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By R. J. Wygal

A method is presented by which homogeneous, isotropic, and reproducible aggregates of particles such as glass beads, sand, ground glass, etc., may be prepared simply and quickly by using a "particle distributor". Several applications of the packing method to the construction of model reservoirs are described. The flow characteristics of these packings me given. Some factors which limit the usefulness of unconsolidated packs for simulating reservoir rock are overcome through artificial cementation of the aggregates. INTRODUCTION Many laboratory studies of porous media are based on experiments using test packs of sand, glass beads and other granular materials. More confidence can be had in the results of such studies if the test packs are known to have uniform properties throughout, to be mechanically stable and to be accurately reproducible. This paper presents a method by which such packs of a variety of shapes and sizes may be constructed using a particle distributor. Some factors which limit the usefulness of unconsolidated packs as a means for simulating reservoir rock are described. A method of overcoming these limitations, which involves artificial cementation of the packs, is presented. DEVELOPMENT Particles packed by the tamping or vibrating techniques are usually either non-uniform or mechanically unstable. An invention by Currie and Gregoryl overcomes these difficulties to some extent. The invention is based on the use of a stream of falling particles impinging on and passing through a distributor matrix of several layers of closely spaced spheres. The particles are well dispersed by numerous collisions with the spheres. They become homogeneously distributed at some distance below the distributor matrix and at that position are collected in the sample container. This apparatus, however, was found to be ineffective when small particles were used since small particles accumulate on the top surfaces of the spheres. An improved distributor matrix was made, first in the form of a stack of five sieves, and then in the more adaptable form of spaced layers of wire screen. Fig. 1 shows a longitudinal view through a five-layered distributor of eight-mesh wire screen. This device gives excellent results.* Fine particles pass through the distributor with very little hold up. The fine structure of the wire matrix with its many deflecting surfaces ensures good particle distribution. There are no dead areas immediately below the screen as were sometimes observed to occur below a matrix of spheres. Using such a device, 20-30 mesh sands were packed to a porosity of 32.1 per cent at rates varying from 0.17 to 6 lb/rnin. The lowest porosity achieved by tamping small increments of similar sands into a container was 33 per cent. Motion pictures at 3,000 frarnes/sec of the surface of a growing pack of 2-3/4 mm glass beads have helped to show the mechanism by which packs of low porosity are obtained. The beads, although packed at the rate of 5 lb/min, were observed to hit the pack surface singly, for the most part, and at well spaced points of impact. Part of their energy is transferred to surface beads which are thereby

By H. G. Warren, E. W. Hough

Correlation of interfacial tension of the methane-n-pentane and methane-n-decane systems was made by Hough and Stegemeier by the use of the Weinaug and Katz equation. The methane-n-heptane and ethylene-n-heptane systems were investigated by the authors. Additional systems taken from the literature were the me thane-propane system by Weinaug. and Katz, the methane-n-butane system by Pennington and Hough, and the n- butane -carbon dioxide system by Brauer and Hough. These systems were analyzed by the Weinaug-Katz equation and the Sugden equation on the IBM 1620 11 computer using a regression analysis program. Parachor and exponent values for each component and system were determined. The experimental work was close to the critical point of the light component in the n-butane-carbon dioxide and ethylene-n-heptane systems, between the critical points of the light and heavy components in the methane-n-pentane, methane-n-heptane and methane-n-decane systems, and close to the critical point of the heavy component in the methane-pmpane and methane-n-butane systems. An empirical formula was developed to find a value of the constant B in the Sugden equation for binary hydrocarbon systems from the parachor values of the two components. Interfacial tension for binary hydrocarbon systems was found to be a direct function of liquid-vapor phase density difference in the same manner as a single-component system. DISCUSSION Interfacial tension may be defined as the measure of the specific free surface energy between two phases having different compositions.1 The interfacial tension (IFT) for a binary system may be predicted for any temperature and pressure if the

Jan 1, 1967

By A. M. Skov, L. F. Elkins

First response to large-scale water flooding in the fractured very low permeability Spraberry sand has led to a new unique cyclic operation. Capacity water injection is used to restore reservoir pressure. This is followed by marly months production without water irzjection and the cycle repeated. Expansion of the oil, rock and water during pressure decline expels part of the fluids but capillary forces hold much of the injected water in the rock. At least with reservoir pressure restored and with partial water flood development, field performance has proved this cyclic operation is capable of producing oil from the nzatrix rock at least 50 per cent faster and with lower water percentage than is imbibition of water at stable reservoir pressure. INTRODUCTION The Spraberry Field of West Texas presents unusual problems for both primary production and water flooding. Extensive interconnected vertical fractures in the fractional-md sandstone permitted recovery of oil on 160-acre well spacing, but they made capillary end effects dominant. Primary recovery by solution gas drive is less than 10 per cent of oil in place. The concept of displacement of oil from the sand matrix by capillary irnbibition of water has led to field techniques which promise greatly increased oil recovery. Free exchange of laboratory research, reservoir information and results of field pilot tests among the various companies has been very important in development of this technology. Five units covering a total of 170,000 acres have been formed for water flooding, and 10 other areas covering an additional 175,500 acres are in various stages of unitization. Part of the Driver Unit reaching fillup first has demonstrated very unusual waterflood behavior and indicated numerous operating problems that will develop within and among the various units. SPRABERRY ROCK AND PRIMARY PERFORMANCE The Spraberry, discovered in February, 1949, is a 1,000-ft section of sandstones, shales and limestones with two main oil productive members: a 10-15 ft sand near the top and a 10-15 ft sand near the base. In part of the field some thinner intermediate sands are oil productive, and others are water bearing. All sands have permeabilities of 1 md or less and porosities of 8-15 per cent. Ordinary core analysis and electric and radiation logs are ineffective in differentiating between oil productive and nonprcductive sands. Sands capable of containing producible oil are best identified by mercury injection capillary pressure measurement and, in some cases, by core water saturation. About 3,500 wells have been drilled in the 500,000-acre trend. Vertical fractures were observed in practically all Spraberry cores. Continuity and interconnection of fractures were confirmed by pressure interference among wells during early development.' Major fractures trend northeast-southwest as indicated by oriented cores and confirmed by five fluid injection tests, by analysis of the pressure transients observed during development,''' and by three interference tests in the Driver Unit Water Flood reported herein. Fracture spac- ing probably averages inches to a few feet. Spraberry wells typically produced 100-400 BOPD initially after hydrauLic fracture treatments. By 1962 oil production had declined to an average of 12 bbl/well/day, near the economic Limits of operation. Reservoir pressure had declined from 2,300 psi initially in the Upper Spraberry and 2,500 psi in the Lower Spraberry to 500-1,000 psi. Partial closing of the fractures with declining reservoir pressure is believed to be the cause of such low oil production rates at these relatively high reservoir pressures. Cumulative recovery of 208 million bbl of oil is 80 to 90 per cent of that recoverable by primary means. Performance of the entire reservoir is summarized in Fig. 1. IMBIBITION WATER FLOODING By 1952 reservoir performance indicated low primary recoveries. Most engineers, expecting serious channeling of injected fluids through the fractures, held little hope for secondary recovery. With its extensive background of research on the fundamentals of fluid flow within reservoir rocks, Atlantic's Research and Development Division on short notice in 1952 conceived that displacement of oil by capillary imbibition of water into the rock might significantly increase Spraberry recovery. Laboratory data reported by Brownscombe and Dyes scaled to probable reservoir conditions showed potential waterflood recovery equal to or greater than primary recovery with a 10-15 year flood life.= A pilot test using three 40-acre injection wells, one central producing well and 18 surrounding observation wells demonstrated technical feasibility of the process. Injection of 1.5 million bbl of water from November 1952 to August 1955 proved water entered the rock and displaced oil

By D. N. Dietz

A method for determining average reservoir pressure is presented, which is simpler to apply than that devised by Matthews. Brons and Hazebroek. For bounded reservoirs, identical results are obtained if stabilized-flow conditions prevail. The present method yields inferior results in the transient state. The method can, with a slight modification, also be used for water-drive reservoirs. INTRODUCTION In the method proposed by Matthews, Brons and Haze-broek' for determining average reservoir pressure in a multi-well reservoir, the cumulative production (the production time of each well) enters twice: once when the buildup is plotted against In (t + t)/At to arrive at p*, and a second time when the correction, p*-p is determined with one of the several formulas for differently shaped drainage areas. Once a steady state has been attained, however, the previous production history should be immaterial. The same pressure distribution could have been arrived at after different cumulative productions of the individual wells. In principle, therefore, it should be possible to determine average pressures without referring to cumulative productions. In the following paragraphs an expression is presented for the difference between the pressure in a producing well and the average pressure of its drainage area. Then the build-up time needed to overcome this difference is indicated. CASE OF A CIRCULAR DRAINAGE AREA AND A CENTRAL WELL PRESSURE DISTRIBUTION BEFORE SHUT-IN The general differential equation of radial flow (see Muskat, Eq. 10.2) may be written In the steady state, the rate of production of the well is equal to the rate of expansion of the fluid contained in the the drainage area; thus

Jan 1, 1966

By D. G. Russell

A simple method has been developed with which flowing bottom-hole pressure data from two-rate flow tests in oil or gas wells can be analyzed to estimate the formation permeability, skin factor and average reservoir pressure. The required pressure data are obtained by observation of the transient bottom-hole pressure behavior after the stabilized producing rate of the well is changed to another, higher or lower, rate. The new method yields the same information as a conventional pressure buildup analysis, but eliminates the need for closing in the well. The analysis of a two-rate flow test is of the same degree of difficulty and requires about the same engineering time for application as a conventional pressure buildup analysrs. The extended closed-in periods experienced with conventional buildups because of long, low-rate afterproduc-tion periods are eliminated by fIow tests. Other anomalous pressure buildup effects, such as "humping" due to weli-bore phase segregation, can be successfully eliminated with the new method. Generally, flow tests of about 24 hours' duration run with conventional pressure measurement equipment are sufficient for interpretation purposes. Field testing of the two-rate flow test method has established that it is a reliable and economical method which can be used in many instances to complement or even replace conventional pressure buildup methods. INTRODUCTION The principal method for estimating formation permeability and well damage, or skin factor, in a producing oil or gas well is the analysis of shut-in bottom-hole pressure buildup data.' This familiar method has been used quite successfully by reservoir engineers for many years. It is based on the solution of the radial flow equation for constant rate conditions, and requires that the well be closed in for a sufficient period of time to obtain a clearly defined linear portion on the plot of observed t + ?t bottomhole pressure vs log t + ?/?t(where At is shut-in A? time, and t is producing time to the instant of shut-in). From the slope of the plot and other normally obtainable data, the permeability, skin factor, and reservoir pressure at infinite shut-in time (if the reservoir were infinite) can be estimated. Over the years several drawbacks have become apparent in the use of conventional shut-in pressure buildups for determining permeability and skin factor. The conventional pressure buildup interpretation theory assumes that a well is closed in at the sand face and that no production into the well occurs after shut-in. In practice, of course, the well is closed in at the surface, and inflow into the well continues until the well fills sufficiently to transmit the effect of closing in to the formation. This adjustment period is commonly referred to as the "after-production" portion of the pressure buildup. In the tight reservoirs, long, low-rate after-production periods frequently occur, and the well must be shut in for several days or, in some instances, even weeks to obtain an interpretable buildup curve.' Obviously, such long shut-in times can cause loss in current income, both from reduced oil production and from the fact that personnel and pressure measurement equipment are occupied with a single well for too long a time. In other cases, even long shut-in periods do not seem to be of much aid in obtaining an interpretable buildup." If there is considerable phase redistribution (liquid fallout or bubble rise) after a well is shut in, then curves with no interpretable portion are often obtained during the buildup. In addition to instances in which wellbore effects cause trouble, there are also cases in which the major objection to use of the closed-in pressure buildup is simply the fact that the well must be shut in. When there is no proration and when the well has limited producing capacity. closing in the well means loss of income. From the foregoing discussion it is apparent that it ib desirable to have an alternative method of obtaining the same information as that derived from a conventional buildup without the need of closing in the well. One possible solution which has been offered for this problem is the use of a bottom-hole shut-in tool' which isolates the major portion of the flow string from the formation face during the buildup. In this paper an alternative method which is frequently successful in avoiding wellbore effects and which does not require the use of special equipment is presented. A new, simple method has been developed with which the flowing bottom-hole pressure data from flow tests in oil or gas wells can be used to estimate permeability, skin factor and the average reservoir pressure. The required pressure data are obtained by observation of the transient bottom-hole pressure behavior after the stabilized production rate of the well is changed to another, higher or lower rate. The need for closing in the well is eliminated, and pressure measurement periods of only 18 to 24 hours arc usually sufficient, even in tight reservoirs. Thus, the new

By S. C. Miller, R. D. Carter, H. G. Riley

A merhod based on theoretical considerations is presented for estimating srabilized performance of gas wells from data obtained from short-term flow tests. Calculated results are presented for II gas wells evaluated with four-to five-hour flow tests. For seven wells on which comparison war available, satisfactory agreement between measured and calculated performance was obfained. Although the final results are favorable, diflerences in flow capacity k,h obtained from flow tests conducted at different rates were observed. This inconsistency did not seriously affect the calculated results in the test wells. INTRODUCTION In order to negotiate the sale of gas from newly drilled gas wells and to plan otherwise for the orderly exploitation of gas reserves, it is necessary to predict the long-term withdrawal rates physically possible for each well. Since low permeability gas wells do not stabilize except after a lengthy period of flow, the operator is unable, without wasting large amounts of gas, to measure directly the stabilized performance prior to contracting for sale of the gas and connecting the well to a pipeline. This problem is not new, but is one which has recently received considerable attention because of the rising importance of natural gas in the economy and new emphasis on the development of gas reserves. Thus, a method is needed for estimating stabilized performance from information obtained from short-flow tests in which stabilized flow is not attained. Such short tests would be suitable for wells not connected to a pipeline since the tests require only small amounts of time, labor and production of gas. Other requirements of such a method are short calculation time and use of a model of the gas well and its reservoir drainage area which is as consistent as possible with present theoretical, experimental and field knowledge. The method presented herein was developed in an effort to meet these requirements. It has been named the "two-Row method". In addition to the section on the evelopment and description of the method, a description of field application of the method and a discussion of well test results are in- cluded. Finally, there is a section on an additional consideration prompted by some of the test results. The Appendix contains certain details of the derivation of the two-flow method and a sample calculation for Test Well 1. MATHEMATICAL DEVELOPMENT AND DESCRIPTION OF THE METHOD The method is based upon the following assumptions: (1) The system can be described as a finite cylindrical gas reservoir produced by a single cylindrical well located at the center of the reservoir. Flow is strictly radial. (2) Flow obeys Darcy's law except very near the well where a quadratic flow law is assumed. (3) The system is also characterized by a laminar flow skin factor. The use of a quadratic or turbulent flow term is more fully justified in Refs. 1 and 2. When such a system is opened at the wellbore to constant rate flow from an initial condition of uniform pressure distribution, the radius or drainage quickly passes through the non-Darcy flow and "skin effect" region imrnediately surrounding the well. After the drainage radius or "transient" has passed beyond this region and before it reaches the external boundary, drawdown can be expressed by'." After the drainage radius reaches the external boundary, stabilized flow exists and drawdown is given by'," Eqs. 1 and 2 form the basis for the method. The validity of these equations will now be examined in the light of some well test data previously available. It has been found by actual measurements on gas wells that the relationship between drawdown and flow rate, that is the performance curve, is closely approximated by' Eq. 3 is a straight line on log-log graph paper. Eq. 2 and, for any lixed value of time, Eq. 1 plot as a curve which is slightly concave upward on log-log graph

By J. K. Elliott, P. H. Kelly

Many engineering functions such as surface metering work and laboratory compressibility check points involve the use of liquid densities of light hydrocarbon mixtures at various pressures and temperatures. However, at the present time, no simple reliable method exists for determining density variation, particularly if the composition of the liquid is unknown. Consequently, a study was undertaken to develop and present a simple and accurate method of predicting density variation of a light hydrocarbon liquid with pressure and temperature, knowing only the density of the liquid at some condition. The experimental liquid compressibility data from API Project 37 by Sage and Lacey' have been considered to be accurate within 0.5 per cent and cover a wide range of pressure (14.7 to 10,000 psia), temperature (100" to 400°F) and molecular weight (up to 150). From these data, a set of liquid density curves, which relate density to pressure, temperature and molecular weight, was developed. These curves make it possible to predict density variation with pressure and temperature. Compared to extensive laboratory compressibility data on a complex, light hydrocarbon liquid, the use of the liquid density curves resulted in an average error of less than 0.5 per cent. Based on the results of this analysis, it is concluded that the set of liquid density curves developed from the data of Sage and Lacey provides an accurate and simple method for predicting the density variation of light hydrocarbon liquids when the density at some condition is known. These curves should be very helpful in many engineering calculations, particularly in the surface metering of light hydrocarbon liquids. INTRODUCTION Many situations arise in field and engineering laboratory work, such as reservoir engineering studies, check of experimentally determined laboratory data and orifice flow-meter formulas, where liquid density factors at various pressure-temperature conditions are required. Also, the need for accurate light hydrocarbon liquid information has become more important with the advent of miscible-type displacements for secondary recovery purposes in oilfield operations. Several reliable methods are available1 - "or determining the density of liquid hydrocarbons if the composition of the liquid is known. However, there is a definite lack of methods for accurately determining the variation of density when the composition of the liquid is unknown. The purpose of this study is to review the various methods for determining hydrocarbon liquid densities and to develop a simple and reliable method of determining variation in density of light hydrocarbon liquids with pressure and temperature when the compositio~n of the liquid is unknown. METHODS FOR DETERMINING DENSITY OF LIQUIDS OF KNOWN COMPOSITION Sage, Lacey and Hicks' have proposed a method to predict the density of light liquid hydrocarbons by using partial molal volumes. Data are available on experimentally developed partial liquid volumes of hydrocarbons over a rather limited range of temperature, pressure and composition. The partial mold volume method has proved satisfactory for determining the density of some hydrocarbon liquids when the composition is known. Within the range covered in the Sage, Lacey and Hicks1 data, the results agree within about 3 per cent of the experimental values. Hanson mentions the limitation of this method to a composition range of approximately 10 per cent by weight of methane, which will not allow this correction to cover most low molecular weight-light hydrocarbon liquids. Standing and Katz2 studied data on light hydrocarbon-liquid systems containing methane and ethane at high temperature and pressure and have presented a method for determining liquid densities, assuming additive volumes for all components less volatile than ethane and using apparent densities for methane and ethane. The compressibility and thermal-expansion curves used by Standing are based on assumptions that compressibility of a hydrocarbon liquid at temperatures below 300°F is a function of the liquid density at 60°F and that thermal expansion of the liquid is affected little by pressure. The information required to use this technique with an example problem is furnished by Standing.' Hanson eports an average error of - 0.5 per cent using the method of apparent densities in calculating liquid densities of several volatile hydrocarbon mixtures. However, as implied, the apparent density method is not applicable for liquid density calculations when the composition of the liquid is unknown. Watson- as presented a method

By W. B. Gogarty, H. R. Bailey

Methods are presented for determining molecular diffusion coefficients by using data from, capillary flow experiments. These methods are based on a numerical solution (presented in a previous paper) of the partial differential equation describing the combined mechanisms of flow and diffusion. Results from this numerical solution are given and compared with the approximate analytical solution of G. I. Taylor. The numerical solution is valid over a much larger time range. These methods are applied to experimental results for the fluid pairs water-potassium permanganate solution and amyl acetate-orthoxylene. Both of these fluid pairs have approximately equal densities and viscosities. Graphical and numerical techniques are presented for deter mining diffusion coefficients from the flow data. Values obtained by these techniques are compared with values obtained by other methods. INTRODUCTION The molecular diffusion coefficient is known to be a variable in determining the amount of mixing in a miscible displacement process. The effect of molecular diffusion on dispersion in longitudinal flow through porous media -has been examined by different investigators. These investigators concluded that at low velocities of flow, the amount of dispersion is approximate1y proportional to the molecular diffusion coefficient. The influence of diffusion on fingering, channeling, and overriding has been mentioned by other investigators.4 Recent studies5,6 have been made on the effects of molecular diffusion in connection with the problem of gravity segregation. Many different methods have been developed for the experimental determination of molecular diffusion coefficients? These methods differ mainly according to boundary conditions selected and analytical procedures used. Nevertheless, all of these methods have the condition in common that the bulk fluids in which diffusion is occurring are stationary with respect to each other. In connection with a series of papers on mixing in capillary flow,8-10 Taylor suggested the use of a flow method for determining molecular diffusion coefficients. Additional studies 11-13 have been conducted on miscible displacements in capillary tubes, but the data from these studies were not used for the specific purpose of determining diffusion coefficients. The flow method proposed by Taylor results in a single value of the diffusion coefficient for the fluid pair used in the displacement experiments. This single value represents the true value for the fluid pair when the diffusion coefficient is independent of concentration. If the diffusion coefficient is a function of concentration, the single value obtained by the flow method gives an average value for the coefficient of the fluid pair. These average values are based on diffusion taking place over the entire range of concentration, i.e., from 0 per cent of one fluid to 100 per cent of that same fluid. In field applications of the miscible displacement process, gradients occur over the same range of concentration as are found in the displacements in capillary tubes. Molecular diffusion coefficients obtained from the capillary flow method should, therefore, be especially relevant to field operations. This investigation was undertaken to evaluate the feasibility of obtaining molecular diffusion coefficients from capillary -flow experiments. In making this evaluation, diffusion coefficients were first determined for two systems from data obtained in capillary flow experiments. These values of the diffusion coefficient were then compared to values obtained by other methods. MIXING IN CAPILLARY FLOW—THEORETICAL The theoretical basis for determining molecular diffusion coefficients from capillary flow experiments is the partial differential equation relating the mechanisms of flow and diffusion. For the case of laminar flow and with a fluid pair having equal densities and viscosities, the equation is given by

By F. Selig, E. J. Couch

The following points serve to supplement the interesting theoretical study on forward combustion by Thomas, and may be of some general interest. The mathematical model presented by the author can be viewed as a first approximation to the problem of calculating the vertical burned height profile resulting from ignition of the interval | z |<h/2 in a very thick formation. However, the assumption of a given minimum combustion temperature T, would give a better approximation to the vertical burned profile if the burning height were not constrained to stay within the ignited layer. This follows from the fact that both fuel and oxygen are present in the adjacent region, |z|>h/2, and for certain combinations of process variables the temperature is sufficiently high for complete combustion to occur. In this case, the vertical coverage will first become greater than the ignited interval before cutting back to zero as the extinction limit is approached as shown in Fig. 1. Such growth of vertical coverage is possible if the minimum combustion temperature is less than the initial source temperature (r = rj) at the edge (z = h/2) of the ignited interval. But this latter temperature is equal to one-half the source temperature for the case of infinite burned height.&apos;,&apos; Thus growth will occur initially if The fact that the vertical burned height can be greater than the ignited interval complicates the picture considerably, as can be seen by referring to Fig. 1. Here burning occurs in Region A, whereas none was possible in the original mathematical model. At the same time, there is no combustion in Region B, contrary to the assumption of burning in the entire layer |z|<h/2. This apparent inconsistency suggests that the vertical coverage derived from this simple model does not adequately represent the vertical height of the burned zone. A more compatible model will result from the actual stepwise calculation of the burning length (as a function of time) that would give rise to the specified combustion temperature. This problem can be more conveniently attacked by using the temperature distribution, given by the author&apos;s Eq. 12, expressed in the integrated form:

By J. E. Warren, J. H. Hartsock

In the receipt of J. E. Warren and J. H. Hartsock&apos;s comments on our paper, "Aspects of Gas Deliverability", We wish to point out that all the physical characteristics we conceived as constant were absorbed in the gas deliverability constant of the individual well&apos;s performance, listed

By B. K. Larkin, H. K. van Poollen, H. C. Bixel

A detailed treatment is given of the transient pressure behavior of a well located near a linear discontinuity. On either side of the discontinuity, the values of permeability, viscosity, compressibility, and porosity are uniform, but may be different from those on the other side of the discontinuity. These results generalize previously published works which assumed the permeability ratio for media on different sides of the discontinuity was zero, one, or infinity. One application of this material would be to facilitate detection of facies changes or fluid-fluid contacts by transient testing. Solutions, presented graphically, show pressure decline for constant rate fluid production. The range of variables examined includes dimensionless time from 0.1 to 100, rnohility ratio from 0.001 to 1000 and specific compressibility ratio from 0.001 to 1000. For dimensionless tinze less than 0.4, the pressure behavior is essentially that of a well in an infinite reservoir. Thus, reservoir properties near the well may be estimated in the usual manner. Using an overlay technique to match an experimental curve with one of the theoretical curves, it is possible to estimate the distance to a discontinuity by substituting the actual time t and the corresponding dimensionless time , at which a match occurs into the equation where is the diffusivity near the well, and may be estimated from data taken at early time. Because of both the complexity of the solutions and limited computer facilities, well interference problems were not examined extensively. However, it was found that well interference may be masked by the discontinuity in some cases. By superposition of the results of constant rate fluid production, the problenz of pressure buildup may be treated. Such results show that for early shut-in times correct values for the transnzissibility are obtained from conventional semilog plotting. However, erroneous values of the static reservoir pressure are obtained unless data at large shut-in times are taken. INTRODUCTION Transient well testing has yielded estimates of such reservoir parameters as static reservoir pressure, permeability. porosity, and well bore effects. In most tests, it was assumed that the reservoirs were homogeneous and iso-tropic. Anisotropic media&apos; and radial discontinuities were considered.&apos;.&apos; Various studies have been made on multiple-layer reservoirs." &apos; Linear discontinuities have been described for problems of one-dimensional flow.&apos; In the present paper, a mathematical treatment is given for a well located near a linear discontinuity in an otherwise infinite medium (see Fig. 1). An actual reservoir analogy would be a well in a relatively thin reservoir near a fluid-fluid contact, or a well located in a reservoir exhibiting a sudden change in rock and formation characteristics, such as thickness, porosity, or permeability. A sudden change in porosity or permeability may be caused by clay fill or other types of facies changes. Sudden thickness changes niay be the result of such geological formations as buried shore lines and channel deposits. In otherwise uniform formations, a linear discontinuity may be caused by fluid-fluid interfaces such as oil-water contacts. 3s-oil contacts, or gas-water contacts.

By J. S. Levine, M. Prats

A homogeneous and uniform cylindrical reservoir containing oil and gas is fractured vertically on completion and is produced at a constant bottom-hole pressure. The fracture has an infinite flow capacity, is of limited lateral extent and is bounded above and below by the impermeable strata defining the vertical extent of the reservoir. Results show that such a fractured reservoir can be represented by a reservoir of circular symmetry having very nearly the same production history. The well radius of this circular reservoir is about 1/4 the fracture length and is essentially the same as that obtained previously for a single fluid of constant compressibility. At the same value of cumulative oil production, gas-oil ratios of fractured reservoirs producing at constant terzinal pressure are larger than those of reservoirs having no fractures. This leads to more inefficient use of the reservoir energy in fractured wells and results in lower reservoir pressures for the same cumulative oil production. The reduction in operating life due to fracturing a reservoir is not as great as that for a slightly compressible fluid. This diflerence can be accounted for by the lower reservoir pressure in the fractured reservoir and its adverse effect on the average mobility and compressibility of the oil. As anticipated, the reduction in operating life increases czs the reservoir permeability decreases. The type of results presented in this report can be used to determine the economic attractiveness of fracture treatments per se, to setect the initial spacing to be used in developing a field, and to compare the relative merits of fracturing available wells and infill drilling. INTRODUCTION The effect of vertical fractures on a reservoir producing either an incompressible or a compressible liquid has already been discussed in the 1iterature.l,2 Those results indicate that the production history of such a reservoir is essentially the same as that of a circular reservoir having an effective well radius of approximately one-fourth the fracture length. The present work reports on the effect of a vertical fracture on a reservoir producing two compressible fluids —oil and gas—by solution gas drive. Because of the empirical nature of the PVT and relative permeability data used to obtain the performance of such reservoirs, results can only be obtained numerically and with the aid of high-speed computers. Since reservoirs lose their radial symmetry when fractured vertically, pressure and saturation can no longer be given only in terms of distance from the well. Two coordinates (such as x and y) must now be used to describe the pressure and saturation within the reservoir, and, since we are dealing with compressible fluids, time is also a variable. Thus the solution of a vertically fractured reservoir requires finding two unknowns (pressure and saturation) in two space variables (say x and y) and in time (t). Since no means are readily and generally available for solving such problems at the present time, we have used the results of previous work1,2 to approximate the effect of a vertical fracture on a reservoir producing both oil and gas by depletion. The purpose of the present wmk, then, is to investigate the possibility of using available numerical techniques (limited at the moment to one space variable) to study the two-space-variable flow behavior resulting from a vertical fracture. Results obtained in the course of this investigation are also reported and discussed. Input and output data of the numerical methods used are given in practical units: BOPD, feet, psi, cp, and md. Results are discussed fist in terms of specific reservoir and crude properties and geometries. Later, dimensionless parameters are introduced in order to extend results to different values of some of the reservoir and fracture properties. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The reservoir has an impermeable, circular, cylindrical outer boundary of radius r,. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below&apos; the producing layer (reservoir). It is assumed that there is no pressure drop in the fracture due to fluid flow. 1 indicates the general three-dimensional geometry of the fractured reservoir. Gravity effects and the effects of differential depletion resulting from variations in hydrostatic head (pressure) will be neglected. Thus, the flow behavior in the fractured reservoir is described by the

By D. R. Wieland, H. T. Kennedy

The object of the work reported here was to determine the content of elemental sulfur in gaseous methane, carbon dioxide, hydrogen sulfide, and in mixtures of these gases, at pressures and temperatwes encountered in natural gas reservoirs. Sulfur content at equilibrium is reported for pure methane, carbon dioxide, hydrogen sulfide, and on three binary mixtures of each of the three pairs of gases at pressures of 1,000, 2,000, 3,000, 4,000, 5,000 and 6,000 psia and at temperatures of 150°, 200° and 250°F. In addition, the sulfur content of three ternary mixtures at the same temperatures and pressures are reported. The results indicate that the sulfur content is higher in the gases at higher temperatures and pressures. The content is highest in hydrogen sulfide, intermediate in carbon dioxide and lowest in methane. INTRODUCTION At ordinary pressures and temperatures, the concentration of a nonvolatile material, such as sulfur in a gas at equilibrium, is a function of the vapor pressure of the material and is independent of the nature of the gas. As the pressure and temperature increase, the gas assumes some of the properties of liquids, including the power to dissolve other liquids and solids, to an extent dependent on the nature of both the gas and the material dissolved. At equilibrium, the content of sulfur in the several gases considered here may thus be considered as solubilities which are fixed for a given composition of gas, temperature and pressure. The study of the solubility of elemental sulfur in gases is of interest because sulfur is sometimes present in reservoirs producing natural gas and must be present in the vapor phase. Upon reduction of pressure and temperature, the sulfur precipitates from solution in the reservoir and in the tubing and fittings. Due to the fact that the greatest pressure drop in the reservoir is around the wellbore, the volume of sulfur precipitated will be greatest in this locality and can cause a substantial reduction in the permeability of the formation in this area. As the natural gas flows up the tubing string, the pressure and temperature are further decreased, causing further sulfur precipitation. If the volume of free sulfur is large and remedial measures are not taken, complete plugging of the tubing can occur. Knowledge of the content of sulfur as a function of composition, temperature and pressure will enable the operator to determine what changes in pressure and/or temperature are necessary to keep the sulfur in solution. If the sulfur solubility is large and can be controlled, the operator may desire to produce the well at a high wellhead pressure and temperature and reclaim the sulfur at the surface. The high solubility of sulfur in hydrogen sulfide (5.6 per cent by weight at 5,000 psi and 200°F) suggests the feasibility of recovering sulfur by injecting this gas into sulfur-bearing reservoirs and expanding the produced gas to precipitate out the sulfur. The present project was designed to measure the solubility of sulfur in carbon dioxide, methane, hydrogen sulfide and mixtures of these three gases at various temperatures and pressures. These gases were the major constituents of the gas well where plugging by sulfur was encountered. A study of the published literature shows no data on the solubility of sulfur in any gas, although Hannay and Hogarth,&apos; in 1880 reported that it was soluble in carbon disdlide above the critical temperature of carbon disulfide. EQUIPMENT AND PROCEDURE Fig. 1 shows the layout of equipment employed to bring gases to equilibrium with sulfur and to isolate the quantity of sulfur contained in a known volume of gas. Pure gases were measured in the charging bomb at known temperatures and pressures and displaced into the reservoir bomb to make up the desired mixtures. When equilibrium was achieved

By J. N. Sicking, F. H. Brinkman

A new method for obtaining equilibrium vaporization ratios (K-values) for reservoir fluids has been developed and tested. By application of the method, complex experimental measurements of liquid and vapor phase compositions are eliminated. This simplified technique reduces the cost of experimental equilibrium-ratio data for reservoir studies of condensates and volatile crude-oil systems. The method is designed for systems of constant composition and, therefore, is best suited for depletion studies where compositional changes at high pressures are minor. The basic data required, in addition to the composition of the initial reservoir fluid, are the relative vapor-liquid volumes and densities at reservoir temperature and variom reservoir pressures. Tests demonstrated that the method predicts equilibrium ratios accurately for condensates. A single test on a crude oil was not conclusive; further testing will be necessary before the accuracy of the method can be determined for crude-oil systems. In addition to determining equilibrium ratios, the calculation method provides information on the physical properties of the "plus" component in the vapor and liquid phases. The "plus" component is that mixture of components heavier than the least volatile fraction analyzed. This information is useful in studies of both natural depletion and cycling operations for condensate reservoirs where the heptanes-plus component in the gas phase is produced from the reservoir. INTRODUCTION As more volatile oil and condensate reservoirs are found, the use of phase behavior techniques to predict their performance is increasing in importance. These techniques have long been used for condensate fields and have more recently been applied to crude-oil fields containing oils of medium-to-high volatility. In these phase behavior methods, equilibrium ratios (K-values) are used to predict compositional changes in the reservoir fluids—thereby accounting for the recoverable oil that exists in the gas phase. The reliability of the predictions depends to a large extent on the equilibrium ratios used. These values must be obtained for each component for the entire pressure range being investigated. Unfortunately, because of the complex nature of hydrocarbon mixtures, accurate K-values are hard to obtain. The equilibrium ratios for a particular component will vary not only with the temperature and pressure, but also with over-all composition of the system. The importance of composition is quite critical at elevated pressures, but becomes negligible at pressures below about 300 psia. Therefore, because most phase behavior problems involve the high-pressure region, each fluid system becomes a special case. Experimental programs to determine characteristic K-values are quite difficult and time-consuming. Thus, it is often necessary to resort to approximations of the K-value data. Charts giving K-values for various mixtures and classes of mixtures are available in the literature. However, there are two major difficulties in using them: (1) the K-values of the "plus" component (that mixture of components heavier than the last one analyzed) must be obtained by extrapolation from the K-values of the other components; and (2) the K-values obtained must finally be adjusted by trial and error to agree with observed volumetric data. To eliminate these difficulties, a new method of determining equilibrium ratios was developed. Briefly, after the composition of the system as a whole has been analyzed, the method uses empirical correlations and the gross fluid properties of the system (relative vapor-liquid volumes and densities) to calculate K-values. Because the calculative procedure is long, it is best solved on a digital computer. About one hour of machine time on an IBM 650 computer is required to develop a K-chart for the fluid being examined. DEVELOPMENT OF THE METHOD OF OBTAINING K-VALUES Equilibrium ratios are defined as the ratio of the mole fraction of a component in the vapor phase to its mole fraction in the liquid phase. This statement is expressed in Eq. 1. A typical plot of equilibrium ratios for a particular system is shown in Fig. 1. It should be noted that, at pressures near the saturation pressure, the K-values appear to converge to a common point. This apparent convergence point is called the convergence pressure and is a characteristic of the system involved. Various empirical correlations of K-values have been noted. It has been observed that an isothermal plot of

By M. R. Tek, R. L. Nielsen

The scaling laws as formulated by Rapoport relate dynamically similar flow systems in porous media each involving two immiscible, incompressible fluids. A two-dimensional numerical technique for solving the differential equations describing systems of this type has been employed to assess the practical value of the scaling laws in light of the virtually unscalable nature of relative permeability and capillary pressure curves and boundary conditions. Two hypothetical systems — a gas reservoir subject to water drive and the laboratory scaled model of that reservoir — were investigated with emphasis placed on water coning near a production well. Comparison of the computed behavior of these par ticular systems shows that water coning in the reservoir would be more severe than one would expect from an experimental study of a laboratory model scaled within practical limits to the reservoir system. This paper also presents modifications of the scaling laws which are available for systems that can be described adequately in two-dimensional Cartesian coordinates. INTRODUCTION Present day digital computing equipment and methods of numerical analysis allow realistic and quantitative studies to be carried out for many two-phase flow systems in porous media. Before these tools became available the anticipated behavior of systems of this type could be inferred only from analytical solutions of simplified mathematical models or from experimental studies performed on laboratory models. To reproduce the behavior of a reservoir system on the laboratory scale, certain relationships must be satisfied between physical and geometric properties of the reservoir and laboratory systems. Where the reservoir fluids may be considered as two immiscible and incompressible phases, the necessary relationships have been formulated by Rapoportl and others.2-5 Rapoport&apos;s scaling laws follow from inspectional analysis of the differential equation describing phase saturation distribution in such systems. It will be recalled that these scaling laws presuppose three conditions: (1) the relative permeability curves must be identical for the model and prototype; (2) the capillary pressure curve (function of phase saturation) for the model must be linearly related to that of the prototype; and (3) boundary conditions imposed on the model must duplicate those existing at the boundaries of the prototype. These three requirements seldom if ever can be satisfied in scaling an actual reservoir to the laboratory system because: (1) The laboratory medium normally will be unconsolidated (glass beads or sand) while the reservoir usually is consolidated. Relative permeability and capillary pressure curves are usually quite different for consolidated and unconsolidated porous media. (2) The reservoir usually will be surrounded by a large aquifer which could be simulated in the laboratory only to a limited extent. (3) Wells present in the reservoir would scale to microscopic dimensions in the laboratory if geometric similarity is to be maintained. In view of these considerations, rigorous scaling of even a totally defined reservoir probably would never be possible. The purpose of this paper is to assess the practical value of the scaling laws in the light of the unscalable variables. This has been done by carrying out numerical solutions in two dimensions to the differential equations describing the flow of two immiscible, incompressible fluids in porous media for a field scale reservoir and a laboratory model of that reservoir. While both the reservoir and the laboratory model were purely fictional, each has been made as realistic and representative as possible. The field problem selected as the basis for the investigation was an inhomogeneous, layered gas reservoir initially at capillary gravitational equilibrium and subsequently produced in the presence of water drive. The laboratory model of this reservoir was designed to utilize oil and water in a glass bead pack. The numerical treatment employed was similar to that of Douglas, Peaceman and Rachford6 and it included both capillary and gravitational forces as

By D. G. Russell

Two techniques have been developed with which the applicability of pressure build-up analyses can be extended to include pressure data which previously have been considered virtually unusable. One of the interpretation methods makes possible the analysis of pressure build-up performance during the wellbore fill-up or after production period which occurs soon after a well is closed in. The other technique is an extension of a method for analyzing pressure build-up performance during the late-time portion of the pressure build-up which occurs after boundary eflects first begin to alter the shape of a conventional pressure build-up curve. With both of these methods it is possible to obtain estitnates of the kh product, the skin factor and the reservoir pressure. In addition, with the late-time analysis technique it is possible to obtain an estimate of the contributory drainage volume of the well being tested. This means that in some cases a check on reservoir limit test and (or) material-balance calculations can now be obtained from pressure build-ups. Both methods are slightly more time-consuming than conventional pressure build-up analysis methods because trial-and-error plots of pressure data must be made. The late-time method for analysis of pressure build-ups is in principle applicable to the late-time portion of a two-rate flow test or a pressure drawdown test. The interpretation formulas and procedures for these types of tests are also outlined. In these cases, as with pressure build-ups, it is significant that an estimate of the contributory pore volume is also obtained. On the basis of limited experience with the new techniques, it appears that satisfactory estimates of the kh product, skin factor, reservoir pressure and, for late-time analysis, contributory drainage volume can be obtained. INTRODUCTION The analysis of bottom-hole pressure build-up behavior in closed-in wells has been a a subject of interest in petroleum engineering circles for many years. In fact, few other subjects have received as much attention as pressure buildup analysis methods have. The cause for this interest is essentially twofold in nature. First, the pressure behavior of a well can normally be measured with a reasonably high degree of accuracy so that good data for analysis can be obtained. Secondly, over a fairly wide range of operating conditions, valuable information as to the quality of the reservoir rock and completion efficiency of the well can be obtained at a nominal cost. In recent years, numerous papers have been prepared on the effects of various operating conditions and reservoir heterogeneities on pressure buildup behavior. Very little work has been done, however, on extension of pressure build-up analysis methods to those pressure data which are not amenable to analysis by the present methods. The theory upon which the analysis of shut-in bottom-hole pressure build-up data is based is derived from the solution of the radial flow equation for a slightly compressible fluid for constant-rate conditions. It requires that the well be closed in for a sufficient period of time to obtain a clearly defined linear portion on the plot of observed bottom-hole pressure vs log (t + ?t)/ ?t (where At is shut-in time, and t is producing time to the instant of shut-in). From the slope of the plot and other normally obtainable data, the formation permeability, the well damage or skin factor, and the reservoir pressure at infinite shut-in time (if the reservoir were infinite) can be estimated. The successful application of this procedure depends on being able to recognize the straight-line section on the basic pressure build-up plot. The presently used pressure build-up interpretation theory also assumes that a well is closed in at the sand face and that no production into the well occurs after shut-in. In practice, of course, the well is closed in at the surface, and inflow into the well continues until the well fills sufficiently to transmit the effect of closing-in to the formation. This adjustment period is commonly referred to as the "afterproduction" or "fill-up" portion of the pressure build-up. During the period that the well fill-up effect is most pronounced, the basic pressure build-up plot is nonlinear. At later shut-in times after the effects of a drainage boundary have been felt at the well, deviation from the straight-line behavior of the pressure build-up plot also results. In many cases either of these effects or a combination of both can make the straight-line portion on the pressure build-up plot difficult to recognize. Obviously, an extension of pressure build-up analysis methods to include the afterproduction period and the period in which boundary effects are being felt would be desirable and might render valuable pressure data which for years have been considered virtually unusable. The principal reference of note concerning pressure buildup analysis during the afterproduction period is a paper by Gladfelter, Tracy and Wilsey.1 In the approach of these authors it is necessary to measure the rate of influx into the well during the afterproduction period. This is done through sonic measurements or through measurement of tubing-head and casing-head pressures simultaneously with

Jan 1, 1967

By D. A. Flanagan, P. B. Crawford

A study has been made of the feasibility of storing liquid meihane at low pressures in undergrohd caverns. Methane liquefies at — 258°F at atmospheric pressure. It is shown that the methane evaporation rates will rapidly decrease and cool the surrounding rock so that at the end of one month they would be between 50 Mcf/hr for a cavern of 25-ft radius and 700 Mcf/hr for a cavern of 100-ft radius. At the end of 10 years, the evaporation rates would be 18 and 100 Mcf/hr, respectively, for caverns of the same radii. The evaporation rates may be reduced by a factor of 2 to 10 by the application of insulation. The cost for mining the caverns is estimated to be $.75 to $1.25/Mcf of storage. This is substantially less than surface storage; it is believed to be safer and to result in lower maintenance, savings in space and savings in strategic materials. INTRODUCTION During the past few years, there has been an increasing interest in the economic feasibility of liquefying methane. Methane liquefaction is being considered for tanker transport; in addition, liquefaction is being reconsidered for shaving peak gas demands. Several articles have described natural gas liquefaction and the progress of the tanker in making trial runs from the United States to Great Britain to determine the feasibility of tanker transport. At the present time, pipelines are not designed to supply peak gas loads during extremely cold periods such as are oiten encountered in the North and Northeast. Gas is being stored in underground reservoirs en-route to its destination, but in many instances satisfactory storage in porous reservoirs has not been practical, especially along the Eastern seaboard where few petroleum reservoirs have been found. In England and other foreign countries, it is unlikely that satisfactory porous structures could be found, and it may be desirable to mine or excavate the rock to obtain storage. By storing the methane near the consumption point, product availability can be increased during periods of need. Methane liquefies at - 258OF at atmospheric pressure, and 1 cu ft of liquid methane will make about 600 cu ft of gaseous methane. Methane liquefaction for shaving peak gas demands was conducted in Cleveland, Ohio in the early 1940&apos;s. The Cleveland plant was designed to liquefy 4 MMcf/D of natural gas.&apos; Development of several low-temperature processes has resulted in some improvements in the design of low-temperature storage vessels. Liquefied-methane storage vessels can be designed for various evaporation rates. The desired rate will depend on the reason for liquefaction. For shaving peak gas demands, the desired evaporation rates would be low, possibly between 0.2 and 1.5 per cent/day. For methane tanker transport and supply, the evaporation rates must be high and must equal the total daily rate of supply. It is apparent that the cost of storage facilities will depend on the objective and will vary appreciably. Surface storage of methane may cost between $3,000 and $20,000/MMscf, depending on the size and design. The failure of the surface storage vessels at Cleveland, with the resulting injury and loss of many lives, has caused considerable emphasis to be placed upon developing new, safer and improved methods of storing liquid methane. Storage of liquefied propane and butane in underground salt domes has been very satisfactory. The per-barrel cost of underground storage for any sizeable capacity is very small compared to the cost of surface storage for LPG mixtures. The larger the capacity of the underground storage, the less is the per-barrel cost. Numerous advantages appear to exist if liquefied methane could be satisfactorily stored underground, the principal advantages being increased safety, lower initial cost, lower maintenance cost, savings in space and savings in strategic materials. The purpose of this paper is to determine the feasibility of storing liquid methane underground in mined caverns. If a large spherical cavern is dug underground and is filled instantaneously with liquid methane, the surface of the sphere may approach the liquid methane temperature almost instantaneously. The temperature distribution for a sphere of radius r. in an infinite medium, initially at one temperature with spherical surface kept at T, from time t = 0, is given by Eq. 1.&apos;

By C. R. Knopp

Gas-oil relative permeability ratio is an important relationship in oil reservoir predictive calculations. A correlation has been developed from 107 gas-flood k/k tests on Venezuelan core samples. The correlating parameter is based on restored-state water saturation tests and&apos; is applicable to both consolidated and poorly consolidated sandstone reservoirs. Data of the correlation show that there are no distinguishah1e differences between the mass-data groupings for the two c1assifications A procedure is recommended for running .sufficient relative. permeability analyses to compute a geometric mean of the sample group. The geometric mean is more representative of the total core, and probably the entire reservoir. For example, while only one in four of the k,,/k,., test curves agreed closely with the resultant correlation of this report, the geometric mean curves of the 16 suites (three samples or more). showed good agreetment ill three cases out of four. INTRODUCTION The gas-oil relative permeability ratio is an important, fundamental relationship in most oil reservoir predictive calculations. Predictive calculations are made to estimate future reservoir production characteristics and ultimate oil recovery. The k1,/k2, relationship is specifically needed to relate the surface gas-oil ratio to the reservoir oil and gas saturation, and to calculate the relative movement of these phases within the reservoir whenever some of the more complex driving mechanisms are present. Laboratory k1/k2, tests are not generally run as a routine analysis. Consequently, k1/k2 data often are not available when needed because the cost of laboratory work could not be justified or the need for such data had not been properly anticipated. When laboratory k1/k2, data are available, they are often very difficult to interpret. For example, wide divergence is sometimes shown in a family of k1,/k1, tests representative of the producing horizon in a single well. With these considerations in mind, a study was made to determine if a relationship might exist between the k1,/k2, curve and some other simple laboratory test criteria. The most probable k1/k2, curve correlation for Venezuela described in this paper is the result of the investigation. The presented correlation defines the most probable gas-flood k,,/k,, curve through the medium of air-water capillary displacement and centrifuge water saturation tests. The laboratory procedures of these tests are. relatively simple, and inexpensive; test data should be. widely available- from routine analysis. DATA AVAILABLE, LABORATORY METHODS The report correlation utilized 107 gas-Hood k1/k2, tests run on sandstone cores of Venezuelan reservoirs. Table 1 is a general tabulation of data pertinent to the tests, while Table 2 summarizes the data. Thetests include 96 from Western Venezuela and 11 from Eastern Venezuela. Eighty-two- of the 107 test samples were sandstones that varied from poorly consolidated to-unconsolidated; 25 were consolidated. The average sample porosity was 26.7 per cent and the average permeability was 1,121 md; these values typify the better sandstone reservoirs of&apos; Venezuela. The Welge gas-flood technique,&apos; based on fundamental Buckley-Leverett frontal displacement theory, was introduced in about 1952 and is widely accepted in the industry. The laboratory procedure is relatively simple, rapid, and can be performed on small core samples. While there have been some minor variations in sample preparation and laboratory procedure in the tests used for the correlation, these tests can be generally summarized as follows. The core sample was first sol vent-extracted and dried. Connate-water saturation was restored by the oil-flushing or evaporation-blow down methods. At the beginning of gas flood the hydrocarbon pore volume was completeiy saturated with the test oil phase. Unsteady-state gas-oil displacement then began with the injection of nitrogen or helium. while the displaced oil and gas phases were incrementally metered at the out-flow face. From the test data, the k,,/k,, curve was calculated by the Welge method.&apos; The individual oil and gas relative permeabilities were also calculated." CORRELATING PROCEDURES In attempting to establish a basis of correlation, we found that broad mid-range sections of 105 of the 107 k,,/k,, test curves could be closely duplicated by a straight line. Only two curves did not show a degree of linearity in this region. Correlation-curve definition parameters were subsequently developed from this observation of consistent mid-range linearity. Possible correlating variables were limited to the physical properties measured on core samples that (1) were widely available as common test data and (2) could be easily and cheaply obtained through future laboratory work. The more obvious possibilities were porosity, permeability and

Jan 1, 1966

By J. M. Smith, G. P. Willhite, J. S. Dranoff

Heat transfer rates were measured in sandstones with flow of gases perpendicular to the direction of energy transfer. Effective thermal conductivities ker ranged from 0.7 to 1.7 Btu/(br)(ft)(°F). The contribution of the solid phase appeared to be the most important in these consolidated materials, although the thermal conductivity of the gas had some effect. The velocity of the gas through the pores of the sandstones had no influence upon ker up to values of 168 lb/(br) (sq ft) in agreement with data obtained for unconsolidated beds of glass beads. The present results indicated that gas mixing, and hence heat transfer by convection in the pores, is less for perpendicular transfer of energy than when fluid flow and energy transfer are in the same direction. HEAT TRANSFER PERPENDICULAR TO FLUID FLOW IN POROUS ROCKS Heat transfer in porous media with pore sizes in the micron range depends upon the fluid in the pores and the geometry of the solid phase. The best characterized system is a bed of solid spherical particles. Heat transfer in this system has been studied extensive1y (Ref. 8 summarizes the literature up to 1959) when the pores contain stagnant fluid. When the fluid is in motion the directions of flow and energy transfer have an effect on the heat transfer rates, as demonstrated by comparing the work of Willhite, et al l3 for perpendicular flow and that of Kunii and Smith9 for parallel flow of energy and fluid. For perpendicular flow, no increase in effective thermal conductivity ke was noted up to mass velocities G of 77 lb/(hr) (sq ft). In contrast, for parallel flow ke, increased with G in the same range of flow rates. These higher values of ke in the direction of flow also have been observed3,4,10 in beds of larger particles, 0.1- to 0.5-in. diameter. For beds of consolidated materials, such as porous rocks, data are not available for these compar- isons, although Adivarahan1 reported results for the parallel case. Hence the primary objective of this work was to measure ke values for perpendicular flow of fluid and energy in porous rocks. Of interest also was the variation in effective conductivity with fluid velocity. APPARATUS AND PROCEDURE In the experimental method, a constant heat flux was applied to the inner surface of an annular section of the porous rock. By cooling the outer wall, a temperature gradient through the annular sample was established and measured with thermocouples placed within the sample at various radial positions, and at three elevations (A,B.C). The location of the 2-in. O.D., 3.75-in. long sample in the apparatus is shown in Fig. 1. Fluid entered the bottom (1) of the 3-in. I.D. (approximate) steel shell, flowed upwards through the sample and out at the top (3). pressure taps (2,4) were used to check the permeability of the sample. The energy flowed radially from the centrally-located electric heater through the sample and was absorbed in the water-cooled jacket. The samples studied were naturally occurring sandstones from different locations with the properties given in Table 1. These materials are identical with those used by Adivarahan 1 for the parallel flow of energy and fluid. Prior to use they were refluxed with toluene to remove hydrocarbons and leached with distilled water to remove soluble salts. Each sample was visually examined and discarded if large nonhomogeneities, such as cracks or stone particles, were noted. Eight copper-constantan thermocouples were inserted in holes drilled radially into the sample with

By W. Hurst

What is entailed here is the extension of the sinzplified material balance formulas to encompass interference between oil fields. As previously reported, the ex-plicitness as so revealed for the cunzulative pressure drop as a function of all factors contributing to its change in the material balance equation, is now transcribed through the inter-conzmunicating aquifer to effect an increased pressure drop on an adjoining field by interference. Such is performed by mathematical analyses and the application of the Laplace transformations. What is accomplished is that the reiteration problems previously associated with interference studies are nullified, since volumetric changes for the fluids in situ are automatically adjusted by the explicitness so expressed, and what pertains to the superposition principle applies only to the impeded water drive upon a su.bject field, which is likewise incorporated in the over-all pressure drop that results. The mathematics treats with the rigorous solution of the problem, as well as with methods easily amenable to numerical interpretation by the practicing engineer. In all cases, however, what is deduced for areal extents, reveal interference between wells when time becomes large that further substantiate the analyses. Probably of equal significance, all variables and for/nation characteristics are accounted for. This applies to the differences in PVT analyses that occur from field to field and the physical parameters associated with the lithology of the reservoirs. The latter is deduced from a rigorous interpretation of the unsteady-state flow problem for sands of different permeabilities in series. Thus, what purports to be a trial-and-error calculation to include variations in sand conditions within the intercommunicating aquifer to define water drive has little importance compared to the designation of such paranzeters at the reservoir to constitute the essential criteria. INTRODUCTION In a recent publication, the reader has been introduced to the simplified material balance formulas,&apos; It has been shown that the complexities formerly associated for identifying and determining reservoir pressure in the material balance relationship can be resolved by treating with the Laplace transformations, the inversion of which reveals pressure as an explicit function of all factors contributing to its change. For an under saturated oil reservoir, this constitutes an integrated effect from the inception of production; and for a saturated oil reservoir, such represents a survey traverse. However, what is most important, for the first time we are able to express the pressure change in a reservoir directly to the oil, gas and water produced, and to the physical parameters of the formation, in the same manner that reservoir pressure has served in the past to define well interference — which opens up this avenue of endeavor for investigating interference between oil fields. The occurrence of interference is by virtue of the juxtaposition of two or more oil reservoirs producing in the same aquifer. Thus, in any one field, the extended production so incurred creates a pressure lowering in the field, as well as in the aquifer, which can now be transcribed by these simplified material balance formulas to the adjoining reservoir to observe the increased pressure change so induced. These effects are retroactive from field to field, to reveal a pressure drop far in excess of normal depletion to evidence the interference pattern that can result. Such is the purpose for the present undertaking — to develop the mathematical physics treating with this phenomenon of interference between oil fields, employing the simplified material balance formulas. Since Ref. 1 often will be referred to in the text, it is identified as "The Simplification", in the same connotation that another paper has been referred to as the "Laplace Trans-formation. However, the amount of material to be covered in these passages is substantial; therefore, the total emphasis will be placed upon developing these fundamental ideas treating with interference rather than attempting to offer work curves, which always can be prepared. For the practical application of these simplified material balance formulas that form the framework of the present development, the reader will find a comprehensive treatment in the Appendix of Ref. 1, illustrating a factual field example. In essence, what is proposed by this presentation is to make available to the practicing engineer a simplified means to treat with interference between oil fields, independent of the complex machine computations that have attended such undertakings in the past. As the problem unfolds, a common pattern will be discernible, whereby the many reiteration processes that previously required computing devices are eliminated.