0.7 to 20 cp at atmospheric pressure. Water floods were conducted at various pressure levels from run to run. The recovery as a function of flooding pressure was found to be different for each fluid, with optimum gas saturations ranging from 7 up to 35 per cent. The data indicate that substantially higher recoveries may be obtained if water floods are conducted at an optimum pressure and that this optimum pressure is a function of fluid properties. The same core was used for all tests, and the reproduction of saturations for various runs indicates that wettability in the predominantly water-wet core did not change. INTRODUCTION A paper was presented by Bass and Crawford' which described an experimental study of the effects of flooding pressure and rate on oil recovery by water flooding. This work was conducted using high-pressure models operated in a manner similar to that of an actual reservoir, with gas saturations being obtained by a solution-gas-drive mechanism. They found that the greatest oil recovery was obtained for the system studied by flooding in the presence of a 5 to 7 per cent gas saturation. Another experimental study simulating field conditions was presented by Richardson and Perkins.' They used an unconsolidated sand pack containing kerosene-natural gas fluid and interstitial water. They flooded at various pressures and flooding rates. For their system it was found that the recovery was independent of the pressure level at which the water flood was performed. Kyte, et al," found that oil recovery by water flooding was increased as the free gas saturation at waterflood initiation was increased. However, after the initial gas saturation was increased above 15 per cent, the increase in oil recovery tended to level off. All of their runs were made at the same pressure using a light oil saturated with helium. The desired gas saturation was obtained by injecting helium into the core. Dyes' made calculations which showed that an optimum gas saturation of 12 to 14 per cent may result in an increase in oil recovery of 7 to 12 per cent over that obtained by flooding at the bubble-point pressure. Others have also found that the presence of a free gas saturation may increase the waterflood oil recovery. In each case shrinkage was small and changes in fluid properties with respect to pressure were small. A careful review of the literature reveals that at the present time there is a wide difference of opinion on the factors affecting waterflood recoveries. This diversity of opinion is probably due to the fact that very little research has been done which has taken into account the many variables existing in an actual field being water flooded. Since the literature contains little information on high-pressure waterflooding studies using various types of reservoir fluids, it was believed appropriate that such a study should be made. EQUIPMENT AND PROCEDURE All tests were made using the same consolidated sandstone core. Torpedo sandstone was used to turn a cylindrical core 13.5-in. long and with a 2.92-in. average diameter. The core had a porosity of 28 per cent and a permeability to brine of 146 md. This brine was made up by adding 20,000-ppm sodium chloride and 30,000-ppm sodium nitrite to distilled water. This was used as connate water and flooding water. No fresh water was ever brought in contact with the core, as tests showed the sandstone contained argillaceous material which swelled in the presence of fresh water and plugged the stone. The core was sealed in a section of 6-in. N-80 tubing with Woods metal filling the annulus. The core was mounted horizontally; an injection well was placed in the center of one end and a production well in the center of the other. Pressure control was maintained by placing a back-pressure regulator (upstream control) on the producing well. The "live" oil was stored in a separate bottle and water was injected into this bottle to displace the oil for saturating the core using a two-cylinder standard-proportioning pump. This same pump was used for water flooding the core at a constant rate. This system was enclosed in water jackets and the temperature was automatically main-

By G. R. Countryman, G. H. Thomas, I. Fatt

Miscible displacement in both thewetting and non-wetting phase has been studied in two-phase systems. Experimental data show that dispersion is a function of saturation and cannot be predicted from dispersion theory developed for single-phase miscible displacement. Some of the effect of saturation on dispersion is a result of the development of dead-end pores or a dendritic pore structure in multiphase systems. The presence of dead-end pores or dendritic structure is confirmed by pressure transient studies. INTRODUCTION Displacement by flooding with a miscible liquid is a possible means for recovering the estimated two-thirds of the oil that remains behind after primary production. The immense economic importance of a process that can recover such large quantities of oil has led to extensive laboratory studies of miscible displacement. As usual in production research, most laboratory studies of miscible displacement have not attemped to reproduce all of the conditions existing in a petroleum reservoir. In the early stages of investigation the need has been for information on the broad, basic principles of the phenomena involved in miscible displacement. The effects of overburden pressure, reservoir temperature, and wettability have been considered of secondary importance. However, one property of a petroleum reservoir which is expected to be of major importance and yet has been omitted from many laboratory studies is the presence of interstitial water. Two possible effects of interstitial water on the displacement mechanism in the hydrocarbon phase immediately come to mind. First, from the general1y accepted theory that capillarity governs the distribution of oil and water in porous rock one would expect that for water-wet rock the water will be in the small pores and oil in the large pores. A miscible displacement of oil carried out in the presence of water is operating in a pore size distribution shown in Fig. lb, whereas if the same test had been performed with only one phase present the pore size distribution is as shown in Fig. la. Although there is not yet a theory of miscible displacement which explains in detail the effect of pore size distribution, one would expect the differences between Figs. la and lb to influence the displacement efficiency. A second factor which may make a multiphase system different from a single-phase system is the presence in the multiphase system of dead-end pores or dendritic structure. Experiments of various kinds on reservoir rock have led to the belief that all pores in the network structure of a porous rock take part in conducting fluid during single phase fluid flow.1,2 There are then no dead-end pores and no fingers or dendritic structures containing stagnant fluid. In a multiphase system, however, the second phase may

By H. D. Outmans

The mechanics of filtration are described by a theoretical-empirical nonlinear diffusion equation which, under certain circumstances, may,be linearized and then solved explicitly. For filtration under static conditions linearization leads to a boundary value problem analogous to a heat flow problem with a known explicit solution. The corresponding solution for static filtration is compared with published experimental data. Under dynamic conditions filtration takes place in three successive stages. During each of these the rate of filtration and/or the thickness of the filter cake are different functions of the filtration time. The proposed mechanism explains the observed high resistance of dynamically deposited filter cakes against erosion and also the connection between filtration rate and viscosity of the drilling fluid. Several of the quantities governing dynamic filtration have no counterpart in the static filtration mechanism. The static filtration rate is, therefore, not a reliable measure for the dynamic rate and vice versa. INTRODUCTION Filtration under simulated borehole conditions, i.e.. either from a static suspension or from a suspension flowing parallel to the filter cake (dynamic filtration), has been the subject of several laboratory studies.1-7 This experimental and theoretical work showed that many aspects of the filtration mechanism cannot be explained by an elementary filtration theory based on the assumption that the filter cake is incompressible. For a more satisfactory theory compressibility should be taken into account. This has been done. at least to some extent, in the filtration theory developed for the chemical industry. Surveying the literature (see Ref. 8 for a bibliography) it becomes apparent that, although several properties of filter cakes deposited from many different suspensions have been measured, including compressibility, the filtration equations are essentially empirical in nature. No attempt has been made, for instance, to develop filtration theory along the lines of consolidation theory.12 This theory, upon which a highly successful development in soil mechanics rests, would appear to be an excellent starting point for a filtration theory since the compressibility concept is an essential part of it. As we will use this theory in the following development it is well to state the four assumptions on which it is based: 1. The fluid flow through a compressible porous medium is governed by Darcy's law. 2. The rate of change in fluid content of an element of the porous medium is proportional to the rate of change of solid pressure between the particles. 3. The solid particles are incompressible within the range of pressures considered. 4. The total pressure on a surface normal to the line of flow is equal to the sum of the fluid pressure and the pressure between the particles (solid pressure) at that surface. To calculate the rate of filtration and other quantities of interest it is necessary to know the filter cake compressibility and the permeability as a function of the solid pressure. It has not been possible to calculate these functions from theoretical considerations and they will therefore be introduced in this paper as empirical expressions. Both are determined in a compression cell where the cake is subjected to a mechanical load. Permeability and compressibility are measured after the solid pressure has stabilized, i.e., after the excess fluid pressure has been dissipated and the uniform pressure is transmitted entirely by the solids. The permeability and compressibility thus determined are not the same as during filtration in the borehole, under either static or dynamic conditions, because then, due to frictional drag, the solid pressure and hence also the permeability and compressibility vary along a line normal to the direction of the mean flow and can only be defined locally. DERIVATION OF THE FILTRATION EQUATION FOR COMPRESSIBLE FILTER CAKES As the filter cake in the borehole is thin compared to the radius of the hole the filtration may be considered as linear. Taking the x-axis normal to the wall, with its origin at the formation, we have, according to Darcy's law,

By F. M. Stewart

A paper by Hurst and van Ever-dingen in 1949 led to the practical solution of many nonsteady-state flow problems.' Subsequently, applications of this material have been discussed by several authors. The purpose of this note is to outline a simplified direct procedure for water-influx pressure predictions for pressures above the bubble point. The prcposed procedure is unique in that after the "time-conversion factor" and the "influx constant" are obtained, predictions are made without trial and error. The elimination of trial and error is made possible by expressing certain PVT characteristics as explicit functions of pressure. The procedure is applicable to fields producing at pressures above the bubble point and under radial water drive from an infinite aquifer. In the usual applications it is required to predict (1) withdrawals given future pressure history or (2) reservoir pressure, given future withdrawals or production. The theory, assumptions and conditions under which the nonsteady-state solutions hold are covered in the references and are not repeated. The simplified direct procedure outlined here has been developed from published solutions for the so-called "pressure case". A similar direct procedure for the same special conditions has been developed for the "rate case"; however, in this instance the procedure is lengthy and offers no advantage over the trial-and-error solution. Consequently, this paper covers only the procedure based on the pressure case. DISCUSSION Under the pressure case prediction of withdrawals from a water-drive field, given the future pressure history, involves the use of a nonsteady-state water-influx term together with the material balance equation. The solution of a particular problem requires the following steps: (1) division of past pressure and production data into equal time intervals; (2) computation of cumulative water influx vs pressure drop for each time interval of past history using the material balance equation; (3) determination of the best values for the "time-conversion factor" and the "influx constant" by equating the nonsteady-state influx term with the influx computed in Step 2 (this requires a trial-and-error procedure); (4) prediction of influx for a given pressure drop using the influx term and the computed factors from Step 3; and (5) calculation of fluids produced using the predicted influx in the material balance equation. A forecast of the water-oil ratio or water production rate is required to complete the calculation. It is assumed that a reliable estimate of original oil in place is available, as well as the production and pressure history, and data on reservoir fluid characteristics. If the original oil in place is unknown, it may be computed simultaneously with the tinie-conversion influx factors2,3. However, the oil in place so computed may be considerably in error3; consequently, it is usually preferable to use oil in place estimated from pore-volume data unless there is considerable doubt as to its accuracy. The Material Balance Equation Above the Bubble Point Above the bubble point the simpli-