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By M. C. Miller, M. R. Tek, D. L. Katz

Previous studies have shown the influence of geometric variations on the drawdown and buildup behavior of a well completed in a homogeneous, porous medium containing a single-phase fluid. Other papers have shown the influence of leakage across caprocks on drawdown tests on water wells and aquifers. This paper presents two additional cases: the influence of an adjacent, expansible source of fluid, and the effect of caprock leakage on the measured in situ permeability and compressibility. Methods are developed for evaluating the influence of these two deviations from a uniform aquifer upon the in situ permeability and compre.ssibility indicated by pump tests. INTRODUCTION In relatively uniform porous media, saturated with a single-phase fluid, the effective reservoir permeability and compressibility of the water porous bed system may be obtained from drawdown and build-up data on wells, such as the initial wells drilled in aquifer storage reservoirs (Homers and Katz et al.7). In considering the drawdown and build-up behavior of wells adjacent to storage reservoirs, or of wells in an aquifer adjacent to an unknown gas field, questions arise as to the quantitative effect of the presence of compressible fluid on the measured permeability and compressibility of the formation. The solution to this problem is presented, using a constant pressure line source at some distance from two wells, one of which is pumped and the other used as the pressure observation well. The solution is presented in terms of the geometry of the wells and their relationship to the constant pressure line. It is known that caprock leakage can interfere with the determination of an aquifer in situ permeability and compressibility (Mantush4, Hantush and Jacob3 and Witherspoon and Muellerll). This paper presents an added case of the effect of leakage through a caprock of known thickness and permeability on well tests starting with a reservoir of constant pressure above the caprock and contributing to the aquifer as a result of the pump test. Again, both a pumping well and an observation well are postulated. These two cases are presented together since both a compressible fluid nearby and. caprock leakage show similar deviations from expected in situ permeability and compressibility results, and the solutions are similar in form. EFFECT OF A NEIGHBORING CONSTANT PRESSURE SOURCE ON IN SITU MEASUREMENTS Bixel, Larkin and van Poollenl presented a paper showing the influence of a discontinuity of the properties of a fluid formation system on drawdown and build-up performance. Their case is related to the present study but may be distinguished from it in that their numerical work and resulting graphs relate only to the performance of the pumping well itself. Incorrect values for in situ reservoir properties may be obtained from pump test data if the skin effect of the well and after-flow are not considered. Analysis of this problem by the authors indicated that the more reliable method of evaluating the influence of an adjacent gas field or expansible source was to use one well for pumping and another well at some distance from it for observing the aquifer pressure performance. An expansible source of hydrocarbon near a pumping well may be approximated by a line of constant pressure for a pump test. Fig. 1 illustrates a pumping well and an observation well near a line of constant pressure. Pressure behavior of an observation well during a drawdown pump test near a line of constant pressure is given by the diffusivity equation:

Jan 1, 1967

By T. K. Perkins, L. R. Kern, R. E. Wyant

Many factors influence the results of hydraulic fracturing to stimulate well productivity. Most of these factors have been studied and their effects discussed in the literature. However, the movement of sand through the crack during fracturing has received little consideration. Exactly where the sand goes in the fracture, how it gets there, and how this might influence results have not been studied, judging from published reports. To get an answer to some of these questions we have studied sand movement through a simulated crack segment in the laboratory. Although the work is preliminary, we have answered some questions about sand movement in a semiquantitative sense. We are reporting our results at this time in the hope of stimulating interest and further study in this important aspect of fracturing. LABORATORY EXPERIMENTS Fig. 1 is a schematic diagram of equipment used in this study. A segment of a vertical crack 0.25-in. wide was formed between two Plexiglas plates, and movement of sand in fluid through the crack was observed under various conditions. Sand and fluid were injected into the crack at constant rates. The top and bottom of the crack were closed so that the fluid flowed horizontally. Because of gravity most of the sand settled to the bottom before moving very far through the crack. If the fluid velocity exceeded a certain critical value, then all the sand injected was washed on through the crack even though it had settled to the bottom. When the fluid velocity was less than this value when sand injection was started, most of the sand which settled in the crack remained in place until enough of the area open to fluid flow was blocked by the settled sand to increase the fluid velocity to the critical value.. At this point all injected sand moved on through the crack even though it settled to the lower portion of the moving fluid section. All sand previously settled remained in place. When fluid velocity was increased at this paint by increasing fluid injection rate (without also increasing sand injection rate), some of the previously settled sand was washed out until enough area was open to flow that the velocity dropped to the critical value. Conversely, when the fluid velocity was decreased by decreasing the fluid injection rate (without changing the rate of sand injection), additional sand would settle until the critical velocity was again attained. In short, the critical velocity is an equilibrium value which the system will automatically seek, if possible. The equilibrium (or critical) velocity was measured under several conditions. The results of these measurements are shown in Pig. 2 where the equilibrium velocity is shown as a function of the amount of sand injected per minute. The points falling in the shaded band shown in Fig. 2 were obtained with water. The higher viscosities were obtained by adding CMC (carboxymethylcellu-lose). Data indicate that the equilibrium velocity is not very sensitive to sand size or transport fluid viscosity, or to sand injection rate (except at very low rates). Points connected by the solid CUrve in Fig. 2 were obtained with water and 20- to 40-mesh steel shot. The equilibrium velocity is apparently fairly sensitive to difference in density between particle and fluid. Only one run was made with a gelled fluid and that was gelled water at zero sand injection rate as shown in Fig. 2. (Measurements of equilibrium velocity at zero sand injection rate were made by filling the crack

By H. T. Kennedy, S. G. Dardaganian, J. L. Clanton

This paper presents equilibrium ratios measured on oil and gas samples taken from the Kelly-Snyder field at the reservoir temperature of 128°F. From these data smoothed curves were constructed for the reservoir fluid up to the bubble-point, 1,874 psia. A material balance calculation, based on the use of equilibrium ratios instead of the usual solubility and shrinkage meayr~rements is made on the field and compared to the con\.entional method. DETERMINATION OF EQUILIBRIUM RATIOS The equilibrium ratios, or K values were obtained by recombining separator samples of oil and gas taken when the reservoir was above its bubble point, and thus represented the reservoir fluid before any change had occurred due to the evolution of gas. The well sampled had a gas-oil ratio of 617 cu ft of high pressure sepalator gas per barrel of separator oil. and a bottom-hole temperature of 128°F. Samples were brought to equilibrium by agitating for 24 hours in a thermostat, maintained at the proper temperature within 0.2ºF, mercury being added when necessary to maintain the pressure constant. Samples of oil and gas were then taken by constant pressure displacement, and analyzed by low temperature frac-tionation. Equilibrium ratios were determined at 128°F and 500, 800, 1,000 and 1,400 psig. At the latter pressure, there were indications that the gas samples were contaminated with small quantities of oil, and only data taken at the first three pressures were included in Fig. I. This figure has been extended to higher and lower pressures by the use of the Fluor charts for 3,000 psi convergence pressure, on which the experimental points fit well. MATERIAL BALANCE EQUATIONS The possible errors involved in conventional material balance equations have been analyzed by Muskat1. Where the errors occur in field measurements, or in obtaining effective averages from variables such as gas-oil ratio and pressure, they will evidently be involved in any attempt to estimate the content of the reservoir. On the other hand, substantial errors may be introduced in conventional material balance calculations by: (1) the neglect of the amount of oil dissolved in gas at high pressures and temperatures in the reservoir, but which condenses under separator conditions; (2) the assumption that the separation of oil and gas in the reservoir may be duplicated by a differential liberation in the

By H. N. Mead

In a reservoir when gas comes out of solution and rises, additional pressure is created because of the change in position of this gar in the bounded volume. If this pressure effect is not taken into account as a pseudo-influx ill mate rial balance calculations on reservoirs in which there is evolving gas, an error is introduced that is directly proportional to the length of oil column and amount of gas evolved and inversely proportional to the reservoir pressure. In reservoirs with small oil column this pseudo-influx is of little importance. The larger the oil column the more important the effect becomes. The most important application, however, will be for reservoirs with declining pressure in which increasing water entry has been indicated by material balance calculations. It could be that this indication is the result of the cited pseudo-influx and not water entry at all. This paper attempts to explain this phenomenon in relation to material balance calculations and prevents an approximate method of determining tile pseudo-influx. INTRODUCTION Only recently has the pressure-volume relationship of gas rising in a liquid appeared in engineering literature, although its effect has been discussed for several years—especially among engineers engaged in drilling operations. The writer published a paper in 1957 in which this phenomenon was examined in relation to blowouts in drilling wells.' Also, in 1957 Stegemeier and Matthews' authored a paper in which the relationship was presented with respect to pressure build-up tests when shutting in producing wells. In this paper the effect will be called "pressure inversion" which is descriptive of the phenomenon. Fundamentally, the effect just mentioned is the result of changing the position of a quantity of energy contained within the gas itself (as a result of its compression) within the bounded volume in which the gas exists. If the gas were allowed to expand "beyond all bounds" there could be no pressure inversion effect. This is because all of the energy contained within the gas as a result of compression would have been re- PRESSURE INVERSION A phase that has not been discussed in the literature is the effect of pressure inversion upon material balance calculations in reservoirs in which the reservoir pressure drops with reservoir withdrawal and is below the bubble point. This action releases solution gas which, after the critical gas saturation has been reached, will tend to rise and collect at the top of the reservoir. The pseudo-influx caused by this effect could be great enough in reservoirs of large oil column and high permeability to render invalid any analyzer studies made on them — if pressure inversion has not been taken into account. Most suspect would be those reservoirs with large oil column in which increasing water entry has been predicted but the bottom wells have not yet produced water. Example 1 Simply stated, this phenomenon exists in oil reservoirs because of the difference in density and expansibility between gas and oil. For example, if a bubble of gas exists at the bottom of a 1,000-ft column of oil having a gradient of 0.30 psi/ft with the pressure at the top of the column zero, the bubble of gas will be compressed to a pressure of 300 psi. If temperature and the combined volume of the oil column and the gas bubble remain constant, as the gas bubble rises in the oil column it will have the same volume and consequently the same pressure at any point. Under these conditions when thc bubble reaches the top of the oil column, the pressure will be approximately 300 psi at the surface and approximately 600 psi at the bottom of the oil column. If it is assumed that the oil is incompressible the pressures will be exactly 300 and 600 psi, respectively.

By M. E. Hassan, J. C. Calhoun, R. F. Nielson

An apparatus was constructed for the measurement of inter-facial tensions over a range of temperatures and pressures. This apparatus utilized the pendent drop method, and resembles in construction similar apparatus recently described in the literature and in use in some petroleum research laboratories. The interfacial tensions of benzene, propane, n-pentane, n-hexane. n-octane. and iso-octane against water were measured at temperatures ranging from 26" to 82" C and at pressures ranging from 1 to 204 atm. Values of interfacial tensions for the henzene-water system and their variations with temperature and pressure are generally in good agreement with value? of previous investigations. The data in all cases showed a slight decrease of interfacial tension with pressure at constant temperature in the range studied. The effect of pressure became less as the pressure was increased, with an indication of a reversal of the effect at higher pressures. There was a decrease of interfacial tension with temperature at constant pressure in all cases, as would normally be expected. This rate of decrease became greater the higher the temperature. A general equation is presented for the interfacial tensions as a function of presPurp and temperature over the range studied, and the constants calculated for each system. A definite trend was found in the effect of molecular weight on the interfacial tension at a given temperature and pressure, for the homologous series from propane to n-octane. Data for n-decane from the literature fitted well into this trend. INTRODUCTION It has been recognized for many years that surface forces play an active part in the production history of an oil reservoir as well as in determining the amount of unrecoverable oil. The magnitudes of these forces are governed by the values of the interfacial tensions. These vary greatly with composition, pressure, and temperature. The effect of composition is greater than that of pressure and temperature over the ranges normally encountered. In fact, because of the variety of compounds present in a crude, no attempt has yet been made to correlate crude composition with its surface tension or with its interfacial tension against water. The amount of dissolved methane greatly affects the surface properties of an oil, and there are "surface-active" compounds in crudes which have a much greater effect on boundary tensions than do equal amounts of hydrocarbons. The purpose of the present work was partly to contribute data that would eventually be used in correlations applicable

Jan 1, 1953

By J. G. Eenink, R. A. Cunningham

One phase has been completed of a laboratory invesrigation of formations with relatively high permeability under conditions of overburden, formation and mud coltrmn pressures. The following statements are based on these tests within the limitations described. Drilling rate decreased when mud column pressure was greater than formation pressure. The decrease was primarily due to a layer of cuttings and mud particles held to the hole bottom by the difference in pressure. Much of the force of the bit was wasted in this layer and was unavailable for penetrating virgin formation. Adequate jet velocities helped clean the filter cake and chips away and resulted in increased drilling rate — the higher the jet velocity the faster the drilling rate. Drilling rate increased slightly when formation pres sure was greater than mud column pressure. The i11(reuse resulted from cleaning the hole bottom as formarion fluid flowed into the borehole. Overburden pressure had practically no effect on drilling rate. INTRODUCTION Some formations are difficult to drill in a wellbore hole but are easily drilled under atmospheric conditions in the laboratory. The reasons for this apparent difference in drillability are not fully understood. An increase in formation compressive strength causes a reduction in drilling rate. It has been shown by Griggs1, Handin2 and others3-1 that formation strength increases under conditions of high hydrostatic stress. Based on these results, a decrease in drilling rate would be expected with increased drilling fluid pressure in formations of low permeability having low formation fluid pressure. Such a decrease has been shown by Murray. et a1 and Eckel" This effect holds even when using nitrogcn under high pressure as the drilling fluid (unrc-ported tests from the Hughes laboratory). Poor bottom-hole cleaning may account for decreased drilling rates. Even though it may not be easily explained, evidence of poor cleaning can easily bc seen. One evidence is the increased drilling rate that accompanies decreased balling obtained with addition of oil in some muds used to drill shales.4,7,3 Many variables can affect drilling rates. These may include mud composition and properties, formation properties, bit types and operating conditions, formation fluid, mud column and overburden pressures. These must be studied a few at a time under carefully controlled conditions which simulate actual conditions as closely as possible. The purpose of this work was to investigate effect of overburden, formation and mud column pressures on drilling rates in permeable formations. These tests were suggested by conditions shown in Fig. I. Represented is a borehole drilled into a formation with relatively high permeability. In the vicinity of the hole bottom there exists a mud column pressure, P a formation fluid pressure, P and an overburden pressure, The mud has low water-loss properties and sufficient weight to more than overbalance the formation pressure. Laboratory investigation included a range with formation pressure greater than mud column pressure. NOMENCLATURE Pm mud column pressure = pressure of drilling fluid or mud at hole bottom Pf, formation pressure = pressure of fluid in intersticies of formation

By J. R. Elenbaas, J. A. Vary, D. L. Katz

The development of gas fields, oil fields and aquifers for storing natural gas is treated from two main vieu.-points: (I) the volumetric storage capacity for gas in a given situation and (2) the prediction of the number of wells required for the delivery of gas. Other experiences in the design and operation of storagc fields are incluclerl INTRODUC TION Storage of natural gas in underground reservoirs near the terminus of long distance pipelines has been the prime factor in opening the space heating market to the natural gas industry. Storagc has permitted a major. increase in both the load and the load factor of pipe-lines; some are now operating at steady load throughout the year. Thus, underground storage has been responsible for the rapid increase in demand for natural gas in recent years. Three types of reservoirs have been used for gas storage: natural gas reservoirs, oil reservoirs, and waterbearing sands or aquifers. This paper presents the factors to be considered when developing gas storage reservoirs of these three categories. There are two prime considerations tor any storage reservoir: (1) the volume of gas which a given reservoir will store advantageously and (2) the number 01 wells needed to provide the required peak deliverability. These two problems will be considered for the three types of reservoirs just noted STORAGE IN PARTIALLY DEPLETED GAS FIELDS Early storage operations consisted of replenishing the natural gas in a depleted gas field situated adjacent to the market. Today, newly discovered fields near the market may be considered for storage, and this discussion applies equally to both types of reservoirs. For reservoirs originally containing gas or oil, the question of the impermeability of the cap rock nor-mally does not arise. However, such fields are likely to have many wells drilled either to or through the reservoir under consideration. Positive assurance must be obtained that such wells are or can be made mechanically tight. Corroded casings may need to be lined or permanently plugged. Abandoned wells should bc reopened and properly cemented. The volumetric capacity for gas storage depends upon space available in the porous rock as well as pressure and temperature of the gas in the reservoir. The production-pressure decline data on partially depleted gas reservoirs without water drive permit calculation of the reservoir space for gas. Isopachous maps of sand volume and porosity data for the reservoir rock provide an alternate method of calculating the pore volume for water-drive reservoirs. The pressure range selected for the storage cycle depends upon ()) the safe upper limit of pressure. 2) the flow capacity of wells and (3) compression requirements when injecting gas into the reservoir or delivering to market. Normally, gas and oil fields have pressures at discovery in the range of 0.43 to 0.52 psi/ft of depth. Pressures of around 1.0 to 1.2 psi/ft of depth appear to lift the overburden1-3 and invite uncontrolled movement of fluids in the porous rock. Some top pressure is normally selected for a storage reservoir ranging from below discovery pressure for deeper reservoirs to 0.65 psi/ft of depth for shallower reservoirs. Pressures to 0.66 psi/ft have been experienced without difficulty. The lower pressure limit is set by water intrusion accompanying low pressures, reduced flow capacity for wells at lower pressures and compression requirements. Depletion-type gas reservoirs often encounter water problems in the later stages of gas production. Such water intrusion may be due to movement from the surrounding aquifer. Accordingly, displacement of this water back into the aquifer by gas pressure and subsequent surges of water corresponding to the gas storage pressure cycle must be considered. Storage fields often produce in four months a volume of gas equal to its initial content. Rapid decreases in reservoir pressure occur, such as 20 psi/day. Accordingly, closed-in pressure observation wells which reflect the pressure in the bulk of the reservoir are required for following the operation of the reservoir. It has been found that a plot of observation wellhead pressures against gas content, Fig. 1, is very useful in observing operation of the field, checking the inventory and predicting future behavior. The plot is based on a given quantity of base or cushion gas in place. The injection and withdrawal curves may spread depending upon the homogeneity of the reservoir rock. permeability of the rock, well spacing and flow rates.

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-

By R. E. Collins, Paul B. Crawford

Potentiometric model studies have been made of ertically fractured reservoirs. All fractures originated at he well and extended into the reservoir for various distances. Studies were made to determine the areal sweep efficiencies for line-drive patterns. The areal sweep efficiency for the unfractured system studied war 70.6 per cent. It was found that if the vertical fracture war paralIel with the direction of flood, the areal sweep efficiency may be considerably reduced, approaching zero in some instances. If the fracture was at right angles to the direction of flood, the areal sweep efficiency may be greater than that for the unfractured system. The sweep efficiency depends on the length and orientation of the fracture and direction of the flood. It is concluded that considerable effort should be made to determine the nature and direction of fractures before initiating a flooding program. Such information is required if the maximum sweep efficiency is to be obtained and channeling is to be reduced to a minimum. INTRODUCTION The recent development and application of fracturing techniques to petroleum reservoirs has served to focus considerable attention on the effect such fractures may have on secondary oil recoveries. The opinion has been expressed that fracturing may not alter appreciably the water flooding possibilities. Others have thought that channeling may occur and that the creation of fractures may serve to substantially reduce the possibilities of conducting a successful water flood in the reservoir The purpose of this study was to obtain quantitative estimates of the effect of vertical fractures on the areal sweep efficiencies of line-drive flooding patterns. In presenting this work on vertical fractures, it is not to be inferred that the present fracturing techniques result in producing a vertical fracture. The types of fracture which are created are not known with certainty in every case. Some are believed to be horizontal, others inclined along bedding planes, and some data indicate vertical fracturing. Each fracture will probably exert a different influence on the sweep efficiency of the secondary recovery program; consequently each fractured well should perhaps be considered as a separate problem, requiring a careful analysis. Preliminary work on the effect of fractures on producing capacity of a well indicated that for practical purposes many fractures may be treated as having a nearly infinite permeability This assumption appears to be valid in some instances where the permeability of the fracture is only 10 or 100 times greater than the permeability of the matrix. It is known to depend on the type, orientation and length of the fracture. In realization that fractures of only 0.01 in. width have calculated permeabilities near 5,000 darcys, it is believed that many fractures may be considered as of near infinite permeability without incurring serious error. Primarily for this reason this study was made with fracture analogs of very great permeability compared to the matrix. FRACTURED SYSTEM ANALOG The pattern sweep efficiencies reported here were obtained with the aid of a potentiometric model. The application of this type model for solving reservoir problems has been described in the literature.1,2,3 The electrolyte reservoir was 20 in. wide by 30 in. long by approximately % in. deep. The liquid depth was constant, which corresponded to a constant permeability-feet product from the injection to the production well. The electrolyte reservoir was built to correspond to a unit in the line-drive pattern shown in Fig. 1. This figure shows part of an infinite array of wells having alternate rows of fractured and unfractured wells. The area within the dashed rectangle shows the area generally studied in the model. In using the model to represent a fractured system, a thin strip of copper was soldered to the well or wells and the length and orientation of the strip was made to conform to the vertical fracture under consideration.

Jan 1, 1955

By J. C. Todd

The volume of air needed to move the combustion wave through each acre-foot of the reservoir is a very important quantity for engineering economic analyses. A new method, which involves backflowing the injection well, has been developed for determining the air requirement .for the forward combustion thermal pilot in the Delaware-Childers field. The method depends on measurement of the total volume of air injected and measurement of the pore volume in the zone swept out by the combustion front. The basic premise of the method is that the pore volume of the swept zone contains air, while the formation around this zone contains flue gas, oil and water. The pore volume of the swept zone is determined by blowing down the pilot section of the formation through the injection well after the pilot is shut down. Oxygen balance is used to differentiate between air from the swept zone and flue gas from the surrounding formation. Data indicate that air'is produced from the swept zone by relatively efficient miscible displacement by the flue gas. The pore volume occupied by the air in the swept zone is determined by simple PVT relationships. The average temperature of the swept zone is calculated by heat balance from the total amount of heat generated during the pilot and losses to the surrounding formation. The new method was applied successfully at the conclusion of the pilot in the Delaware-Childers field; the air required for the reservoir conditions involved was calculated to be 19.2 MMscf/acre-ft of reservoir swept. INTRODUCTION One of the most important pieces of information to be found from a forward combustion pilot is the air requirement; that is, the volume of air needed to move the combustion wave through each acre-foot of the reservoir. Compression costs per barrel of oil recovered, which are a direct function of the air requirement, are a large portion of the expenses incurred in the thermal recovery process. Thus, the profitability of thermal recovery depends largely on the air requirement, which is a measure of the amount of fuel deposited and burned in the thermal recovery process. Several attempts have been made to correlate crude oil properties with fuel deposition and, hence, air requirement.l*2 Other attempts have been made by our laboratory to find the air requirement by material balance, gravity survey, pressure transient analysis,3 magnetic survey and the poten-tiometric model. Better methods are needed to predict more accurately the economics of fieldwide application of the process. Another method of finding the air requirement consists of operating a pilot thermal project for several years. This is followed by an extensive coring program to define the volume swept by the combustion wave. The air requirement is found from the total injected air divided by the volume of the swept region found from the coring operation. This method is costly; an adequate coring program usually costs several hundred thousand dollars. A thermal pilot was operated in a watered-out portion of the Delaware-Childers field, Bartlesville sand, from Nov., 1960, until May 3, 1965. One of the primary purposes of this pilot operation was to determine the air requirement for economic calculations which determine the advisability of expanding the project to the entire 128,000 acre-ft field. Barnes4 has described the result of this pilot operation,, including calculation of the air requirement from limited coring data. The value obtained, as calculated, was 15.8 MMscf/acre-ft. One purpose of this report is to describe a new method of finding the air requirement from a thermal pilot operation. A second purpose is to describe the application of this method to the Delaware-Childers thermal pilot. THEORY AND DESCRIPTION OF THE METHOD The combustion zone in the forward combustion process is propagated from injection w producing

Jan 1, 1970

By L. D. Mullins, E. B. Elfrink, W. L. Wahl

In the past few years several articles and papers presenting results of solution gas-drive depletion calculations have appeared in the lit-erature. Such calculations are of interest to the oil industry, for investment decisions must often be made before much is known about a reservoir. At other times, an estimate of the possible benefits to be realized from alternate production methods is desirable, and theoretical depletion calculations can serve as a floor or reference level from which to work. In any case, an estimate of ultimate oil recovery based upon engineering data is commonly required. An engineer confronted with the problem of obtaining, for a specific reservoir system, an estimate of ultimate oil recovery by solution gas-drive depletion usually will be forced to perform the calculations himself. This is despite the quantity of data in the literature. Rarely will either experience or the literature provide results from a reservoir system similar in all important respects to the one under consideration, and calculated results are not so plentiful that satisfactory interpolation procedures can be devised. Performing the calculations, however, is a tedious, time-consuming task unless an electronic computer is available, and, in practice, time and manpower are not always available for this purpose. A quick, simple, consistent method was needed for reducing the uncer- tainty in estimated oil recovery from solution gas-drive reservoirs when only minimum information about the reservoir system is available. PROCEDURE Method of Calculation The usual requisite assumptions were made so that the material balance equation could be used to calculate data for the charts. The following assumptions were made: (1) the reservoir is homogeneous and isotropic; (2) oil recovery is due entirely to solution gas drive and neither a gas cap nor a water drive nor gravity drainage is present; (3) the initial reservoir pressure is the bubble-point pressure of the reservoir fluid; (4) initial total liquid saturation is 100 per cent of pore space; (5) interstitial water saturation remains at the initial value as the reservoir pressure declines from the bubble-point pressure to atmospheric pressure; (6) equilibrium gas saturation is 5 per cent of pore space; and (7) oil and gas saturations are uniformly distributed throughout the reservoir at all times. There are no saturation gradients due to a wellbore, nor is the geometry of the reservoir system considered. The material balance equation was written in the form of a differential equation' which was integrated to determine the change in oil saturation for an assigned pressure drop. Formal integration was not possible, so recourse was made to the Runge-Kutta method' of numerical integration. All computations were performed on IBM equipment. Numerical integration yielded the change in oil saturation within the reservoir as the pressure. declined from the bubble-point pressure to atmospheric pressure. The initial oil saturation minus the change in oil saturation yielded the oil saturation at atmospheric pressure. The oil originally in place was obtained by dividing the initial oil saturation by the initial formation volume factor (differential liberation) while the oil in place at atmospheric pressure was obtained by dividing the final oil saturation by the formation volume factor at atmospheric pressure. U1timate oil recovery, expressed as a percentage of the initial oil in place, was obtained by dividing the difference between the oil initially in place and the oil in place at atmospheric pressure by the oil initially in place and multiplying by 100. PVT Data Charts were based upon 135 solutions to the material balance equation. PVT properties of the reservoir fluids were variables in this equation and had to be known as functions of pressure. The required PVT data might have been obtained from either actual reservoir fluid systems or correlated data. However, correlated PVT data were developed and employed in the calculations for the following reasons: (1) it is doubtful that 135 sets of PVT data could have been obtained for the values of variables investigated in this study, and (2) results of recovery calculations from randomly obtained PVT data could not be correlated as well as rssults from selected PVT data. The PVT data used to develop the

By R. C. McFarlane, T. D. Mueller, J. E. Walstrom

In evaluating uncertainty, experiments are usually performed repeatedly and then conclusions are drawn from the distribution of results. With the advent of high-speed electronic computers, it is possible to perform experiments using mathematical models constructed to simulate complex experiments or operations. Statistical methods are then applied to the results of the simulated experiments. This procedure forms the busis of this paper. Demonstrated is the need for properly accounting for uncertainty in petroleum engineering problems. How uncertainty affects solutions is evaluated in three example illustrations. The method used to evaluate uncertainty in petroleum engineering studies is the Monte Carlo simulation procedure.'-" INTRODUCTION The solution to most technical problems may be derived from interrelationships among several quantities called variables or parameters. There may be only a few variables or several hundred. Interrelationships among parameters may be explicit or implicit, well established or only approximate. Some variables that fully or partially depend on the magnitude of others are called dependent variables. Input variables for most practical problems are not precisely known; there is usually an uncertainty in their value. The degree of uncertainty may vary from one variable to another. Variables that are known accurately are called determinates.' For instance, the gravity of crude obtained from a particular pool may be known precisely, and therefore is a determinate. The degree of precision with which a quantity can be determined increases as data describing the pool are accumulated during the development of the field and the producing life of the pool. The uncertainty of a parameter may result from difficulty in directly and accurately measuring the quantity. This is particularly true of the physical reservoir parameters which, at best, can only be sampled at various points, and which are subject to errors caused by presence of the borehole and borehole fluid or by changes that occur during the transfer of rock and its fluids to laboratory temperature and pressure conditions. Uncertainty may also result in attempting to predict future parameter values. This type of uncertainty is particularly evident in investment analyses involving future costs, prices, sales volumes and product demand. Uncertainty in the solution to investment problems is often called risk, and its study is called risk analysis.' Uncertainty also enters into biological and sociological analyses in which indeterminate factors are often important due to limited control of the experimental material. It is customary, in evaluating uncertainty, to perform repeated experiments and to draw conclusions from the distribution of the results of these experiments. With the advent of the high-speed electronic computer, it is possible to construct mathematical models which simulate complex experiments or operations and to perform the experiments repeatedly, utilizing the models. Statistical methods are then applied to the results of the simulated experiments This method forms the basis of the investigation reported here. PROBABILITY DISTRIBUTIONS FOR VARIABLES The uncertainty in the value of a variable may be indicated by a probabilistic description accomplished by expressing the quantity by a probability distribution. Many recognized probability distributions can be used to describe physical quantities. Recent studies used various types of distributions to describe core analysis data.',' However, for the examples in this paper, the uniform and triangular distributions are believed to reasonably approximate the data used (Fig. 1). The uniform distribution confines the variable between an upper and a lower limit. The variable may lie anywhere between the two limits. This distribution is used when no one range of values for a variable is more probable than any other, but information or intuitive reasoning indicates the variable will lie somewhere between the chosen limits. The triangular distribution is used for a variable when more data are available to indicate a central tendency of distribution. This allows postulating a "most likely" value to the distribution and upper and lower limits. In this case, as for the uniform distribution, the variable is not expected to assume a value less than the lower limit or greater than the upper limit. However, with improved quality of data it can be postulated that the variable will tend to assume a value close to the most likely value, and that there will be a decreasing probability for values away from the most likely value. The area under either of these probability distributions is equal to unity since it is assumed that there is a 100 percent probability that the variable will lie somewhere under the curve. An ordinate erected at any particular value of the variable divides the area under the curve into two parts: the area to the left of the ordinate represents the probability that the value of the variable will be equal to or less than the value of the variable at the position of the ordinate, and vice versa. The probability is zero that the variable will have any specific deterministic value. If two ordinates are drawn for any two values of the variable, the probability that the variables will have a value lying between these ordinates is equal to the area under the curve lying between the ordinates.

By A. S. Odeh

During the last few years, several authors1-6 have advanced mathematical solutions, both exact and approximate, to the pressure behavior of a well producing from a region bounded by a circular discontinuity. The most recent solutions were reported by Carter, and by Bixel and van Poollen. Varter solved the appropriate equations analytically. Bixel and van Poollen reported a numerical solution and provided correlation charts for the interpretation of flow tests; their method uses an overlay technique and is valid for wide ranges of mobility and storage capacity ratios. The mobility ratio is defined as the ratio between k/p of the two zones, and the capacity ratio is defined as the ratio between $c of the two zones. We have developed independently an analytic solution, programmed for the CDC 1604, that agreed with Carter's.'1 The same differential equation, assumptions, boundary, and initial conditions were used. We ran 30 drawdowns and 30 buildups of a unit capacity ratio and of varied mobility ratios. We also varied the radial distances to the discontinuity, and the radii of drainage. We examined the results and obtained correlations that were simple to apply and useful in flow test interpretation. The results and the correlations reported here in graphical form allow the determination of the mobility of the two zones and the radial distance to the discontinuity. They eliminate the need to fit field data to a curve as is required in the overlay technique.6 Thus, the advantages of the present correlations are that they require less effort and they provide a unique answer. They also provide guidelines for selecting the parts of the flow test plot to be used in the interpretations. Moreover, the results obtained from the programmed analytic solution define the time intervals where the approximate solution reported by Hurstl and discussed by Larkin2 are applicable. Normal buildup and drawdown plots resulted in an early straight line, a long transition zone and a late (second) straight line. However, the second straight line will not occur if the drainage radius is not at least 10 times the radial distance to the discontinuity. The mobility of the imer and outer zones can be calculated from the slopes of the first and second straight lines, respectively, by the normal methods of the flow test analysis. We found that the radial distance to the discontinuity may be determined from the abscissa value for the point of intersection of the extensions of the two straight lines. Results similar to some of these were reported by Adams et aL7 The approximate solution compared favorably with the exact one during the first and second straight-line periods. However, during the transition zone, deviation occurred between the two solutions. Discussion of Figures Figs. 1 through 7 summarize the results of this investigation. On Figs. 3 through 7 there appears the equation of each respective plot. In this paper the mobility ratio M is defined as klp2/kzpl, where subscripts 1 and 2 refer respectively to the inner and outer zones, and dimensionless time = t, = 6.32 X In the investigation, M ranged from 0.25 to 50;

Jan 1, 1970

By B. L. Landrum, J. Simmons, J. M. Pinson, P. B. Crawford

Patentiometric model data have been obtained to estimate the effect of vertical fractures on the areas swept after breakthrough in water flooding and miscible displacement programs such as gas cycling where the mobility is near one, The data are presented for the case of the fire-spot pattern in which the cemer well is fractured various lengths and orientations, the data indicate that for 10-acre spacing, fractures extetidirrg over 1300 ft in either directior1 from the fractured well may re.srrlt in reductions in sweep efficiencics from 72 to approximately 34 per cent. However. the area swept after break through may be quite largr and only 10 or 12 per cent 1ess than would be obtained if the reservoir were trot fractured. For the specific case when the volume of fluid injected is equivalent to 100 per cent of the pattern vol-unie, the swent area may vary from 80 to 88 per cent, depending on the lenght of the fracture. The former value is that which occurs when the break through or sweep efficiency was orrly 34 per cent and the latter figrrre of 88 per cent is that which is obtained if the reservoir were unfrac-ttm'd. It is pointed out that although the sweep efficiency may he very low in vertically fractured five-spot patterrz.s, the area swept at low water-oil ratios may be only 5 to 10 per cent less than those achieved if the reservoir were unfractured. INTRODUCTION Since the initiation of commercial reservoir fracturing techniques it has been desirable to determine the effect of fractures on the areas swept after breakthrough. Most water flooding or gas cycling projects are continued for substantial periods after the brcakthrough of the injected fluid. Although the sweep efficiency serves as one criterion for rating various flooding patterns. the area swept after breakthrough for various water-oil ratios or percentage wet gas, if cycling. is of perhaps more importance than the sweep efficiency alone. Sweep efficiency data on the vertically fractured five-spot have been presented3. Previous work on the line-drive pattern has shown the effect of vertical fractures on the area swept after breakthrough for the case in which the distance between injection and producing wells divided by the distance between adjacent input wells was equivalent to 1.5 (see lief. 2). The data indicated that for the line-drive pattern it may be desirable to flood or cycle substantially perpendicular to the fractures in order to achieve the greatest recovery for the smallest volume of fluid injected. For this study the center well of a five-spot is assumed as the fractured well. All fractures were assumed to originate at this well and extend into the reservoir for various distances and orientations. All the fractures are straight and are of large permeability compared to the matrix proper. These data are presented to aid the engineer in estimating fractured five-spot pattern performance. ANALOGY The potentiometric model was used in making this study. The model used was 20 20 in. by approximately 1-in. deep. For certain portions of the study one corner of this model was considered to be an injection well and the opposite corner a production well. To simulate vertical fractures a copper sheet was soldered to the wire well and made to conform to the desired length and orientation. In other studies the same model was used except that the four corners of the model might be considered as the corner wells of a five-spot pattern and a fifth well was placed in the center of the model. The well placed in the center of the model was fractured. The total fracture length is L and the well spacing. d. The complimentary fracture angles will be obvious from Figs. 3 and 4. The data obtained on the potentio-metric model assumes the pay to be uniform and homogeneous, the mobility ratio is one, steady-state conditions exist and gravity effects arc neglected. The permeability of the fractures is very great compared to that of the matrix proper. The po-tentiometric model has been used widely both in water flooding and gas cycling projects, and may be used for miscible displacement; how-ever. it is believed that the poten-tiometric model data are more properly applicable to gas cycling than water flooding because the model as-

By A. R. Gregory, M. R. J. Wyllie

The literature reveals that scant attention has been paid to the systematic experimental determination of the formation factors of unconsolidated porous media. No experiments appear to have been made on the effect of increasing cementation on the formation factor of an initially unconsolidated porous medium. Measurements have accordingly been made of formation factors as a function of porosity for aggregates of spheres in the porobity range 12-56 per cent and of cubes, cylinders, discs and triangular prisms in the porosity range 30-45 per cent. The results are examined in the light of the theoretical equations of Clerk Maxwell. Fricke and Slawinski. Aggregates of unconsolidated spheres and beach sands have been artificially cemented with silica and the formation factor-porosity relationships determined. A theory is outlined which seeks to explain the results obtained and which postulates that formation factor and porosity data for consolidated sandatones may be used to determine the original porosity of the unconsolidated sands from which the consolidated sandstones were derived. It follows also that for consolidated sandstones F' = C where F is formation factor, fractional porosity and C and k are constants. INTRODUCTION All schemes of electric log interpretation implicitly or explicitly require the use of formation resistivity factors or? as they are now generally called. formation factors. 'The originator both of the concept of formation factor and of the use of formation factor data in electric log interpretation was Archie.' More recently it has been suggested that formation factors are of value in describing fluid flow in porous media. A full discussion of this problem together with adequate reference.: has been given by Wyllie and Spangler.' In spite of the theoretical and practical importance of the subject, there appear to be remarkably few basic formation factor data in the petroleum literature. A wider search reveals that this situation is nut confined to the literature of the to industry. at various times over the last century, attempts have been made to compute the theoretical relationship between the formation factor and porosity of simple systems of unconsolidated particle.. However. the formulae derived are not all identical and the experimental evidence adduced in support of each i; not wholly convincitng. Most of the experimental work has been carried out on system. of unconsolidated spheres, although some attention has been given to systems of spheroids and cylinder. Other simple shapes have received neither theoretical nor experimental investigation. The writers feel that a better understanding of the formation factors of complex systems of practical importance requires a firm foundation of reliable experimental data applicable to simple systems. Accordingly, they have reviewed the relevant literature and have determined experimentally the relationship between the formation factor and porosity of unconsolidated aggregates of particles of regular geometrical shapes. Some experiments have also been carried out to investigate the effect of cementation on the formation factor-porosity relationship. PREVIOUS WORK Although the term formation factor was not used. the relationship between the resistivity of a solution containing dielectric solids and the resistivity of a similar solution not containing solids is an old and important problem. The formation factor of a system of dispersed spheres as a function of porosity was considered by Clerk Maxwell. Maxwell arrived theoretically at the following expression: where F is formation factor and fractional porosity. Lord Rayleigh' also examined the problem of the formation factor of systems of spheres and cylinders and concluded that for spheres Clerk Maxwell'.; expression gave a value of F whirl]. at low porosities particularly, was too small. However,

Jan 1, 1953

By B. K. Larkin, H. R. Bailey

One of the most complicated and potentially one of the most promising secondary recovery methods is that of underground combustion. A number of field tests1,2,3 have been performed recently, apparently indicating that the underground combustion process is technically feasible. Relatively little has been published concerning laboratory or theoretical investigations of the underground combustion process. A few results of laboratory tube runs have been published4,5 and these data are valuable in understanding the process. The tube run data also emphasize the complexity of the thermal recovery process. Under various conditions, problems in heat transfer, multiphase fluid flow and combustion rates arise. For example, at low temperatures the combustion process proceeds so slowly that the front cannot be propagated. It is a problem in heat transfer to predict what injection rate of air is necessary to sustain combustion in a particular reservoir. Certain fluid flow aspects of underground combustion have been studied by Wilson, et a13. In the development of some of the flow equations, assumptions of adiabatic combustion were made. Vogel and Krueger1 have devised an analog model of the heat conduction process for a constant temperature cylindrical source moving radially outward in a homogeneous conducting medium. Jenkins and Ramey7, in a discussion of the Vogel and Krueger paper, present an analytical solution of a heat conduction problem related to problems considered by Vogel and Krueger. In the present paper an idealized model of the heat conduction problem is considered, which is more general than the model of Vogel and Krueger or Jenkins and Ramey. In particular, initial well heating, vertical heat losses and arbitrary frontal velocities are included. Equations for the temperature are obtained in terms of integrals. These integrals have been evaluated numerically for the case of frontal movement corresponding to a constant air injection rate and the case of constant frontal velocity. These results are presented graphically. From these graphs one may estimate conditions under which nearly adiabatic temperatures are obtained. The application of these results requires some knowledge of reservoir thermal properties, combustion rates and residual fuel content. Assuming no vertical losses it is possible to obtain explicit evaluations of these integrals in the linear con- stant velocity case and in the radial case corresponding to a constant air injection rate. PRESENTATION OF RESULTS The geometry of the radial system is illustrated in Fig. 1 where the reservoir is represented as a slab of thickness h. By making an energy balance for an infinitesimal volume, one obtains the following equation. T= x T+F(z.r).........(1) where F (z,r) =HWv1/pc (r-r1), if z h/2 = 0, if z >h/2 . In Eq. 1, V- represents the Laplacian operator and 8 is the Dirac delta function. Eq. 1 is the well-known diff-sion equation with a source term and may be applied to a linear system. For example, if the source is moving in the direction of the x axis, then the argument of the delta function is x - 1,. In deriving Eq. 1 the following approximations were made concerning the actual combustion process: (1) all heat transfer is by conduction in a homogeneous medium, (2) the formations bounding the reservoir have the same thermal properties as the reservoir, (3) the combustion fuel is consumed instantaneously as the front arrives, (4) there is a constant fuel concentration

By B. E. Eakin, A. L. Lee

Atmospheric pressure viscosity values For light hydrocarbons were calculated by a simple mixing rule based on the Sutherland constants for pure components. This method was further extended so that the atmospheric pressure viscosity of a mixture may be calculated based on the molecular weight information only. INTRODUCTION A proven method for estimating the gas-phase viscosity of a mixture is the one by Herning and zipperer.l This mixing rule has the following form: One method for calculating the pure component viscosity i is the Sutherland equation,2 which has the following form: The Sutherland constants B and S are determined from experimental data. A set of Sutherland constants for light hydrocarbons based on the data of several investigators3-5 is presented in Table 1. APPLICATION The Herning-Zipperer equation and the Sutherland equation were used to obtain an equation for calculating the gas-phase viscosity of light hydrocarbons. This equation has the form, Eq. 3 was used successfully in the study of methane-n-butane mixtures.6 Further analysis indicated that the Sutherland constants are a function of molecular weight.7 The gas-phase viscosity may be calculated based on molecular weight information only. The following equation was developed: The atmospheric pressure viscosity of ethane, propane, n-butaneand four methane-propane mixtures calculated on the basis of Eq. 6 are compared with the data of other investigators in Tables 2, 3, 4 and 5, respectively. A plot of atmospheric pressure viscosity vs molecular weight based on Eq. 6 is presented in Fig. 1.

Jan 1, 1965

By D. W. Peaceman

In previous papers by this author on numerical calculations of multidimensional miscible displacement, some simplifying assumptions were made in writing the dispersion term of the differential equation. It was assumed that the flow vectors were essentially parallel to the x-axis. However, in most multidimensional miscible displacements, the flow vectors are not so simply oriented with respect to the coordinate axes. This paper derives the correct dispersion term /or the more general case; gives a difference approximation /or the dispersion tern; and derives the stability criterion for the corresponding explicit difference eguation.

Jan 1, 1967

By R. F. Jones, N. E. Truitt, M. Prats

In most published discussions and theories of in situ combustion, the combustion fronts are assumed to be vertical. However, evidence from field tests leaves no doubt that combustion fronts often advance more rapidly along the top than near the bottom of a formation as a result of difference in density between injected air and formation liquids. The approximation proposed in this paper to determine the movement of the resultant tilted combustion surfaces states that the vertical rate of movement of combustion surfaces is proportional to the horizontal oxygen flux. Where self-ignition is possible, the proposed method demands that a secondary combustion surface exist around production wells which produce some oxygen. These secondary combustion surfaces may be formed long before the primary combustion surface can advance from injection to production wells. Heat liberated near production wells at these secondary combustion surfaces can contribute to an early increase in production rate. Results indicate that significant oil recoveries cannot be obtained from the usual flood patterns (five-spots, seven-spots, etc.) without producing large volumes of unused oxygen. Ideally, to increase oxygen- consumption efficiency, well patterns should allow oil production from a first line of production wells and gas production from more distant lines of producers. However, it may be desirable to produce some gas at all wells to support (and benefit from) active secondary combustion surfaces. Results indicate that the well spacing through which combustion can be advanced is larger than that predicted by other methods. A large number of production wells may still be desirable to take quick advantage of gravity drainage. From a comparison with results at South Belridge field, California, it appears that this method adequately describes oxygen concentration and temperature histories and combustion-front shapes. However, this method does not accurately locate the most advanced point of the combustion surface. There is some field evidence to substantiate the actual presence of secondary combustion surfaces at South Belridge. Use of the proposed method appears warranted at this time when lay-over of the combustion surface can be anticipated. INTRODUCTION The assumption of vertical combustion fronts has been embodied in all previous publicationsl-6 which use the movement of combustion fronts away from injection wells to determine the temperature and fluid distributions in the reservoir. The only paper concerned with a mathematical model of the combustion process in which a nonvertical combustion front is used was written by Gottfried.11 Actually, nonvertical combustion fronts have been observed in most in situ combustion field tests for which adequate data are available.7, 8, 6 In practice, the typical vertical extent of the burned zone decreases with distance from the injection well, and this burned zone is at or near the top of the sand body. In some cases, such as at South Belridge field near Taft, Calif., the combustion surface is almost horizontal over a very large area.8 Thus, for some years an obvious and serious gap has existed between theory (vertical fronts) and practice (tilted fronts). This is indicated in Fig. 1. Tilted combustion fronts such as observed at South Belridge8 sometimes result from the natural tendency of injected gases to rise to the top of an oil sand. An initially present gas cap, even if thin, would make apparent the effect of gravity segregation from the first, particularly since formations considered for in situ combustion field operations are generally selected for their

Jan 1, 1969

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-