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By H. J. de Haan, L. Schenk

Scope for Thermal Recovery in Shell's Heavy Oil Fields in Venezuela The main heavy oil reservoirs on the East coast of Lake Maracaibo (Fig. I), known as "Bolivar Coast", initially contained some 20 billion bbl of oil in place with gravities in the range of 10 to 15° API. The current total production rate is about 400,000 B/D. Since the recovery to date is on an average only 12.5 percent of the initial oil in place, these fields offer a vast potential for secondary recovery methods. The reservoirs are characterized by moderate depth (generally 1,000 to 3,000 ft; maximum 5,000 ft), good formation properties such as net oil sand thickness (50 to 300 ft), high porosity (30 to 40 percent) permeability (1 to 3 darcies) and oil saturation (initially about 80 percent) and high oil viscosity (100 to 10,000 cp in situ), all of which are favorable for the application of thermal recovery processes. Since 1957, steam drive, steam "soak" and underground combustion have been tested or are being tested in this area.1-3 This paper deals with a major steam drive test in the Tia Juana field. The test was commenced in Sept., 1961, on the basis of encouraging results of laboratory experiments and pilot field tests, which will be discussed briefly. Early Laboratory Investigation of the Steam Drive Process In 1956 a series of model experiments was carried out in the Koninklijke/Shell Laboratorium, Amster- dam, to investigate the displacement of heavy oil by steam.' The prototype studied was a horizontal, un-consolidated heavy oil reservoir, subjected to a linear steam drive. The model, representing a slab from this reservoir along the main flow direction, consisted of a steel tube filled with oil-saturated sand, provided with layers of rock at the top and bottom to simulate conductive heat losses to cap and base rock. Lateral heat losses, perpendicular to the main flow direction, were minimized by insulating the sides of the model. Heat flow and viscosity and gravity forces were scaled by taking the steam injection rate and sand permeability inversely proportional to the geometric reduction factor of the model, and using materials (oil, steam, cap rock, etc.) identical with those of the prototype. The main feature of the process was a frontal displacement mechanism that was observed within wide ranges of injection pressure and rate, oil viscosity, initial oil saturation and sand permeability. The high stability of this front can be attributed to the high volumetric steam flow rates, combined with the reduction in oil viscosity. High recoveries were obtained due to the low residual oil saturation of about 15 percent in the zone swept by steam. The latter value pertains to the specific test conditions (heavy oil, uncon-solidated sand) and may be lower for light oil, due to the effects of steam distillation. The applicability of this experimental approach is limited, as it does not take into account the fact that

Jan 1, 1970

By P. B. Crawford, W. L. Huskey

A study has been made to determine the effects of random vertical fractures existing in a reservoir matrix on the effective matrix properties and producing characteristics of a well. The study indicated that short vertical fractures may distort the isopotentials by 1 percent in the vicinity of the fracture. A correlation was developed which showed the effect of fracture density on the producing capacity of a well. The correlation indicated that fracture shape had little effect on producing capacity, but that total fracture length was closely correlated with the producing capacity. Effective values of reservoir permeability were found to correlate closely with fracture density when measured parallel to the direction of flow. INTRODUCTION In reviewing the petroleum literature, it is found that in most analytical solutions of hydrocarbon reservoir behavior the assumption is made that the rock is uniform throughout although core analysis and well logging surveys indicate that substantially all reservoirs are heterogeneous and some are fractured. Only in recent years have investigators studied the effects of reservoir hetrogeneity. Russell1 discussed the effect of fractures in reservoirs. The literature indicates that fractures greatly affect the exploitation of man reservoirs. This is described by Elkins and Skov2 and Little-field et al. 3 Several types of reservoir heterogeneity have been studied by unsteady-state methods. In 1959, Landrum et al. 4 showed that transient phenomena in hydrocarbon reservoirs could be studied by thermal models. pickering5 studied the flow of heat in stratified linear reservoirs consisting of plates of different metals separated by sheets of rubber which were suddenly exposed to a low-temperature source at one end. Cotman6 studied both laterally and vertically heterogeneous thermal model reservoirs with special reference to cross flow. Another type of reservoir heterogeneity involves the presence of highly permeable, circular lenses around or near the wellbore which was studied by Miesch7 using a thermal and a steady-state model. Many studies have been made of the effects of induced hydraulic fractures on well performance. 8-10 Work regarding vertical fractures that exist initially in the reservoir was reported by Givens, et al.11 Since there appears to be little published work on the importance and influence on reservoir performance of fractures that exist out in the rock matrix, this study was conducted to determine the effects of vertical fractures on pressure distribution, producing capacity and the effective permeability of the reservoir rock. The analogy between the flow of electrical current through a conducting body and the flow of fluid in a porous medium provides a convenient method of studying various aspects of petroleum reservoirs. For electrical radial steady-state flow,12 For steady-state fluid flow, With the above analogy, it is only necessary to establish the ratio of the values of the electrical units in terms of the corresponding flow units to make the analogy quantitative. The ratios are shown below.

By R. W. Parsons

The over-all apparent single-phase permeability of fracture-rock systems was studied using two different two-dimensional models. In a strict sense the results are applicable only to these models, yet may give some insight into real reseruoir behavior. The regular fracture-matrix model can have any number of regular fracture sets. All fractures within each set have the same orientation, width and spacing. The gross behavior is amlytically equivalent to that of an anisotropic permeable medium. In the heterogeneous fracture systems, individual fracture conductivities vary within a two-dinensioml network pattern. A computer program calculates the overall permeability after the conductivities are randomly placed in the network. The calculated permeability depends upon the distribution of these conductivities and the particular fracture pattern into which they are placed. Although detailed information on actual reseruoir fracture systems is not available, this study shows how some of the possible variables may affect obserued gross reservoir characteristics. INTRODUCTION Characteristics of reservoir rocks are determined by direct (core analysis) or indirect (logging and production tests) examination. Both methods, singly and together, are widely used in reservoir evaluations. For fractured reservoirs a problem exists in relating characteristics of cores to the in situ reservoir properties. Knowing fracture characteristics can be of value in determining production mechanisms. Because of the limitations of observation, the intimate details of a fractured rock system will never be known. One approach in analyzing fractures is to study the behavior of some conceptually simple models. The two models analyzed contain features which may be realistic, yet are mathematically tractable. Flow is considered in two dimensions only, with the plane of interest at right angles with the fractures. This concept represents the horizontal flow in a reservoir system with the fractures oriented vertically. Only single-phase, laminar flow is considered. The regular fracture-rock system permeability is calculable from a simple analytical expression. The heterogeneous system requires a computer program for calculating overall permeability.

Jan 1, 1967

By C. F. Weinaug, R. W. Farley, J. F. Wolfe

A rapid, accurate method for predicting the dew points of gas condensate systems and their subsequent normal and retrograde phase behavior with pressure decline has been developed. The method predicts dewpoint pressure, volume percent liquid, condensate content of produced gas, and the liquid and gas compositions at pressures below the dewpoint pressure. Use of a modified form of the BWR equation of state, which was established by regression calculations on experimental data from several gas condensates, provides depletion behavior predictions that are significantly improved over predictions using K-values based on convergence pressures and Standing's correlation for density. INTRODUCTION Prediction of complex hydrocarbon-mixture thermodynamics is fundamental to reservoir and gas process engineering. In particular, accurate descriptions of phase behavior are needed to plan production operations for gas-condensate reservoirs. Although analysis of reservoir behavior during producing and cycling operations can be made for simple reservoir models, extensions of these studies to more complex reservoir systems are still difficult, both with regard to numerical techniques and the phase behavior of the fluids. This report presents a mathematical method for handling the complex phase relationships encountered in retrograde condensates; it represents an advance in reservoir behavior technology in that it will aid the development of closed-form mathematical representations of total production systems. A typical phase diagram for a complex hydrocarbon mixture is shown in Fig. 1. When the reservoir temperature T, is higher than the critical temperature Tc, and lower than the critical-condensation temperature Tct' the mixture is a gas condensate. Decreasing the pressure on a condensate while maintaining nearly constant temperature will result in two phases forming at the dew point. Further decrease in pressure will cause additional liquids to be formed. This phenomenon is called retrograde condensation. As pressure continues to decrease, normal phase behavior regions are encountered (liquid vaporizes as the pressure decreases) until the low-pressure dew point is encountered. When a gas-condensate reservoir is produced, gas is withdrawn and the volume of fluids in the reservoir remains nearly constant as the pressure declines. Heavier components tend to accumulate in the liquid phase as a result of retrograde condensation, while lighter components tend to concentrate in the vapor phase. Thus, as the gas is produced the composition of the gas and the condensate

Jan 1, 1970

By H. N. Hall

Reservoir and producing characteristics can govern the decision to use either a one-, two- or three-dimensional model for making predictions for gravity-drainage reservoirs. Examples of conditions requiring one-, two- and three-dimensional calculations are given. In 1961 the author presented a method for predicting one-dimensional gravity-drainage performance. This work has been exrendcd to obtain a three-dimensional model, which is described and for which a sample problem is presented. Introduction Many papers on gravity drainage or gas-cap drive'-3 have been presented for conditions where the reservoir is treated as a single areal segment so that, in effect, only one-dimen-sional flow is considered. Other authors'-Q ave given evidence showing that the one-dimensional concept would be unsatisfactory to predict reservoir performance for certain types of reservoirs. Experience has indicated that specific reservoir and producing conditions govern whether one-, two- or three-dimensional concepts must be used as a basis for reliable reservoir performance predictioils. Previously the author presented a paper outlining a method for predicting the performance of gravity-drainage reservoirs.' That paper treated the reservoir as a single areal segment so that it was applicable for only one-dimensional flow. The work has been extended to obtain a three-dimensional model that can be used to predict performance of gravity-drainage reservoirs. Although capable of being three-dimensional, it can be used equally well for one- and two-dimensional predictions. This paper discusses those reservoir and producing conditions governing the choice of model, and also describes a three-dimensional model suitable for predicting reservoir performance in the more complex situations. Discussion The Choice of Model To predict reservoir performance, it is necessary to represent the reservoir by a physical model. A mathematical model, based on the assumed physical model, then is used to predict the movement of oil and gas within the reservoir. The word "model" is used here to connote the combination of both a physical model and a mathematical model based on the physical model. Certain types of reservoir situations may be predicted suitably with a one-dimensional model, whereas two- and three-dimensional models may be required for other situations. This section illustrates reservoir conditions associated with the use of a specific type of model. Many types of reservoir structures are such that gravity drainage possibly could be an important factor in oil production. Production practices, as well as the nature of the structure. can influence the type of model that should be used. obviously, it is impossible to discuss all combinations of reservoir structure and production practices that might be encountered in gravity-drainage reservoirs. The following are typical examples of conditions where a choice must be made among a one-, two- or three-dimensional model to predict reservoir performance satisfactorily. Conditions Requiring a One-Dimensional Model Fig. la shows contours for an asymmetrical anticline in which gravity drainage might be expected to be important. Fig. Ib depicts a cross-section along Line A-A in Fig. la. Existence of a gas cap is a good clue that gravity drainage will be an important factor in producing oil from this type of reservoir. However, it is not essential since a gas cap will form as oil is produced from the reservoir. If the proper combination of reservoir permeability and withdrawal rates is present, the reservoir will behave like a

Jan 1, 1969

By W. A. Klikoff, I. Fatt

The concept of fractional wet wattability is examined. Fractional water wettability of a reservoir rock is defined as the fraction of the internal surface urea that is in contact with water. Capillary pressure and relalive permeability of unconsolidated sand are shown to be functions of fractional wettability. INTRODUCTION The petroleum industry has long recognized that wettability of reservoir rock has an important effect on multiphase flow of oil, water and gas through reservoirs. As early as 1928 the American Petroleum Institute sponsored a study of wettability as part of API Project 27 at the U. of Michigan.' Despite 30 years of research, there is still little exact knowledge of the wettability of reservoir rocks. There are two parts to the wettability problem. After agreeing to a uniform nomenclature in regard to wettability,' the first question to be answered is, "What is the in situ wettability of a given reservoir rock?" If this can he answered the next question is, "What part does wettability play in determining the characteristics of multiphase fluid flow through the rock?" This paper represents an oblique attack on the problem of wettability. No attempt is made here to answer the basic question of wettability in situ. instead the consequences of the concept of fractional wettability are examined. Multiphase flow in sandpacks is shown to he highly influenced by fractional wettability. Jennings' has given 3 definition of wettability and the other terms used in discussing wettability. These terms must be applied to the physical situation existing In reservoir rock. A survey of the pertinent literature from 1928 to 1956 indicates that the concept of a contact angle was applied to reservoir rock in the same way it would be applied to a flat, homogeneous surface. Attempts were made to state quantitatively the wettability of a reservoir rock in terms of a contact angle which was presumably constant at all points on the very rough and heterogeneous interior surface of a porous rock. Calhoun, et al,3-6 prepared synthetic consolidated and un-consolidated porous media in which they claimed there was a known uniform contact angle. They then showed the effects on the capillary pressure and relative permeability characteristics of varying this angle. The API Project 47 at the U. of Texas' and others' have made extensive studies of an indirect approach to the contact angle through the use of heat of wetting data. Even if successful. however, this approach also states the wettability of porous rock in terms of a contact angle which is uniform over the entirc surface. If the angle varies from one part to another on the internal surface, there is no way of determining from the measurements the area distribution of contact angles. In 1956 Brown and Fatt5 suggested that the concept of a contact angle, as applied to reservoir rock, be abandoned. This suggestion was made because it is known that the internal surface of most reservoir rocks is composed of many different minerals, cach with a different surface chemistry and a different capacity to adsorh surface active materials from reservoir fluids. Furthermore, the operation of a contact angle in determining the form of a fluid-fluid interface is difficult to picture in the very complex geometry of a pore. Brown and Fatt proposed that the wettability of reservoir rock be stated in terms of the fractional internal surface area that is in contact with water or oil. All surfaces on which there is water are called water-wet; surfaces on which there is oil are called oil-wet. The fractional water wettability is then stated as a number which represents the fraction of the internal surface that is in contact with water. A symmetrical statement can be made for the fractional oil wettability. The concept of a fractional wettability as previously stated has in its favor the recognition of the heterogeneous mineral composition of most reservoir rocks. Another point in its favor is that fractional wettability can be measured quantitatively with relative ease. Hol-brook and Bernard10 se a simple dye adsorption test to obtain fractional wettability of reservoir rocks. Amott11 uses a combination of imbibition and displacement to arrive at a wettability index of reservoir rocks which seems related to fractional wettability in the range 0.25 to 0.75 fractional water wettability. Jennings" has shown the changes in relative permeability that take place when a porous material is changed from unity to zero fractional water wettability. He also shows that reservoir rocks in their natural state, but at room temperature and atmospheric pressure, have relative permeability characteristics which would indi-

By A. B. Dyes

In the operation of a water-driven reservoir, a free gas saturation can he established by maintaining production rates fast enough to cause the reservoir pressure to decline below the bubble point. The benefit of such a procedure on the displacement efficiency of the oil by water is illustrated for two types of sandstone samples from one reservoir. Those rock samples showing the poorest recovery to water drive in the absence of a free gas saturation give the most improvement in the presence of a free gas saturation. Employing one type of reservoir fluid (low .shrinkage) the benefit of evolved gas on oil displacement is calculated at several reservoir pressures below the bubble point. This demonstrates the presence of air optimum pressure for water displace,rnent which is .several hundred pounds per .square inch below the bubble point. Displacement of oil at the optimum pressures results in the removal of 7 to I? per cent more oil than would be obtained by displacement at the bubble point pressure. In some instance. this magnitude of increase may be worthy of consideration in establishing the maximum efficient rate of production from many water-driven fields. INTRODUCTION For many years the belief that the use of gas could be combined with that of water to result in a better recovery operation than the use of either of these phases alone has been prevalent in the oil industry. This is evidenced by the several field trials which have employed the injection of gas and water in combination.1.2.v In most cases the small amount of any additional recovery, in combination with the limited knowledge available on the older reservoirs, has left uncertain the influence of gas on recovery in these earlier attempts to combine the use of gas with water. Research studies in recent years have demonstrated through laboratory. floods the magnitude of the benefit of the presence of gas on the displacement of oil from sandstones by water.5,6,7,8,9,10,11 In addition, studies of the distributions of the phases which are present during the displacement process have developed concepts concerning the mechanism by which this benefit is obtained."." It now remains for the engineer to incorporate these findings into the general technology of reservoir operations. It is the purpose of this paper to illustrate that the production of a water-driven reservoir at pressures below the bubble point is one such operation whereby improved oil displacement efficiency can be obtained as a result of gas evolution. The establishment of a gas saturation by evolution of dissolved gas was one of the techniques used by several investigators in the laboratory studies. The possibility of obtaining the benefit of gas evolution in water-driven reservoirs by producing at pressures less than the bubble point has 3150 been pointed out previously." The desirability of this operation has undoubtedly

Jan 1, 1955

By H. J. Ramey

Conventional calculation of total system isothermal compressibility for a system containing a free gas phase involves, among other things, evaluation of the change of oil and gas formation volume factors and the gas in solution with pressure. Preferably, this information should be obtained from laboratory measurements made with particular oils and gases. Often, experimental measurements are not available. In this case, it is necessary to obtain pressure-volume-temperature relationships from general correlations such as those of Standing' for California oils. In order to speed estimates of compressibility, generalized plots have been prepared of the change of both oil formation volume factors and gas in solution, with pressure from Standing's correlations. A generalized plot for estimating the change in the two-phase (oil and gas) formation volume factor with pressure is also presented. Usually, the effect of gas dissolved in reservoir water upon the total system compressibility is neglected for gas saturated systems, due to the low solubility of gas in water. Results of this study indicate that the increase in total system compressibility caused by solution of gas in water is often as large as the compressibility of wafer, and can be magnitudes larger for low pressure systems. Generalized results for estimating the change of gas in solution in water with pressure are presented in tabular and graphical form. INTRODUCTION During the past decade, pressure build-up and drawdown techniques have gained an important place in reservoir engineering. Build-up and drawdown analyses are only two special applications of the broad field of transient fluid-flow theory. All solutions of transient fluid-flow problems contain a parameter called the total system isothermal compressibility. This property of fluids and porous rock is a measure of the change in volume of the fluid content of porous rock with a change in pressure, and it may vary considerably with pressure. Evaluation of total system isothermal compressibility is not difficult, but it is tedious and time-consuming. Often compressibilities are estimated roughly, or transient flow methods are neglected completely. The benefits of using accurate system compressibility in properly-executed build-up or drawdown analyses are: 1. Better planning of pressure build-ups may be achieved to avoid unnecessary loss of revenue due to excessively long shut-in periods, or to shut-in periods too short to yield useable data. 2. Better and more reliable estimates of static formation pressures for reserves estimates and rate performance estimates. 3. Reliable information for evaluation of well completion effectiveness, and planning and interpretation of well stimulation efforts. The purpose of this paper is to clarify the nature of the total system isothermal compressibility, and to present useful methods for estimation of compressibility, particularly for systems containing a gas phase. DEVELOPMENT Numerous publications have presented solutions to transient single-phase flow of slightly compressible fluids, stressing pressure build-up applications. In transient flow, a compressibility* term arises to permit volume content of fluids in porous rock to change as pressure changes. The basic nature of the compressibility term is usually taken for granted. Problems arise in practical applications of transient fluid theory because most published works consider only one flowing fluid—in an ideal porous system containing only one fluid. In 1956, Perrine2 presented an intuitive extension of single-phase flow pressure build-up methods to multiphase flow conditions. Later, Martin- established conditions under which Perrine's multiphase build-up method had a theoretical foundation. Perrine has shown that improper use of single-phase build-up analysis in certain multiphase flow situations can lead to gross errors in estimated static formation pressure, permeability and well condition. It is likely that, much pressure build-up data for oil wells should be analyzed on the basis of multiphase flow. For both single-phase and multiphase build-up analysis, the isothermal compressibility term in dimensionless time groups often should be interpreted as the total system compressibility. All real reservoirs contain one or more compressible fluid phases. In addition, rock compressibility can contribute in an important way to the total system compressibility. The proper total system compressibility expression may contain terms for compressibility of oil, gas, water, reservoir rock and terms for the change of solubility of gas in liquid phases.

Jan 1, 1965

By W. T. Weller

In Part I, a study of pressure build-up curves calculated for conditions under which both oil and gas flow led to the conclusion that the presence of a dispersed free gas phase in an oil reservoir must be taken into consideration to estimate accurately average reservoir pressure and permeability from build-up curves. Familiar methods based on the assumption of no free gas can be extended to the two-phase case by using total compressibility and mobility in place of single-fluid compressibility and mobility. These methods give correct values for average pressure and permeability when gas saturation is small. Errors become larger as the gas saturation increases. However, for the use to which the results will be put, the methods are satisfactory for reservoir engineering purposes. An improved method of calculating the performance of depletion-type reservoirs is presented in Part 2. Because the mathematical relationships describing simultaneous flow of oil and gas are quite involved. simplifying assumptions are made to provide means of obtaining approximate solutions of reasonable accuracy. One such approximate method now in use is the constant-GOR solution. It involves the assumption that at any instant, the ratio of total gas flow rate (both free and disolved) to oil flow rate is the same at all points in the reservoir. The approach is not applicable unless the free gar saturation in the reservoir is everywhere greater than the critical gas saturation. This paper presents a modification which, by avoiding the constant-GOR assumption, makes the method applicable to all reservoir conditions, and so far appears to be more accurate than the constant-GOR solution and to he comparable in required compuctation time. PART I— BUILD-UP CURVE ANALYSIS INTRODUCTION Pressure build-up characteristics of shut-in wells have been used for many years by engineers to evaluate average reservoir pressure, effective permeability thickness of the pay section, effectiveness of well completion (skin effect) and reservoir size. A number of methods of analysis have appeared from time to time."" Without exception, these methods are based on the assumption that the reservoir contains but one fluid of constant small compressibility and constant mobility. It has been suggesteda" hat these single-fluid methods may be applied to data from reservoirs containing both oil and gas by substitution of some effective total properties of the multiphase system in place of the corresponding single-phase properties. The present investigation was undertaken primarily to evaluate this approach. METHOD A number of theoretical build-up curves were calculated for conditions of two-phase flow, under the assumption of certain reservoir and fluid properties, and were analyzed by single-fluid methods with appropriate total compressibility and total mobility values for the corresponding single-fluid properties. Results of the analyses were compared with the assumed conditions. The theoretical build-up curves were completed by procedures similar to those of West, Garvin and Sheldon." Since these calculations require a considerable amount of computer time, an attempt was made to derive an approximate calculation method. The attempt was unsuccessful for calculating build-up curves, but the effort did result in a new approximate method of calculating the performance of solution gas drive reservoirs, which appears to be an improvement over the constant-GOR method" used previously (see Part 2). The West, Garvin and Sheldon calculations involved the following assumptions: (1) the reservoir is circular and completely bounded, with a completely penetrating well at its center; (2) the porous medium is uniform and iso-tropic, with a constant water saturation at all points; (3) gravity effects can be neglected; (4) compressibility of rock and water can be neglected; (5) the composition and equilibrium are constant for oil and gas; (6) the same pressure exists in both the oil phase and the gas phase; and (7) no afterproduction occurs, i.e., the well is shut in at the sand face for build-up. These assumptions make it possible to describe two-phase flow of oil and gas by the partial differential equations:

Jan 1, 1967

By D. K. Atwood

Experiments resulted in a satisfactory laboratory method for restoring permeability to clay-containing cores damaged by fresh water. Clay contents of a number of field cores were measured, and permeabilities of plugs from these same cores were then deliberately reduced with fresh water. This damage is attributed to swollen and dispersed clays occupying the pore space. After damaging, a number of experiments were performed to meaJure the amount of damage and to establish some means by which permeability could be restored. The experiments included flooding the damaged cores with water-miscible fluids such as salt water, acetone, isopropyl alcohol and ethanol. Permeability was not successfully restored in these experiments. However, part of the damage was repaired by flooding with oil; when water was removed by distillation in the presence of immiscible fluids such as air or toluene, permeability was completely restored. This evidence suggested that swollen and dispersed clays could be collapsed to their original volume by strong interfacial and capillary forces. It was further postulated that the required forces could be generated by flooding the damaged cores with a solvent partially miscible with water. The flooding experiments were repeated using n-hex-an01 as the partially miscible solvent. Permeability was restored to five of six damaged cores and substantially increased in the sixth. A large fraction of the restored permeability was retained even after water saturation was raised to its original value with 12 per cent salt water. INTRODUCTION Sharp reductions in permeability often occur when relatively fresh water contacts clay-containing formations during drilling and workover operations. These permeability losses are caused by removing inorganic ions from the environment surrounding the clay, and consequent swelling and/or dispersion of clay minerals into the available pore space.' This phenomenon is generally termed clay damage, fresh-water damage, or simply formation damage; it causes large losses in current revenue by preventing oil wells from making their allowable production. Attempts to repair the damage and restore permeability by flowing salt water solutions or brines through clay-damaged cores containing montmorillonite have been unsuccessful.' This irreversibility is thought to result from formation of brush-heap, or edge-to-face, structures when the dispersed clay is flocculated. The brush-heap structures occupy much more space than the close packed domains present before damage.' One solution of the problem is to destroy the clay-water brush-heap and thus restore permeability. Because no satisfactory method existed for restoring permeability to clay-containing formations damaged by fresh water, the work described in this paper was under taken. The laboratory experiments generally consisted of deliberately damaging fresh cores containing clay and then attempting to repair this damage by various means. Results indicate that generating strong interfacial forces within the pore space of damaged cores collapses the clay brush-heap and restores permeability. These forces are most conveniently generated by flowing partially water-miscible solvents, such as n-hexanol, through a core. THEORY OF THE DAMAGE PROCESS The most common clay mineral groups known to cause permeability damage to formations are the mont-morillonites, kaolins, chlorites and illites. These clays are constructed of particles which can adsorb water on their surfaces and edges and, in the case of montmorillonite, between layers of the basic particle itself. This adsorption increases as water salinity decreases. At low salinities the particles disperse into the aqueous phase. When the clays present in the formation are kaolin, chlorite and illite, dispersion accounts completely for permeability damage to porous media. However, unlike the other clays, montmorillonite particles can imbibe water and adsorb ions between layers of sub-particles, or platelets. These platelets have net negative charges on their faces and are held together by exchangeable (or removable) cations such as Na and Ca decrease in ion concentration (salinity) in the fluid surrounding a particle causes migration of water into the clay layers and disperses the basic particle, while diffusion removes the original exchangeable ions from between the platelets. Once these ions are removed, the facing negative platelets repel each other, causing the montmorillonite to swell until, for all practical purposes, the individual platelets are dispersed. For this reason, fresh-water* damage is much more severe in sands containing montmorillonite than it is in sands containing other clays. Many investigators have shown that even trace amounts of montmorillonite can be responsible for marked reduction in the permeability of reservoir sands in the presence of fresh water." ." Monaghan and others have shown that fresh-water damage in montmorillonite-containing cores cannot be

Jan 1, 1965

By W. L. Martin, J. N. Dew, H. B. Steves, M. L. Powers

This paper presents and discusses the results obtained during a pilot test in the Loco field in southern Okla homa. The test was conducted in a 2%-acre pattern that was part of a 20-acre conventional waterflood pilot area. The conventional flood was well past its economic limit when the hot waterflood was initiated to obtain technical information and operating experience. Temperature data from nine observation wells showed that about 75 percent of the pattern was affected by heat, and that heat losses were severe in the pilot pattern. About 60 percent of the injected heat was lost to overburden and underburden zones during hot-water injection. Wellbore heat losses were held at tolerable levels at the shallow depth of the test by providing a low-pressure air annulus between the injection tubing and well casing. Hot water provided water injectivity increases of 200 to 400 percent. The hot water channeled across the lower portion of the oil sand in three directions through zones of relatively high water saturation. There was no conclusive evidence that natural fractures or pressure partings aflected the flow of fluids in the pilot pattern; however response undoubtedly was aflected by localized pressure gradients and by injection-production rate ratios. The results showed that hot waterflooding increased oil recovery in a reservoir containing 600 cp oil. The total tertiary oil production from the pilot pattern area was 3,896 bbl or about 156 bbl/acre-ft swept by heat. The corre.sponding WOR was about 34:I. Description of the Hot Waterflood Process To hot waterflood an oil reservoir, water is injected that has been heated to a temperature substantially higher than the original reservoir temperature, but lower than the vaporization temperature of water at the prevailing pressures. In the reservoir the hot water flows continuously into cooler sand and rapidly loses heat to the sand until it has been cooled to the original reservoir temperature. Thus, a heated zone and a region or "bank" of cooled water begins to accumulate immediately after hot water injection is started. This bank of cooled water continues to grow ahead of the heated zone, which also grows, but at a slower rate. This occurs because heat transfer is almost instantaneous, and the ratio of heat capacities of water to rock is such that two or three unit PV of hot water must be injected to heat a given unit bulk volume of the reservoir. The primary displacement mechanism is the same for both hot and conventional cold waterfloods; i.e., "piston" displacement occurs at the original reservoir temperature. The incremental benefits of hot waterflooding usually occur long after the breakthrough of cold water at producing wells, and the increased oil recovery necessarily is accomplished with high WOR's (water-oil ratios). Heat decreases the viscosity and density of oil and water. These effects result in more rapid and increased recovery of secondary or tertiary oil. If the cost of the required heat is low enough, the ultimate oil recovery of a hot waterflood should be increased substantially over what would be expected at the economic limit of a conventional cold waterflood. The economic benefits of any hot-fluid injection project depend primarily upon the cost of the heat required to produce more oil at an increased rate. This cost depends in part upon the amount of heat lost to surrounding formations. Heat loss depends upon reservoir thickness, water injection rate and temperature, the depth of the formation, well spacing, and the characteristics of the reservoir rock and surrounding formations. In general, percentage heat losses decrease as injection rate and reservoir thickness increase. Although it is an old idea, hot-water injection has not received widespread field application in the oil industry as a drive process. Much of the original field work with hot water was done in Pennsylvania fields where water permeabilities and injection rates are low. In these cases, hot-water injection was used primarily as a means of increasing in-jectivities — not as a recovery process. Ramey' recently has published an excellent review of the development of hot fluid injection processes. Recent publications"' and our own experiences now indicate that hot water or steam injection also can effectively increase the oil recovery from reservoirs containing viscous crudes. However, the economics of these methods as displacement processes have not been established. Several theoretical predictive techniques have been published (for a review see Ramey'), but simplifying assumptions make

Jan 1, 1969

By H. H. Rachford, J. Douglas, D. W. Peaceman

A numerical solution of equations describing two-phase flow in porous media shows promise in providing a technique for predicting the displacermet from satlds of oil by water or gas. The description includes the influence of relative permeability, fluid viscosities and densities, graviry, and capillary pressure, and, tholrgh tested ortly for two-dimensional cases, should be eqlmlly applicable in three-dimensional geometry. Two memorical methods ore presented: the first method is quite general in its applicability; the second method can be used only with certain types of boundary conditions but requires less computing. Comparisons of computed results with data from Inhoratory models are presented. These data were taken on a water flood of a strarified model and on water floods of a five-spot model for favorable and unfavorable mobility ratios. On the stratified model, excellent agreement with recovery at breakthrough was obtained; agreement with recovery after breakthrough was poor. In the five-spot model, good agreement was obtainecl with recovery at and after breakthrough. The purpose of this paper is twofold: first, to present a reservoir engineering method requiring knowledge only of rock geometry and the normally measured rock and fluid properties for calculating the multi-dimensional flow of water displacing oil from porous, water-wet rock containing connate water; and, second, to investigate the validity of the method by comparing results of calculations with previously observed displacements in laboratory models. The availability of such a verified technique of reservoir analysis would afford major advantages. First, it would demonstrate that the fundamental macroscopic concepts of two-phase fluid mechanics, i.e., relative permeability and capillary pressure, yield an adequate description of the physical process. Also, since most reservoir rocks appear to be water-wet and contain connate water along with oil, it would provide a method of technical and economic value for calculating the course of oil displacement by water directly from measured reservoir and fluid properties. It is important to emphasize that the calculation is subject only to the limitations of detail and precision of reservoir information, but not to limitations introduced by simplifying assumptions. Further, it would providc almost the only means of examining the influence of factors such as the size and extent of reservoir inhomogeneities and uncertainties in the basic reservoir data. Knowledge of this influence is essential in stating quantitatively the detail required to define a reservoir and in establishing the relation between the uncertainty in the definition of reservoir properties and the reliability of predicted performance. Moreover, in the process of solving a particular problem much detailed information would become available about the displacement process. For example, at each point in the reservoir for all stages of the displace ment. the calculations would yield not only the water and oil saturations but also the direction and magnitude of fluid velocities and the local fluid mobilities. Such dctail is potentially of great value in giving insight into the mechanics of particular displacements. The development of such a method has long been the goal of research in the application of numerical an-alysis to petroleum reservoir enginecrinz. The most complete treatment of the displacement process published to date is that of Douglas. Blair, and Wagner,' which, however, was limited to flow in a single dimension. The method presented in the present work is hascd on the numerical solution of a finite difference analogue of the multi-dimensional differential system describing the displacement process. Although current work has considered only displacement of oil by water from water-wet sands, the differential system for other immiscible displacements, such as gas displacing oil with which it is in phase equilibrium, is quite similar. It should therefore be expected that the technique described will be applicable to displacement by gas as well. The description of the simultaneous flow of fluids through porous media in terms of relative perrncability and capillary pressure has been adequately discussed in the literature and standard textbooks. Sec, for exampje, Muskat. Chapter VII.'

By J. D. Rice, S. C. Pitzer, C. E. Thomas

Interpretation of pressure build-up data obtained in the conventional manner has often been difficult because of the deviation from theoretical behavior. Major causes of this deviation have been attributed to damage and afterflow, and to fluid redistribution in the wellbore which, in extreme cases, can result in a pressure hump in the early portion of the build-up curve. Theoretical investigations show that bottom-hole pressure is definitely influenced by phase redistribution in the tubing column during surface shut-in tests, and that the magnitude of the effect is sensitive to the producing gas-oil ratio and stabilized rate of flow in the well. For field experiments a wire-line tubing packer, which can be run in the tubing against a stabilized flow rate, was developed for bottom-hole shut-in tests. By use of this bottom-hole .shut-in method, pressure humps previously observed in surface shut-in tests were completely eliminated and the effects of afterflow minimized. Analog and digital computer studies have been made to obtain theoretical curves for comparison with field curves, and remarkable agreement between the results of bottom-hole shut-in tests and theoretical curves has been obtained. INTRODUCTION Pressure build-up data from shut-in wells have been used by the petroleum industry to determine the permeability of the formation, to estimate wellbore damage, to evaluate the static reservoir pressure and to speculate reservoir volume. Calculation of these factors is based on methods of analysis developed for theoretical systems whose build-up curves have a charac- teristic shape when the wells are shut-in at the sand faCe.1,2,3,5,8,10 However, field curves obtained by conventional surface shut-in methods do not always exhibit this characteristic. Such factors as stratification, rock heterogeneities and irregular reservoir geometry can cause the character of a build-up curve to deviate from that predicted by theory for a simple system. In addition, all field build-up data are affected by the methods utilized in obtaining the measurements. Conventional surface shut-in techniques allow two effects to occur which contribute directly to the manner in which pressure builds up at the sand face. The first, afterflow, has been recognized for some time and methods are available for estimating the magnitude of its effect and for correcting its influence.' The second, phase segregation in the tubing column after shut-in, has been reported only recently and appears to exert a considerable influence on the character of field build-up curves.!' In extreme cases, phase segregation can produce "pressure humps" in the early portion of the build-up curve. Such curves have been considered anomalous and have defied analysis. It is the purpose of this paper to discuss the effect of phase segregation in the tubing string during build-up and to present a bottom-hole shut-in technique for obtaining field build-up data which minimizes the influence of afterflow and eliminates phase segregation effects. It will be shown that reliable reservoir information can be calculated from build-up measurements obtained using this method, even though conventional surface shut-in tests on the same wells yield anomalous data. DISCUSSION OF PHASE REDISTRIBUTION EFFECTS Some of the effects of phase redistribution on pressure build-up curves have been described previously by Matthews and Stegemeier" who have presented evidence that phase segregation is responsible for the pres-

By M. W. Clegg

The application of the Laplace transformation to problems in the flow of compressible fluids in porous media has provided a large number of exact solutions. For plane radial flow, however, these solutions are either complex integrals or infinite series and are of little value to the field engineer. In the case of production at constant well pressure, the available approximate solutions me valid for large times only. In this paper it is shown that an approximate inversion formula for the Laplace transform, developed for the solution of viscoelastic problems, is applicable to radial flow problems and provides simple analytical solutions to constant terminal pressure problems. The method may be used to obtain approximate solutions to many problems, including media with radial permeability discontinuities, multi-layer formations and pressure buildup in wells after shut-in. The results are compared with the few available computer solutions as well as the large time solutions, and it is shown that this approximate method greatly extends the time interval over which a simple analytical solution is acceptable. INTRODUCTION The study of transient problems in the flow of fluids through porous media has benefited greatly from the application of transform methods. The use of the Laplace transformation for solving parabolic equations has been widely discussed in the field of heat conduction and diffusion as well as in the petroleum literature. Removal of the time variable with the Laplace transformation generally reduces the problem to a boundary value problem which may be solved by standard techniques. A much more formidable problem then faces the engineer, however, for frequently the transform does not possess a simple inverse. The result is that the general inversion integral must be used and this leads to either an infinite integral or an infinite series, both of which are difficult to handle from a computational standpoint. Asymptotic approximations for the inverse have been known for some time and these yield approximate inverse functions that are valid for very large or very small times — but frequently the times of interest lie somewhere between these two extremes. Therefore, some acceptable approximation valid over a larger interval of time is desirable. During the past few years a number of methods for achieving this have been developed and some of these are discussed briefly in this paper. The relative merits of the various methods are not evaluated here, but some general conclusions reached by other authors are given. One of these methods has been applied to problems associated with the radial flow of compressible liquids to producing wells. In the case of production at constant well pressure, the method leads to simple analytical solutions for a number of standard problems; e.g., homogeneous formation, permeability discontinuities, pressure buildup. These solutions greatly extend the range of validity of the asymptotic ones (valid for large times only) and should be of value in studying the behavior of wells producing under constant pressure conditions. METHOD FOR THE APPROXIMATE INVERSION OF LAPLACE TRANSFORMS Much of the work that has been done on the approximate inversion of Laplace transforms has originated in viscoelastic problems. Apart from the asymptotic approximations,1 one of the first approximate methods was due to Alfrey.2 If (s) is the Laplace transform of f(t) defined by

By R. C. Earlougher

Jan 1, 1970

By B. S. Gottfried

This paper is concerned with possible transport mechanisms which occur during segregated burning (i.e., burning in an oil reservoir in which the oil-bearing formation is overlain by a "clean" porous zone containing only gas). ,Two analytical models are presented. In the first, burning is assumed to occur at the gas-oil interface, and the dominant mechanism is taken to be the rate at which oxygen diffuses to the interface. The second model assumes that combustion occurs within the clean zone, with the dominant mechanism being the mechanism of fuel deposition from the oil-bearing formation to the clean zone. ,Comparison of the results of the analyses with limited experimental data indicates that the latter mechanism is much more plausible. Both models are idealized representations of the actual process and are somewhat conjectural, however, owing to the lack of more definitive laboratory or field data. INTRODUCTION In recent years there has been considerable theoretical and experimental research directed toward understanding the recovery of crude oil by underground combustion. Although two different recovery schemes have been proposed, known as forward and reverse combustion, it is only forward combustion that shows promise of becoming a practical recovery technique.17 Most of the analyses of the forward combustion process have considered it to be pistonlike; i.e., oil is displaced by a vertical combustion zone which, with its ensuing gases, pushes die oil forward as the flame front is propagated through the reservoir.l-8,ll,l2,14-16 However, information gained in field pilot studies indicates that, at least to a certain extent, forward combustion proceeds in a segregated manner in oil reservoirs.10 An ideal segregated system is one in which the oil-bearing formation is overlain by a "clean" porous zone containing only gas. The location of the combustion zone in such a system is not immediately evident, although combustion is believed to occur either at the gas-oil interface or within the clean zone. In either case, the combustion zone may be horizontal and it moves in parallel rather than in series with the produced oil. The purpose of this paper is to explore the likely mechanisms by which segregated burning can occur, and to determine the sensitivity of the resulting models to changes in various physical parameters. Two governing mechanisms are considered: the case of a horizontal combustion zone at the gas-oil interface with the rate of combustion governed by the rate at which oxygen diffuses to the interface, and the case of combustion within the clean zone with the combustion process controlled by the transport of fuel from the oil-bearing formation to the clean zone. Both models are somewhat hypothetical and involve considerable simplification of the actual process. However, the results allow some preliminary understanding of segregated burning which is useful in the absence of more definitive laboratory or field pilot data.

Jan 1, 1967

By S. T. Yuster

An analysis has been made of the factors responsible for tilted fluid contacts in petroleum reservoirs. The factors are static and dynamic with the former being controlled by those variables responsible for the capillary rise of fluids, and the latter include rate of formation tilt and hydraulic circulation. The explanation that rate of formation tilt may be responsible for tilted contacts appears untenable. The role of some of these factors in the migration, fractionation. accumulation and possibly the prospecting for oil is indicated. INTRODUCTION While it is generally considered that the segregation of gas, oil, and water in a petroleum reservoir results in approximately horizontal planes separating these phases. there are factor; which modify this simplified picture to a very great extent. Some of these factors are static. and the departure of the fluid interfaces from horizontal may represent a static equlibriun. Some of these factors may be dynamic and the interface positions may be non-equilibrium and represent a steady .state condition. Of course combinations of these two situations may also exist. The reasons most commonly given for this behavior are hydraulic circulation of fluids, change in character of the formation, and a rate of tilt of the formations more rapid than the rate at which the fluids may re-establish an equilibrium. Russell's Principles of Petroleum Geology' contains an extensive discussion of tilted fluid interfaces and their importance in the migration and accumulation of petroleum. Russell cites examples of tilted water tables at Lance Creek, Wyo.; Rock River, Wyo.: Cut Bank. Mont.; Cushing, Okla.: and Graham; Okla. Also given was the Wasson Field of West Texas which has saucer shaped oil-water and gas-oil interfaces with the saucer being deeper in the case of the oil-water contact. The Bradford Field in Pennsylvania has a tilted water table. San Ardo. Coalinga None and some other California field.; have tilled contacts. A recent paper by King Hubbert discuses the effect of hydraulic circulation of the water underlying the oil in a given stratun and it- implications in the accumulation of and drilling for the oil. The foregoing would indicate that tilted fluid interfaces are not uncommon and that it would be of importance to underbtand the forces which are responsible. Further. it would be of interest to examine certain idealized models of situations which may cause tilting with the thought in mind of deriving relatinnships between the important variables. These relationships may only hold exactly for the model* assumed but the results may apply semi-quantitatively to actual formations and at least indicate trends. The discussion will be divided into the effect of static factors and the effect of dynamic factors. STATIC FACTORS It has already been mentioned that one reason given for tilted contacts is a variation in formation properties. Specifically this is a change in the pore size of the formation capilaries. If the internal surface of the reservoir is preferetially wet by water, capillarity will cause the water 10 rise to a greater extent above the free water table in a small capillary than in a large one. Consequently the water-oil contact will be higher in the lower permeability sands than in the higher permeability sand.;. The free water table may be defined as the position the oil-water interface would take if capillarity were absent. Since the phenomenon is the capillary rise of water in an oil environment, a derivation of this shold give a relationship between the variables. Fig. 1 is an idealized model of the situation with a uniform bore circular capillary ubstitttting for the tortuous non.circular formation pore. The upward force.; on the column of water will be equated to the forces downward to give the equilibrium situation. forces upward = force downward where r is the radius of the capillary in ceritimetera. a is the interfacial ten4on oil-water in dynes per centimeter, 0 is the

Jan 1, 1953

By P. J. Closmann

Steam zone growth as a function of time has been calculated for the case of constant rate steam injection into a preheated reservoir. To simplify the calculation a linear temperature profile has been assumed in the cap and base rock at the start of steam injection. The results indicate that at early times augmentation of steam zone growth due to preheating should be greatest. At longer times the steam zone development becomes close to that calculated with no preheating. INTRODUCTION With the increasing application of thermal recovery processes to recover viscous oil, cyclic steam injection has become important in many large-scale projects. In this process, the cycle of steam injection followed by oil production is repeated a number of times. At the beginning of the second and later cycles, steam is injected into a reservoir that has already been heated but that has lost part of its heat both in produced fluids and by conductive heat loss away from the injection zone. In such a case, the temperature level of the injection zone and the temperature distribution of the surrounding rock will affect the growth of the steam zone developed during the subsequent steam injection. Knowledge of the size of steam zone developed is important in determining the amount of oil displaced and the extent of heating in the reservoir. It is also useful to be able to compute the size of the steam zone for cases where thief zones take most of the injected steam. This paper presents a fairly straightforward method of estimating the steam zone developed in a preheated formation, based on certain simplifying assumptions. Some cases of steam injection into a reservoir at its original temperature have already been considered elsewhere.1-3 Because it is difficult or impossible to obtain an accurate representation of the temperature distribution in the reservoir some time after initial heating has taken place, in this work, a linear temperature profile in the cap and base rock is assumed (Figs. 1A and 1B). The linear profile has the advantage of yielding heat flow equations that

Jan 1, 1969

By M. P. Tixier, F. Segesman

Explanations bused on theory and laboratory tests tire offered for certain unusual features of SP logs which may befog their interpretation, namely the "sawtooth" SP and the reduction or increase of SP with time. The sawtooth SP is frequently seen in SP recordings of thick, very permeable watcr. sands of the Louisiana Gulf Coast. Using SP data from a resistor network analog, this anomaly is explained on the basis of a thin shale streak within the sand. under which the invading filtrate accumulates by gravity segregcitiorr. The electromotive force then appearing across the shale streak gives rise to the anomalous SP. Reduction of the SP on successive logging runs Ira often been observed in very permeable sands when as indicated by the resistivity curves, the invasion depth has been reduced to practically zero. Laboratory results verify that this phenomenon may be due, to the presence of air electromotive force, across the mod rake when the fluids on either side differ in activity. Also, there is, in some cases; an increase of SP deflection on successive logging runs, because the emf initially present across the mud cake disappears as the somewhat saline water just behind the mud cake is progressively flushed out by the invading filtrate.. Field example and laboratory data are presented. I N T R ODUCT1ON Some unusual features are frequently observed on certain SP logs which make interpretation difficult. The purpose of this paper is to offer explanations of several cases of interest. All of these arc related to the conditions of invasion. The cases to be discussed are as follows. 1. The sawtooth SP anomaly which is frequently seen within the thick, highly permeable saltwater sands of the Louisiuana Gulf Coast 1. Reduction of the SP deflection from the usual sand line in very permeable sections when the depth of invasion is practically nil. As will be explained, such a reduction is attributable to a membrane electromotive force across the mud cake. 3. An increase of the SP with time just after the iormation is drilled as a result of progressive flushing of the invaded zone. FUNDAMENTALS OF SP PHENOMENA An explanation of SP phenomena must begin with the following basic ideas. 1. The SP log derives its character from ohmic potential differences in the hole produced by currents flowing through the mud.' These currents are driven by electromotive forces which are primarily electrochemical in origin. Other possible types of SP-producing electromotive forces may be disregarded for the purposes of this paper. 2. The electrochemical emf's associated with permeable formations are now understood to consist of liquid-junction emf's and membrane emf's.',' As schematically shown by thc battery symbols in Fig. 1, the liquid-junction emf, E,,,, exists at the contact between waters of different chemical activities, a,,,, (mud filtrate) and tr,,. (formation water); the shale membrane emf, lS.sn. exists across the adjacent shales whcn they separate waters of different activities. The emf's just cited are additivc in producing SP currents (arrowed circles of Fig. I). Thus, neglecting for the moment any possible membrane emf across thc mud cake, the total electrochemical emf, E,, will be given by their sum, where K,., -! K,q,, = 12 + 59, or 71 at 75 °F. In the log interpretations the SP of a thick sand (i.e.. the static SP) is generally related to the activities of the filtrate and formation water through Eq. 1. by setting the static SP equal to - E,.

By Robert G. Nisle

The classic theory of pressure build-up in shut-in oil wells as developed by Horner and van Everdingen is based on two-dimensional radial symmetry in the well-reservoir system. Such symmetry does not exist in the case of a well which parfiially penetrates the producing formation. As a result of this lack of symmetry the use of the classic theory in such cases become questionable. In this paper the mathematical theory has been extended to include the case of partially penetrating wells. Numerical solutions illustrating the up-dication of the equations are presented. The effect of partial penetration on pressure build-up is shown by a comparison of synthetic pressure build-up curves derived from the numerical solution of the equation for partially penetrating wells for various degrees of penetration. It is shown that partial penetration is detectable from the characteristic shape of the pressure build-up curve and that formation productivity may be calculated from the pressure buildup data in a manner identical to that described by the classic theory. INTRO DUCTION The use of pressure build-up data on shut-in oil wells is a well-established technique for the measurement of reservoir productivity. The work of Homer' and van Everdingen' is well known. Since the publication of their pioneering work, others"' have elaborated and extended the technique. These authors have dealt exclusively with the case of wells that completely penetrate he producing formation. Two-dimensional radial symmetry is achieved, thereby, and the problem is greatly simplified. This ideal situation is seldom encountered in practice. But, in spite of this, the technique is often employed anyway. The question, therefore, arises: how much confidence can be placed in the results of such calculations when the data are taken on wells that only partially penetrate the producing formation? It is the purpose of this work to answer this question. Only the single-phase fluid case will be considered. van Everdigen' has presented a rigorous solution of the two-dimensional case, but because of mathematical complexity, this solution has not proved to be particularly useful. Horner' utilized Kelvin's point source solution in two dimensions and showed that it is a sufficiently good approximation for oil reservoir studies. The geometry of the well and the producing formation is illustrated in Fig. l. Geometrically the formation may be described as an infinite slab. The infinite slab is defined as the solid (the porous medium in this case) bounded by two parallel planes. Thus, the slab extends to very great distances in both the positive and negative directions of two of the coordinate axes, but is finite in