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By John P. Dolan, Charles A. Einarsen, GiIman A. Hill

his paper discusses how the following formation characteristics can be determined mathematically from drill-stem test pressure charts: true formation pressures, effective permeability of the entire section tested, well productivities, wellbore damage, and possible detection of barriers (faults, pinchouts, etc.). This paper also presents a practical method for im-mediate determination of eflective permeability and well-bore damage from successful double shut-in pressure tests. A list of recommendations for improving the reliability of drill-stem rest pressures is also presented. INTRODUCTION A drill-stem test is a temporary completion designed to sample the formation fluid and to establish the possibility of commercial production. Early pressure recording devices were used merely to verify proper operation of the testing tool. Until recently the accuracy of the pressure gauges has been insufficient for any reliable quantitative use of the recorded pressures. In view of the need for more reliable formation evaluation and as a result of the recent interest in exploration work employing the concept of hydrodynamic en-trapment,1,2,3,4 better pressure recording gauges are now in use. These devices can record pressure within 1 per cent above 1,000 psig and can detect differential pressures as low as 1/2 psig. In addition to formation pressure, several other reservoir characteristics can be determined from DST charts; namely, well productivity, formation permeability, wellbore damage, and the possible existence of bar- riers (faults, pinchouts, changes in permeability, etc.). This paper presents a practical method to interpret DST pressure charts for formation pressures and many other reservoir properties, a method that has been developed in analyzing approximately 4,000 DST charts during the last five years. The techniques used are a composite of published articles on drill-stem testing5,6, together with well-known pressure build-up analysis methods. THEORY It has been shown7,8 that the following equation may be used for analysis of pressure build-up curves: Po - Pw = 162.6 qu/kh log ( to + ?/?) .....(1) When this equation is applied to the curves obtained in drill-stem testing, the assumptions and boundary con-ditions are more nearly realized than in conventional flow and build-up tests on producing wells. Zak and Griffen9 have recently discussed in detail the use of this equation in analyzing DST charts. One of the problems with DST curves is the lack of reservoir data for precise analysis. Therefore, it is necessary to develop empirical rules and field methods for analyzing DST charts in quantity. For this reason, the empirical methods presented in this paper have been developed, and their derivation is found in Appendices A and B. METHOD USED FOR ACCURATE READING OF DRILL-STEM TEST CHARTS In order to apply the pressure build-up theory to DST charts, it is necessary to obtain a digitized expression for the pressure and time data recorded by the pressure gauge. These data may be either provided by the service company or interpolated by projecting a

Jan 1, 1958

By J. R. Jargon, H. K. van Poollen

Non-Newtonian fluids may be injected into a reservoir during secondary recovery operations. The non-Newtonian fluid used in this work is a power-law type of fluid that is, the viscosity of the fluid decreases as the flow rate or rate of shear increases. This paper presents equations for steady-state linear and radial flow of such fluids, transient behavior results from a finite difference model of a radial system, and transient behavior results from a field test. The equations that describe the flow of a non-Newtonian fluid are non-linear and are solved numerically. Finite difference solutions are presented as curves of dimensionless pressure drop at the wellbore us dimensionless time for a constant injection rate. Solutions were obtained for 5-percent, 10-percent and 100-percent PV of a non-Newtonian fluid for injection rates of 1, 10, 100 and 1000 cc/sec and for a 5 percent PV of non-Newtonian fluid located at T = Tw' 3, 10, 20, 50 and 100 ft for a flow rate of 1 cc/sec. The buildup curves do not exhibit a stmight-line portion as is the case for Newtonian flow through porous media. Correlations also are shown for the productivity index vs rate for the computer model study and the field tests. INTRODUCTION During various secondary recovery operations non-Newtonian fluids are injected. Such fluids, in general, include thickened water and gelled fluids. The term "non-Newtonian" implies that the viscosity is not only dependent upon temperature and pressure, but also on the rate of shear that is applied to the fluid. For example, water, which is a Newtonian fluid, will have essentially the same viscosity no matter what rate of shear is applied. A pseudo-plastic fluid exhibits a decreasing viscosity when higher rates of shear are applied; a dilated fluid has an increasing viscosity with increasing rates of shear. The objective of studies performed and described in this report is to obtain relationships and mathematical and empirical descriptions of the flow of non-Newtonian fluids through porous media. Simple mathematical relationships, computer studies that include the unsteady-state behavior of such fluids, and field tests are used. PREVIOUS LABORATORY STUDIES Laboratory studies have been performed by several investigators. The fluids used in one investigationl were surfactant-stabilized dispersions of water in hydrocarbons. Its Fig. 6 is reproduced as our Fig. 1. This effective-viscosity vs frontal-velocity diagram shows that this fluid is of the power-law type over a fairly large range of frontal velocities. Gogartyl developed an equation for the effective viscosity as a function of shear rate that reduces to Eq. 2 for frontal velocities greater than about 2 ft/D. The fluid characteristics used in the present study are similar to those reported by Gogarty. Christopher and Middleman2 performed experiments with power-law fluids flowing through pipes packed with glass beads. They used a modified Blake-Koteny

Jan 1, 1970

By Stephen G. Dardaganian

Pore volume compresibilitieu measured in the laborutory on core .samples are reported for typical reservoir .sarrrl.rtorres at reservoir pressure. These cornpressibilities ore different for each reseriloir sample and cannot be correlated to porosity. The compressihilities are a1so functions of presslire. In recent years several authors have pointed out the importance of including pore volume compressibility of the reservoir rock in certain reservoir engineering calculations. Hall' showed that in calculating the hydrocarbon volume of an undersaturated reservoir from the production per unit change in reservoir pressure, the neglect of rock compressibility could, in the extreme case of low porosity rock, lead to results in error by a factor of two. Hawkinsl' and Hobson and Mrosovsky showed that both rock and interstitial water compressibilities must be included to achieve satisfactory accuracy in this calculation. Although rock compressibility is an important reservoir property in these calculations, very few data are available. Hall' made compressibility measurements on a few samples in a small pressure range. No other direct pore volume compressibility measurements have been reported. Geertsma' discussed the elasticity of reservoir rock but did not present any new data. In this paper, rock compressibilities suitable for reservoir calculations are reported for a variety of sandstones typical of U. S. reservoirs. The measurements were made through a range of pressures which is also typical of reservoirs. EXPERIMENTAL PROCEDURE Plugs 1 in. in diameter and 2- to 3-in. long were diamond drilled, 14 from cores from seven oil-bearing or potentially oil-bearing sandstone, one from a quarried sandstone, and another from a shallow, oil-free, pure orthoquartzite. The plugs were extracted in toluene, vacuum dried, and then weighed. The plugs were then saturated with kerosene and reweighed. This gave pore volume. They were then again extracted in toluene. vacuum dried, and jacketed in 0.004-in. thick copper foil. A short length of -in high pressure tubing led from the jacket. The jacketed sample was then saturated with kerosene and placed in the hydraulic pressure cell as shown in Fig. 1. Movement of the mercury slug showed the change in pore volume of the sample under a given pressure change. Temperature was about 72" F for all measurements. Before taking any measurements, the external pressure on the sample was raised to 10,000 psig and kept there for several hours. This formed the thin copper jacket around the plug. 7-he pressure was then removed and the sample left at least overnight, during which time it recovered its original pore volume. An external pressure of 12,000 psig was then applied and held constant. The change in pore volume was noted as the internal pressure was raised in 1,000 psig increments to 10,000 psig. The internal pressure was then reduced to zero, again noting pore volume changes,

By J. Douglas, W. T. Ford, G. E. Henderson, I. F. Roebuck

An implicit numerical method is presented for simulating the differential and algebraic relations governing one-dimensional three-phase flow in porous media. The method is based upon compositional representation of the hydrocarbon system. Variable physical properties and water-oil capillary forces are included in the formulation; however, the effect of gravity is ignored. The effects of changing composition and mass transfer are considered through the use of known phase behavior concepts and correlations. The differential and algebraic equations and the numeric approximations to these equations are presented, and the computing algorithm is discussed in detail. This method is used to simulate a single-phase, two hydrocarbon-component gas displacement laboratory experiment. The method is also applied in simulating a solution gas drive and a gas injection problem using a reservoir oil represented as a nine-component mixture. For comparative purposes, the latter two problems also are examined with a one-dimensional volumetric model. The results of the simulation of the laboratory experiment demonstrate the relevance of this method to problems wherein the effects of mass transfer significantly enter the displacement process. The comparison of the results of the volumetric and compositional methods applied to a reservoir system illustrates not only the advantages and benefits derived from examining the hydrocarbon system composition, but also the necessity of this approach in application to a general reservoir problem. INTRODUCTION The advent of large memory, high-speed digital computers enabled reservoir engineers and mathematicians to pool their knowledge in the development of sophisticated techniques for the prediction of oil and gas reservoir performance. These techniques have progressed from volumetric and trend analysis calculations to the development of "mathematical models", or reservoir simulators. These latter models, in general, utilize finite difference approximations to the rather complex partial differential equations that mathematically describe the physics and thermodynamics of fluid flow in porous media. Many forms of this "model" type are available today, and much of the recent literature has been devoted to either improving known methods or developing new techniques for use in these prediction methods. The more advanced techniques in use at this time consist of the solution of differential and algebraic relations that describe a reservoir geometry in one, two or three dimensions and that incorporate reservoir hydrocarbon and water fluid systems that obey specified physical laws, having properties that might easily be measured in the laboratory. This type of model utilizes formation volume factors and other volumetric data, expressed directly as functions of pressure, to represent the reservoir fluid system. Experience gained with this type of simulator, commonly known as a "beta-factor model", has indicated that the approach has many outstanding defects, most of them due to the assumption that those properties measured in the laboratory completely describe the behavior of the fluid system throughout the life of the project under study. The effects of varying hydrocarbon composition and mass transfer between the phases of this hydrocarbon system must, by definition, be ignored. These defects, which may often be ignored in

Jan 1, 1970

By R. G. Jones, W. L. Kinney, R. E. Schilson, R. S. Wilson, G. A. Clark, H. Suralo

This paper describes the field opeturluns at the Fry in situ combustion project in Crawford County, 111. Field operations may be divided into three phases—-preliminary engineering studies of 1960 and 1961, ignition, and, finally, operation of the test. The Fry combustion project was a pilot test of cocur-rent in situ combustion on a 3.3-acre inverted five-spot. In the summer of 1960, air injection tests were conducted using old wells in the area and a new air injection well drilled especially for the test. In the summer of 1961, four new oil wells were drilled on a 3.3-acre pattern for the pilot test. All old wells in the near vicinity of the pilot test were abandoned to preclude any possibility of gas escaping to adjacent formations. A compressor plant was also installed at the test site in 1961. Air injection tests and other engineering studies were made. At the conclusion of these tests, ignition was accomplished with a 40 kw electrical igniter. Operation of the Fry combustion test was little different from the usual oilfield operation with the exception of the high rate of air injection. No difficulty was found with the openhole completion. There were no unusual corrosion or emulsion problems until bottom-hole temperatures exceeded 150F. Four cores were drilled into the burned-out portion of the reservoir with no mechanical difficulties. INTRODUCTION A cocurrent or conventional in situ combustion test was conducted on the Fry unit of the Marathon Oil Co., in Crawford County, Ill. Besides testing combustion in the Fry type of reservoir, the test was intended to evaluate various operating procedures. This paper presents a discussion of the Fry combustion test field operations. The geology of the reservoir and the test performance are discussed in other papers.'.' HISTORY OF THE FRY UNIT The Fry unit consists of 11 leases totaling 337 acres in an area in which oil was originally discovered in 1906. Production from this reservoir, as from all Robinson sand reservoirs in the area, was very high during the first few years. As the dissolved-gas energy was dissipated, early attempts to retard the decline consisted of vacuum, air, and gas repressuring, and recently, most of the area has been subjected to water flooding. Following early air repressuring, the Fry waterflood unit was formed and water flooding initiated in July, 1952. In the area in which the Fry combustion test was conducted, a peripheral water flood of a 160-acre area was developed as shown in Fig. 1. Between 1956 and 1960, it became apparent that the water flood was unsuccessfd. In looking for a suitable site for an in situ combustion test, the Marathon Oil Co. selected the Fry unit as possessing those features most desirable for such a research project. Field operations associated with the Fry combustion test began in the spring of 1960.

Jan 1, 1966

By R. G. Jones, W. L. Kinney, R. E. Schilson, R. S. Wilson, G. A. Clark, H. Suralo

This paper discusses the results of the Fry conventional or cocurrent in situ combustion test, which was conduct-ed in a 3.3-acre inverted five-spot. The depth of the formation was between 880 and 936 ft; the oil had a specific gravity of 28.6° APl and a viscosity of about 40 cp at the reservoir temperature of 65°F. Preceding the combustion test, air injection tests were conducted which, in conjunction with geological studies, were used to evaluate the characteristics of the reservoir. Combustion was initiated on Oct. 13, 1961, with ignition being accomplished by a 40 kw electrical heater. The test phase of the project ended on Oct. 1, 1963. During the test, the average air injection rate was 1,520.-000 scf/D. Throughout the test, production of all fluids— gas, oil, and water—was monitored. Cumulative oil production credited to the project was 100,586 bbl. The cumulative air-oil ratio was 11,500 scf/bbl oil, and the oxygen uti1ization efficiency was 87 per cent. INTRODUCTION The Marathon Oil Co. conducted a successful in situ combustion test, beginning Aug. 22, 1961, at the Fry unit, Crawford County, 111. The purpose of the project was to test the feasibility of cocurrent in situ combustion as a means of oil recovery in the Fry type reservoir. Interest in in situ combulstion as an oil recovery tool has been stimulated mostly by the existence of large reserves of heavy viscous crudes with low expected recovery, usually less than 10 per cent. These are the so-called unrecoverable reserves, and most combustion tests to date have been conducted in this type of reservoir.1-4 In contrast, the Fry combustion test was conducted in a reservoir with a relatively high gravity oil having a relatively low viscosity. This paper discusses the performance of the test. The geology of the reservoir and the field operations are discussed in separate papers.5 TEST SITE The Fry combustion test was carried out in a 3.3-acre inverted five-spot portion of a lenticular body of Robinson sandstone. Net sand was 50 ft thick, porosity averaged 19.7 per cent, oil saturation was 68 per cent of pore volume, and water saturation was 20 per cent of pore volume. The oil in place was estimated at 1,040 bbl/acre-foot, or 171,600 bbl within the 3.3-acre pattern. The water in place was 326 bbl/acre-foot, or 53,800 bbl. The oil has a specific gravity of 28.7° API and a viscosity of 40 cp at the reservoir temperature of 65°F. AIR INJECTION PERFORMANCE Air injection took place in two phases, the phases separated by ignition of the reservoir. In the pre-ignition phase, air injection tests were conducted in the summers of 1960 and 1961. These indicated that the Fry reservoir was confined, and a high return rate of injected air could be expected. This proved to be an outstanding characteristic of the project, as cumulative gas production was 95.3 per cent of the air injected. The difference between the cumulative air injected and the total gas produced can be largely accounted for by the quantity of air stored in the reservoir. An estimated 16 X 10 cf of air remained in the burned-out portion of the reservoir and an indeterminate amount of gas was stored in the unburned but pressurized reservoir. Hence, a complete material balance of gases would account for nearly 100 per cent of the air injected. At times during the test, the daily gas production rate was 98 per cent of the daily air injection rate. Fig. 1 presents the air injection history of the Fry com-

Jan 1, 1966

By C. H. Hewitt, J. T. Morgan

The Fry cocurrent in situ combustion project was carried out in a 3.3-acre portion of a lenticular body of Robinson sandstone of Lower Pennsylvanian age. This particular sand body is about 12,000 ft long and 3,500 ft wide; it varies in thickness from 0 to 55 ft, trends northeast-southwest, and occurs at depths between 880 and 936 ft. The reservoir is made lip of three distinct types of sandstone, each having characteristic reservoir properties. (See Table I.) Textural gradients, sedimentary structures, erosional basal contact, absence of marine fossils, and geometry of the sand body all indicate that the sandstone is of fluvial origin and part of an extensive complex of river-deposited sedirnents. Geographically oriented cores reveal that the direction of depositing currents was to the southwest, parallel with the sand-body trend. Directions of maximum pernleability also parallel the northeast-southwest trend. The layered nature of the sandstone reservoir and its directional properties have controlled the areal distribution of produced gas, the ignition behavior, and the vertical profile and horizontal extent of the conbustion zone. INTRODUCTION This paper is the first of a series of three papers that describe Marathon Oil Co.'s Fry in situ combustion test. This first paper contains the results of a detailed study of the geology and reservoir characteristics of the Robinson sandstone reservoir in which the Fry test was conducted. The second and third parts of this series report on the field operations' and performance. The geologic work reiated to cores cut through the combustion zone, and to the effect of reservoir properties on the combustion process is included in the third paper.' GEOLOGIC STUDY OF THE FRY RESERVOIR The close spacing of continuous cores in the Fry combustion site provided an unusual opportunity to study the details of distribution of lithology and reservoir properties within a petroleum reservoir. The objectives of the study were: 1. To determine the distribution of rock types within the Robinson sandstone in the Fry area; 2. To determine the relationship of porosity and permeability to rock properties and establish the distribution and homogeneity of reservoir properties; 3. To determine the origin of the Robinson reservoirs so that attempts could be made to predict the size, shape, distribution, extent and continuity of Robinson reservoirs; and 4. To determine what effects, if any, the reservoir rocks might have on the combustion process, so that the process might be better evaluated. AVAILABLE MATERIALS Continuous cores were cut from 14 wells in the vicinity of the Fry site. Ten of the cores are from within the 3.3-acre pattern; six of the 10 were cut prior to combustion, and four were cut through the combustion zone. Cores from the Emma Fry No. 15, Wampler 0-2 and Fry B wells (Figs. 1, 2 and 9) were geographically oriented by the Eastman Oil Well Survey Co. Maps, electric

Jan 1, 1966

By H. Kazemi, G. W. Thomas, M. S. Seth

The double-porosity model of Warren and Root7 for examining pressure drawdown and buildup phenomena in naturally fractured reservoirs has been extended to intetpret interference test results. Both analytical and numerical solutions are presented. The practicality of some of the assumptions in the Warren-Root model was also investigated. The standard procedure for interpreting interference tests in naturally fractured reservoirs beretofore employed an equivalent homogeneous reservoir model. A comparison of our results with the equivalent homogeneous model indicates the latter is inadequate for early time responses. For late time responses, however, the simpler equivalent homogeneous model yields favorable results. Finally, the effects of well spacing and the magnitude of the interporosity flow parameter are briefly cited. INTRODUCTION Interference tests can be used to provide reservoir information not obtainable from conventional pressure drawdown and buildup tests. The technique involves measuring the response in one or more wells arising from a pressure perturbation introduced into another well. It has been. successfully applied to single-layer, homogeneous, unfractured, isotropic and anisotropic formations to arrive at values of well storage capacity and a measure of directional permeability.l The techniques of interference testing usually can be extended to heterogeneous systems, provided the permeability contrast between any existing rock layers is not excessive.9 In such cases, the practical interference effects in an observation well of an interconnecting heterogeneous reservoir can be predicted using an equivalent homogeneous model. The calculated properties are then average values. Russell and Prats2 have demonstrated the utility of this approach for a single-phase, two-layer system. For naturally fractured reservoirs, some different approaches usually are necessary. The major departure from homogeneous or nearly homogeneous reservoir treatment is to characterize the fractured system as an entity having "double-porosity"; namely, we suppose the existence of both primary and secondary porosity, where the former is contained within a systematic array of identical rectangular matrix blocks and the latter is contained within an orthogonal system of continuous, uniform fractures (see Fig. 1). It is sufficient to assume that the permeability of the rock matrix is so small that movement of reservoir fluids to the producing wells is entirely through the fracture network. This assumption has been validated by Barenblatt3 and .employed by others?-7 Warren and Root7 used the double-porosity model to develop an interpretation of pressure drawdown and pressure buildup tests in naturally fractured

Jan 1, 1970

By James E. Berryman

A method is presented for calculating the pressure depletion performance of a gas-condensate reservoir from the original reservoir fluid composition using the laboratory determined pressure-production history and retrograde liquid accumulation as guides. Equilibrium vaporization ratios for this calculation were modified from published data until the calculated dew point pressure and retrograde liquid accumulation agreed with the values observed in the laboratory. The calculated results are compared with the produced gas composition determined by low temperature analysis during constant volume laboratory experiments of pressure depletion behavior. This comparison shows that the calculation procedure is adequate for determination of gas phase composition. Data from field separation tests in low temperature separators are presented which verify the applicability of published equilibrium vaporization ratios for a wide range of separator conditions. INTRODUCTION A reliable prediction of the pressure depletion performance of a gas-condensate reservoir is necessary in determining reserves and evaluating field separation methods. The predicted performance is also used in planning future operations and studying the economics of projects for increasing liquid recovery. Because the flow of fluid in a volumetric gas-condensate reservoir is usually single phase flow, the pressure-depletion per-formake is primarily a function of the phase behavior of the reservoir fluid. The properties of a gas-condensate system of major importance are: (1) compressibility factor at reservoir temperature and pressure; (2) dew point pressure; (3) composition changes of the vapor phase with pressure depletion; (4) recovery of original in-place hydrocarbons at any pressure; and (5) retrograde condensate accumulation. These properties are most generally determined by a pressure depletion-type experiment on recombined separator gas and liquid samples in the laboratory. ldeally the pressure-depletion performance can be calculated from the composition of the original reservoir fluid if the reservoir temperature and initial pressure are known.1,2,3 It has been reported "... comparison of such computations with depletion-type ex-periments on recombined samples in the laboratory have yielded results that appeared to be entirely adequate.". The success of such calculations is determined by the equilibrium vaporization ratios chosen, especially those for the higher boiling point hydrocarbons. It is the purpose of this paper to present a method for computing the change in composition of the vapor phase during pressure depletion by production using laboratory determined properties of the system to guide the calculation and to illustrate the use of the method with data from the Cockfield "D" sand of Washington field, St. Landry Parish, La. Results of field tests and laboratory analysis are included. As a result of this reservoir fluid study, a cycling program has been evaluated, the fidd has been unitized, and the cycling started in July, 1956.

Jan 1, 1958

By William Hurst

Muskat's depletion performance equation is here derived considering the expansion behavior of the reservoir hydrocarbon system and a simple fractional-flow equation. This nietkod of derivation leads logically to the two extensions that follow. The first of these is concerned with gravity segregation in a depletion-drive reservoir. The second is concerned with including an empirically determined tern? for water influx in the performance equation. The more general equation for gravity segregation when there is a primary gas cap and empirically-determined water influx is stated for completeness. These equations have been found useful in reservoir performance calculations in Eastern Venezuela. A discussion on the methods of solving these equations follows, and considers firstly the effect of taking finite intervals in the numerical integration, and sec-ondly, methods of incorporating the time functions involved in segregation in with the expansion behavior. The paper concludes with a brief general discussion on further extensions to the depletion performance equation. INTRODUCTION The two fundamental sources of energy by which oil is produced from a reservoir result from pressure depletion inside the boundaries of the reservoir and fluid encroachment across the boundaries of the reservoir. The wells in either case form low pressure outlets through which oil and gas may be produced by the expansive force of the reservoir fluids and associated encroaching fluids. When the reservoir pressure is higher than the bubble-point pressure of the oil, so that there is no free gas in the reservoir, these expansive forces are the only ones available for the production of oil. However, when the reservoir pressure is less than the bubble-point pressure of the oil, free gas is vaporized as the pressure falls. With both oil and free-gas phases present, the additional forces of gravity and capillarity may operate on the gas-oil system, as they have previously operated on the oil-interstitial water system. Gravity tends to segregate the free gas from the oil due to their density difference. Capillarity opposes and eventually balances gravity as the more extreme free gas and oil saturations are reached, preventing the independent move- ment of free gas until it is above a certain saturation, and the independent movement of oil when it is below a certain saturation. The type of depletion performance equation chosen for predicting the future performance of a reservoir depends on the amount of past history available. When the reservoir is somewhere past the halfway mark in depletion, some form of decline curve is often used. With less past history, material balance equations which incorporate empirical factors based on the past performance are often used. When, however, the amount of past history is small, the Muskat depletion performance equation will usually be used. The distinguishing feature of this type of equation is that empirical factors based on the over-all or macroscopic reservoir behavior are almost or entirely absent. Each parameter affecting the reservoir performance is ascribed an independent set of values based on measurements made on laboratory samples; that is, incorporating microscopic empirical factors. In establishing Muskat-type depletion performance equations, it is necessary to consider the reservoir as consisting of a number of associated blocks, in each of which the saturations and pressures may be considered uniform, and in each of which all substances have uniform pressure-volume characteristics. Thus, a primary gas cap can usually be considered as one block and an aquifer as another. Gravity segregation may be negligible for practical purposes when the rock and oil properties are adverse and/or the dip or thickness of the reservoir is too small. In this case the whole oil leg may be considered as one block, except in very large reservoirs. In very large reservoirs the fluid and rock properties may vary enough, particularly in the dip direction, for it to be necessary to divide the oil leg into a number of blocks, in each of which the relevant quantities may be considered uniform. When gravity segregation of the oil and free gas is not negligible, it is necessary to consider the space occupied by the initial oil leg as divided into two blocks, a secondary gas cap and an effective oil leg, in each of which saturations may be considered to be uniform. The total volume of these two blocks is thus constant, but the secondary gas cap grows continuously at the expense of the oil leg. Muskat1 derived depletion performance equations for the basic case of an oil reservoir with closed boundaries and without segregation, and for the case of an oil

By M. Sheffield

This paper presents a technique lor predicting the flow of oil, gas and water through a petroleum reservoir. Gravitational, viscous arid capillary lorces are considered, and all fluids are considered to be slightly compressible. Some theoretical work concerning the fluid flow in one-, two- and three-space dimensions is given along with example performance predictions in one- and two-space dimensions. INTRODUCTION Since the introduction of high speed computing equipment, one of the goals of reservoir engineering research has been to develop more accurate methods of describing fluid movement through underground reservoirs. Various mathematical methods have been developed or used by reservoir engineers to predict reservoir performance, 193-5,7,8 The work reported in this paper extends previously published work on three-phase fluid flow (1) by including a rigorous treatment of capillary forces and (2) by showing certain theoretical mathematical results proving that these equations can be approximated by certain numerical techniques and that a unique solution exists. DISCUSSION The method of predicting three-phase compressible fluid flow. in a reservoir can be summarized briefly by the following steps. 1. The reservoir, or a section of a reservoir, is characterized by a series of mesh points with varying rock and fluid properties simulated at each mesh point. 2. Three partial differential equations are written to describe the movement at any point in the reservoir of each of the three compressible fluids. All forces influencing movement are considered in the equations. 3. At each mesh point, the partial differential equations are replaced by a system of analogous difference equations. 4. A numerical technique is used to solve the resulting system of difference equations. Capillary forces have been included in two-phase flow calculations.1,4,8 The literature, however, does not contain examples of prediction techniques for three-phase flow that include capillary forces. Where capillary forces are considered, each of the three partial differential equations previously discussed has a different dependent variable, namely pressure in one of the three fluid phases. Therefore, three difference equations must be solved at each point in the reservoir. Where large systems of equations must be solved simultaneously, an engineer might question whether a unique solution to this system of equations actually exists and, if so, what numerical techniques may be used to obtain a good approximation to the solution. It is shown in the Appendix that a unique solution to the three-phase flow problem, as formulated, always exists. It is also shown that several methods may be used to obtain a good approximation to the solution. The partial differential equations and differenceequations used are shown in the Appendix. Matrix notation has been used in developing the mathematical results. Two sample problems were solved on a CDC 1604. They illustrate the type of problems that can be solved using a three-phase prediction technique. SAMPLE ONE-DIMENSIONAL RESERVOIR PERFORMANCE PREDICTION A hypothetical reservoir was studied to provide an example of a one-dimensional problem that can be solved. Of the several techniques available, the direct method of solution as shown in the Appendix was used. The reservoir section studied was a truncated, wedge-shaped section, 2,400 ft long, with a 6' dip. (A schematic is shown in Fig. 1.) This section was represented by 49 mesh points, uniformly spaced at 50-ft intervals. The upper end of the wedge was 2 ft wide, and the lower end was 6 ft wide. The section

Jan 1, 1970

By V. J. Kniazeff, S. A. Naville

The problem of unsteady-state condensate-gas flow through porous media leads to a set of second-order non-linear partial differential equations. Such a set of equations is numerically solved in the case of radial two-phase flow around a well, taking into consideration both the thermodynamical properties of the fluid and the mechanical properties of the reservoir. The fluid properties, reflecting the PVT relationship of the gaseous and liquid phases, are expressed by using the partial specific masses of the two main separator products in these phases. The flow properties of the reservoir rock are expressed by the generalized Darcy's law for the liquid phase and by a quadratic relationship between the rate of flow and the pressure gradient for the gaseous phase. The numerical solution of the equations for pressure and saturation US radius and time is worked out through programs written for a computer. The evolution of bottom-hole pressures, well productivities or- deliverabilities and effluent compositions with the depletion of the reservoir is easily derived. The application to the Saharian gas-condensate field Hassi R'Mel led to a better understanding of the drainage mechanism. A zone of fairly high liquid saturation develops around the wells, reducing the effective permeability, and represents a loss of condensible products in addition to the PVT-like retrograde condensation. lnside this zone, near the well, the deviation from Darcy's law for the flow of the gaseous phase governs the well deliver-ability. A back-pressure test has been computed and correlates with the field results. INTRODUCTION Two-phase flow of volatile hydrocarbons, like condensate gas or light crude oil, may be treated as the flow of a binary mixture by an arbitrary division of the chemical components into two groups. This is translated into two equations of mass continuity, which constitute a set of relation- ships for the pressure and the saturations vs the space coordinates and the time. The equations contain the laws governing the composition and the motion of the phases. The problem so defined is solved with the assumption that the compositions of the phases at any pressure are respectively the same as those observed in a PVT measuring cell under differential liberation. In a first series of computations, it was assumed that the flow obeys the generalized Darcy's law. A satisfactory representation of the retrograde condensation around the well was thus obtained. In addition, the trend toward decreasing effective permeabilities was obtained, and the computed composition of the effluent checked the laboratory values. However, it has not been possible within this basic assumption to represent the non-linear relationship between the production rate and the bottom-hole pressure drawdown as observed for gas wells in the field. Following the advice of A. Houpeurt it was decided to consider the relative permeability to gas as a function of the velocity of the gas Phase.1 The necessary physical determinations were made by E. COstaséque using a method devised by A. Houpeurt and R. Iffly. As numerical processing of the equations progressed, several difficulties were encountered which were overcome through collaboration with the computer manufacturer. This mathematical model of two-phase flow in porous media had been primarily intended for and extensively applied to the case of the Hassi R'Mel gas-condensate field, operated by SN Repal for SEHR, a joint subsidiary of SN Repal and CFP (A). The programs have also found their applications to forecast the behavior of several fields in the Sahara area containing light and volatile hydrocarbons. BASIC EQUATIONS We will consider a zone in the porous reservoir where the flow properties and the in situ composition of the fluids can be assumed to be uniform. A part of the equations of transient flow can be written with the specific gravity of the fluids being taken equal to the sum of the contributions due to the

Jan 1, 1966

By L. J. Health

Synthetic rock with predictable porosity and permeability has been Prepared from mixtures of sand, cement and water. Three series of mixes were investigated primarily for the relation between porosity and permeability for certain grain sizes and proportions. Synthetic rock prepared of 65 per cent large grains, 27 per cent small grains and 8 per cent Portland cement, gave measurable results ranging in porosity from 22.5 to 40 per cent and in permeability from 0.1 darcies to 6 darcies. This variation in porosity and permeability was caused by varying the amount of blending water. Drainage-cycle relative permeability characteristics of the synthetic rock were similar to those of natural reservoir rock. INTRODUCTION The fundamental behavior characteristics of fluids flowing through porous media have been described in the literature. Practical application of these flow characteristics to field conditions is too complicated except where assumptions are overly simplified. The use of dimensionally scaled models to simulate oil reservoirs has been described in the literature. 1-3 These and other papers have presented the theoretical and experimental justification for model design. Others have presented elements of model construction and their operation. In most investigations the porous media have consisted of either u ncon soli da ted sand, glass beads, broken glass or plastic-impregnated granular substances—materials in which the flow behavior is not identical to that in natural reservoir rock. The relative permeability curves for unconsoli-dated sands differ from those for consolidated sandstone. The effect of saturation history on relative permeability measurements is discussed by Geffen, et a1.4 Wyga15 has shown quite conclusively that a process of artificial cementation can be used to render un consolidated packs into synthetic sandstones having properties similar to those of natural rock. Many theoretical and experimental studies have been made in attempts to determine the structure and properties of unconsolidated sand, the most notable being by Naar and Wygal.6 Others have theorized and experimented with the fundamental characteristics of reservoir rocks. This study was conducted to determine if some general relationship could be established between the size of sand grains and the porosity and permeability in consolidated binary packs. This paper presents the results obtained by changing some of the factors which affect the porosity and permeability of synthetically prepared sandstone. In addition, drainage relative permeability curves are presented. EXPERIMENTAL PROCEDURE Mixtures of Portland cement with water and aggregate generally are designed to have certain characteristics, but essentially all are planned to be impervious to water or other liquids. Synthetic sandstone simulating oil reservoir rock, however, must be designed to have a given permeability (sometimes several darcies), a porosity which is primarily the effective porosity but quantitatively similar to natural rock, and other characteristics comparable to reservoir rock, such as wettability, pore geometry, tortuosity, etc. Unconsolidated ternary mixtures of spheres gave both a theoretically computed and an experimentally observed minimum porosity of about 25 per cent.7 By using a particle-distribution system, one-size particle packs had reproducible porosities in the reproducible range of 35 to 37 per cent.6 For model reservoir studies of the prototype system, a synthetic rock having a porosity of 25 per cent or less and a permeability of 2 darcies was required. The rock had to be uniform and competent enough to handle. Synthetic sandstone cores were prepared utilizing the technique developed by Wygal.5 Some slight variations in the procedure were incorporated. The sand was sieved through U. S. Standard sieves. A 18-20 sand denotes that the sand passed the Number

Jan 1, 1966

By J. C. Melrose

The contact angle is one of the boundary conditions for the differential equation specilying the configuration of fluid-fluid interfaces. Hence, applying knowledge concerning the wettability of a solid surface to problems of fluid distribution in porous solids, it is important to consider the complexity of the geometrical shapes of the individual, interconnected pores. As an approach to this problem, the ideal soil model introduced by soil physicists is discussed in detail. This model predicts that the pore structure of typical porous solids will lead to hysteresis effects in capillary pressure, even if a zero value of the contact angle is maintained. The model is generalized to situations in which the contact angle takes on values between zero and 40 degrees. For the imbibition branch of the capillary- pressure function, the model predicts a considerable departme ftom the usually assumed cos 0 relationship. In fact, according to the model, it is possible that a displaced wetting phase will not be able to reimbibe, even when the contact angle does not exhibit hysteresis. INTRODUCTION The subject of capillarity in porous media has long been of interest in many branches of engineering and applied science. The earlier investigators were those concerned with the physics of soils. 1-5 More recently, petroleum engineers and others dealing with the problems of petroleum production from reservoir rock have given much attention to the subject.6-10 Also, important additions to the literature of capillarity have been contributed from the field of chemical engineering.11-12 These attest to the wide range of industrial applications in which capillary phenomena play a role. The present paper is concerned with the role which wettability plays in capillary action in porous media. As is well known, capillary-rise (or capillary pressure) phenomena have frequently been interpreted with the aid of a model employing the concept of a cylindrical capillary tube. This approach has enjoyed a certain degree of success in correlating experimental results.13 The generalization of this model, however, to situations which involve varying wettability, has not been established and, in fact, is likely to be unsuccessful. In this paper another approach to this problem will be discussed. A considerable literature relating to this approach exists in the field of soil science, where it is referred to as the ideal soil model. Certain features of this model have also been discussed by Purcel114 in relation to variable wettability. The application of this model, however, to studying the role of wettability in capillary phenomena has not previously been attempted in detail. In the present paper, additional features of the model are introduced. These features are critical in determining the quantitative behavior of the model. GENERAL FEATURES OF CAPILLARY HYDROSTATICS BASIC PRINCIPLES When the interstices of a typical porous solid are occupied by two or more immiscible fluid phases, the fluids are microscopically commingled. Hence, fluid-fluid interfaces are found within a certain fraction of the pore openings. The fundamental equation of capillarity specifies the configuration of these fluid-fluid interfaces. This is known as the Laplace equation, when derived from mechanics, and as the Gibbs-Kelvin equation when derived thermodynamically.15 Given two fluid phases, a and ß, in hydrostatic equilibrium, the Laplace equation states that the respective fluid pressures in regions close to the interface are related by Here oaß is the surface or interfacial tension and rl and r2 are the principal radii of curvature of the surface or interface. The pressure difference, Pa-Pß, is the capillary pressure, PC. As Buff15 has shown, Eq. 1 states the condition for hydrostatic equilibrium within the two-phase confluent region, which is referred to as the interface. It thus can

Jan 1, 1966

By T. G. Richardson, F. M. Perkins

In the recent paper of Richardson and Perkins entitled "A Laboratory Investigation of the Effect of Rate on Recovery of Oil by Water Flooding,"' the authors found very little appzrent effect on oil recovery of having free gas saturation present prior to water flooding in unconsolidated sands. We agree that the effect in this particular system was small but do not feel that the data presented should be regarded as disproving work of others which has shown an appreciable effect. The authors do not make such a claim, but neither do they advance an explanation of the difference. In our experience the following factors must be carefully considered when studying the relation between residual oil saturation after water flooding and initial free gas saturation: (1) the magnitude of the inherent residual oil saturation, i.e., the residual oil saturation with no free gas; (2) the increase in oil-water viscosity ratio with pressure lowering if the gas saturation is obtained by solution gas drive; and (3) the oil shrinkage which occurs with pressure lowering and gas evolution. The system studied by Richardson and Perkins gave residual oil saturations as low as 20 per cent without free gas present, and one should perhaps not expect much further lowering. As an illustration of the magnitude of the effect when the residual oil saturation is inherently higher we submit the accompanying figure. These data were obtained on an unconsolidated sand under elevated pressure in a system similar to the one described by Richardson and Perkins. Gas saturation was established by gas drive before water flooding. The viscosity ratio was 600:1 under the conditions of the displacement. Here the gas was extremely effective in reducing residual oil saturation. This fact was also indicated by Kyte, et al.2 Their plots of residual oil saturation vs initial gas saturation show trends of greater free gas effect with higher oil viscosities. The pressure-dependent effects were not discussed by Richardson and Perkins. They should be separated from the free gas effect if the objective of the study is a correlation of residual oil saturation vs initial free gas saturation. The increase in oil viscosity when gas is released from solution tends to increase residual oil saturation. This effect is rather small in the viscosity range of kerosene. A Buckley-Leverett calculation, assuming an increase in viscosity ratio from 1.2 to 1.8, indicates an increase in residual oil saturation of about 1 per cent of pore volume. Thus, one would expect this system, if operated at constant viscosity ratio, to show a slightly greater free gas effect. As Dyes- as shown, oil shrinkage can offset the beneficial effect of free gas on waterflood oil recovery, if the gas saturation is established by depletion. Since

Jan 1, 1958

By G. M. Webster, G. E. Dawsongrove

The use of percussion-type side-wall cores as an aid in detection and evaluation of hydrocarbon shows and in examination of rock properties has become increasingly popular in recent years. Sidewall coring has an advantage over normal bottom-hole coring methods in that the zones may be surveyed with the electric log prior to selection of sample depths. This allows an efficient sampling of all zones of interest, a procedure not possible with bottom-hole coring techniques unless the costly procedure of continuous coring is utilized. Percussion sidewall coring presently provides the most rapid and lowest cost method of obtaining sidewall cores. It has been recognized for some time that certain discrepancies exist between core analysis data obtained from percussion cores and data obtained from other types of core Samples. This paper presents existing field and laboratory data that were available to the authors. Several examples of comparative core analysis data obtained from various fields are presented along with photomicrographs of thin sections of conventional and percussion cores. The results of this investigation are presented to throw more light on the possible disturbed condition of percussion sidewall cores. SAMPLE DISTURBANCE AND FRACTURING The principal cause of discrepancy between conventional core analysis data and percussion sidewall core analysis data is believed to be the shattering and re-adjustment of the grains due to the large impact shearing forces produced by mechanics of percussion coring. Figs. 1 through 5 are photomicrographs of thin section slides of formation samples obtained by means of diamond cores, conventional cores, and percussion sidewall cores. Figs. 1 and 2 show samples of the

By N. Mungan, J. L. Thompson

Laboratory displacement tests were performed to study oil recovery elficiency by gravity drainage in fractured systems under miscible conditions. The porous media used were cylindrical Berea and Blue Jacket sandstone cores containing a number of well-defined, artificially formed, vertical and subvertical fractures. Butane and Soltrol 130 were the two miscible fluids used. The purpose of the study was to examine the influences of the displacement rate, fracture density, fracture orientation, fracture permeability, matrix permeability, crossflow, core length and connate water on the oil recovery. It was found that displacement rate, matrix permeability and the subvertical fractures affected oil recovery most. The critical flow rate, based on the matrix permeability, was found to be a significant factor in the process. For displacement rates below the critical flow rate, the oil recovery efficiency appeared to be unaffected by the density of the subvertical fractures. At the high displacement rates, the fracture density becomes important, with the recovery being the most efficient in the core having the greatest number of the subvertical fractures. The magnitude of the fracture permeability, the fracture orientation, the core length and the connate water have little effect on the oil recovery efficiency. INTRODUCTION Gravity drainage under miscible conditions from relatively thick reservoirs can be a very efficient recovery process, especially at low flow rates where the gravity forces are dominant and, consequently, the adverse viscous fingers associated with the unfavorable viscosity ratio are minimized or eliminated. Such a process might involve injection of an LPG or some solvent bank at the crest and then driving the bank with dry gas. For some favorable combination of reservoir temperature, pressure and oil composition, there may be no need to inject any solvent bank, since enrichment of the dry gas by the light ends of the crude creates an in-situ solvent bank. Slobod and Howlett2 have studied the effects of gravity segregation in vertical unconsolidated porous media under miscible conditions in the laboratory. The main variables in their study were the viscosity ratio, the density differences and the rate of flow. In this study, the objective was to find the influence of fractures on gravity drainage under miscible conditions. A greater number of related variables were also studied. LABORATORY STUDY CORE DESCRIPTION Displacement tests were performed mostly in cylindrical Berea sandstone cores. In a few cases in which a much lower matrix permeability was desired, Blue Jacket sandstone cores were used. The vertical fractures were formed by cutting the cores with a thin circular blade or by parting them along a bedding plane. The vertical fracture plane always contained the axis of the cylindrical core. The subvertical fractures, cut with a saw, were inclined 45' from the horizontal. The line of intersection between the planes of the vertical and subvertical fractures was parallel to the circular faces of the cores in all cases except one, as depicted by VF3HF-2 in Fig. 1. Fracture geometry of the other cores is also given in Fig. 1. Tables 1 and 2 give the physical properties of the solid and fractured cores, respectively. The suffixes and prefixes shown for each core have been used throughout the paper so that the reader may discern the fracture geometry from the core numbers.

Jan 1, 1970

By Charles R. Stewart, William W. Owens

Reservoir performance predictions based on laboratory core test data assume that fluid flow is laminar for the laboratory test. A study has been made to determine the validity of this assumption for laboratory tests on various types of porosity found in producing limestone formation. Data are presented which show that turbulence and slippage can occur during laboratory tests on hetero-geneous-type porosity limestones, thus causing serious errors in measured single-phase permeabilities and two-phase relative permeability characteristics. In single-phase flow tests it is possible to eliminate turbulence and correct for slippage or to eliminate both factors by controlling test conditions. It is not always possible to control test conditions and thereby eliminate turbulence and slippage in two-phase .flow tests. A correction method is presented which can be used to calculate the true two-phase laminar flow relative permeability characteristic even though furbulence and slippage exist. .INTRODUCTION It is customary to make use of Darcy's law and modifications of this law, together with laboratory data on formation core samples to predict the performance of producing reservoirs. Such predictions are based on an assumption that fluid flow is in the laminar or streamline region for the laboratory test. It was the purpose of this inves- tigation to determine the extent to which turbulent flow may occur in laboratory fluid flow tests on hetero-geneous porosity limestones. Considering that turbulent flow conditions might exist in some laboratory fluid flow tests, additional emphasis was placed on the development of a method to correct for turbulence when laminar flow conditions could not be attained. FLUID FLOW CONCEPTS FOR POROUS MEDIA The Influence of Pore Geometry on Fluid Flow One of the more important factors influencing fluid flow in porous media is the geometry of the pore space which includes such characteristics of the pores as size, shape, distribution, roughness, uniformity, etc. In general, oil- and gas-producing formations can be divided into two broad types on the basis of pore geometry. One has been called sandstone-type porosity media, which is characterized by a small range in pore size, uniformity in shape of the pores, smooth pore surfaces and a regular and uniform distribution of pores. The other type has been called heterogeneous porosity media and is usually limited to the dolomites and limestones. This type is characterized by a wide variation in the size, shape, and distribution of the pores and rough, irregular pore surfaces. It is therefore apparent that conditions are much more favorable for turbulent flow* in heterogeneous-type porosity media than in sandstone-type porosity media. Interrelationship Between Turbulence and Gas Slippage In studying the problem of turbulent flow in laboratory tests on porous media, it is necessary to be aware of the interrelationship between slippage and turbulence for gas flow. As a result of slippage or the Klinken-berg effecta, apparent perrneabilities to gas are greater than the true value because there is no stationary layer of gas in contact with the walls of the flow channels. Gas slippage decreases as the mean free path of the gas molecules decreases. Since the mean free path of any gas decreases with increasing density, increases in static pressure result in lower apparent gas permeabilities. However, a reduction in gas permeability can also be due to turbulence. Therefore, in studying only turbulent flow in porous media, it is necessary to hold gas density, and slippage, constant or to reduce slippage to a negligible value by operating at high static pressures. Presentation of Laminar and Turbulent Flow Data A graphical relationship between permeability and a pseudo-Reynolds number, N,, will be used to show the two types of fluid flow, i.e., laminar and turbulent. The usual graphical method for such a description has been the use of friction factor-Reynolds number charts4. On such a logarithmic diagram, the laminar region appears as a straight line having a slope of 45 degrees. As the friction factor decreases and the Reynolds number increases, the turbulent region is reached and appears as a deviation from the 45-degree slope line. However, in petroleum engineering literature resistance of por-

By L. L. Handy

The most common method of identifying hydrocarbon-bearing strata in a well that penetrates many different formations involves measurement and interpretation of the electrical properties of the formations as determined by electrical logs. Even though this method is used extensively, and even though in a great many instances it is capable of indicating presence of oil or gas, situations arise for which it is extremely difficult, if not impossible, to deduce the presence of hydrocarbons. These situations may involve the following. 1. A thin formation, bounded by highly resistive formations, in which it is impossible to obtain the actual resistivity of the uninvaded zone with existing logging devices. 2. A formation in which invasion has been so extensive that a value for the uninvaded zone resistivity cannot be obtained. 3. A very shaly formation in which the resistivity index, I, is lower than that usually associated with productive formations. 4. Laminated formations comprised of thin productive sands separated by thin shale streaks in which the individual sand and shale streaks are too thin to permit measurement of uninvaded-zone resistivity with existing logging devices. 5. Productive formations in which the water saturation is high. To extend the utility of electric log interpretation to identification of hydrocarbons in all types of formations, there is strong incentive to find a method not subject to these limitations. Some time ago, in connection with research on the wettability of reservoir rock, an investigation was conducted in which the resistivities of cores were measured shortly after they were removed from a core barrel, and again after they had been extracted and restored to their original oil and brine saturation.' The resistivities after extraction were generally lower. Other tests made on the cores indicated that they were more nearly water wet after they were extracted; thus, it was assumed that the observed changes in resistivities were due to a change in wettability of the cores. Other experiments'," have shown that resistivities of rock samples are sharply dependent on wettability. These experiments have shown that oil-wet samples are more resistive than water-wet samples. To obtain an understanding of how the wetting properties of the surfaces of core material affect electrical resistivity, a series of experiments was conducted. Two groups of core samples were prepared for testing. One group contained brine, but no residual oil. The other group was saturated with brine, flooded with oil to a low water saturation, then flooded with brine to a final residual oil saturation. Resistivity measurements were made on each group. Both groups were then flooded with the original brine to which a chemical had been added that renders sand and clay surfaces preferentially oil wet, a so-called reverse-wetting agent. very little change in resistivity was observed in cores containing only water. The group containing residual oil, however, showed resistivity increases of 100 to 200 per cent. These experiments showed that the resistivity of a core containing oil could be altered by changing wet-tability of the core. Moreover, the possibility was introduced that reverse-wetting agents might be employed as the basis for a logging method for identification of oil-bearing strata. Since behavior of a porous rock containing gas and water might be expected to be similar to that of a rock containing oil and water, such a method should also be applicable to identification of gas-bearing zones. In principle the wettability of the invaded zone could be reversed without altering conductivity of the interstitial water or the hydrocarbon saturation therein. Those strata showing significantly increased invaded-zone resistivities would, therefore, contain hydrocarbons; those with no significant change would be filled only with water. Addition of a reverse-wetting agent to a hydro carbon -bearing zone which is, by nature, already preferentially oil wet would not result in an enhancement of its resistivity. It is generally believed, however, that most hydrocarbon-bearing strata are preferentially water wet.

By R. C. Hassinger, D. V. von Rosenberg

Transverse dispersion has received considerably less treatment in the literature than has longitudinal dispersion. Different methods for determining transverse dispersion coefficients have been used in different investigations, and the results obtained have not been consistent enough to permit accurate generalizations as to the effect of various physical parameters on the magnitude of these coefficients. A numerical solution to the differential equation describing transverse dispersion in the absence of longitudinal dispersion was obtained to enable one to calculate the dispersion coefficient from experimental results. The more general dispersion equation including longitudinal dispersion also was solved numerically to give quantitative limits of a dimensionless group within which the assumption of negligible longitudinal dispersion is justified. Possible experimental procedures were examined, and one utilizing a cylindrical packed column was chosen for the determination of transverse dispersion coefficients. Values of these coefficients were determined for a system of two miscible organic fluids of equal density and viscosity, for two sizes of packing material over a wide range of flow rates in the laminar regime. The dispersion coefficient was found to decrease, for a constant value of the product of packing size and interstitial velocity, as the size of the packing material particles increased. INTRODUCTION Longitudinal dispersion has received extensive treatment in the literature, and consequently is better understood than its orthogonal counterpart, transverse dispersion. Many mathematical models of dispersion processes assume that transverse dispersion is rapid enough to damp out any radial** concentration gradients and therefore may be neglected. Laboratory and production results, however, indicate that this is a poor assumption.l,2 Various experimental procedures for determining transverse dispersion coefficients have been used in previous investigations, but the results have generally been expressed by similar correlations. The transverse dispersion coefficients obtained, however, have often varied considerably for given values of the correlation parameters. We feel that further experimental determinations of transverse dispersion coefficients will help alleviate some of the inconsistencies in these empirical correlations. One assumption implicit in all previous investigations is that of negligible longitudinal dispersion in the experimental system. An attempt to justify this assumption often is made using intuitive reasoning, but it is apparent that this reasoning must break down as the condition of zero flow rate is approached. A mathematical examination of the equations describing the system yields physical limits outside of which the assumption of negligible longitudinal dispersion is invalid. BACKGROUND In a porous medium, the "effective molecular diffusivity" De is less than the molecular diffusivity D measured in the absence of a porous medium, due to the tortuous path which a diffusing molecule must travel. Various authors3-5 have reported values of the ratio De/D in the range of 0.6 to 0.7. When there is fluid flow within the porous medium, mass transfer occurs by convective dispersion as well as by molecular diffusion. These are separate phenomena and can be treated as such on a microscopic scale. However, the mathematical complexity is such that only extremely simple geometries could be considered, and the results hardly would be applicable to the complex geometries existent in actual porous media. A more

Jan 1, 1969