Search Documents

By F. J. Fayers, S. A. Hovanessian

The one-dimensional displacement equation for a homogeneous porous medium, including the effects of gravity and capillaty forces, has been solved by a numerical method. A finite-difference scheme is developed for obtaining saturation, pressure and fractional flow profiles in waterflood recovety problems. From the numerical examples given, it is concluded that the gravitational forces have a pronounced effect on the saturation profiles and the pressure distribution curves of the system. INTRODUCTION Within the past 20 years, a number of papers have appeared in the literature dealing with the quantitative treatment of waterflood recovery problems. In their celebrated paper, Buckley and Leverett1 described a method for calculating saturation profiles when the effects of capillary pressure and gravity are excluded. Terwilliger, et al,2 included the effect of gravity in their theoretical and experimental investigation of oil recovery problems and obtained close correlation between experiment and theory. welge3 described a simplified method for obtaining the average saturation and the oil recovery from an oil reservoir. The effect of gravity can be included in this calculation. More recently (1958) Douglas, et al,4 presented a method for calculating saturation profiles which includes the effect of capillary pressure. The authors start with the one-dimensional displacement equation (which is nonlinear in the derivative) and, by a change of variable, transform this equation to a semi-linear partial differential equation. This equation is then solved by a finite-difference method on a high-speed digital computer. Fayers and Sheldon5 obtained the solution of the one-dimensional displacement equation by directly replacing the differential equation by a finite-difference equation. The effects of capillary pressure and gravity were in- cluded in their solution. Here, since the movement of the foot of the saturation front is governed by a separate equation, the elapsed time in attaining a particular saturation profile cannot be obtained. The results of these two papers indicate that the inclusion of capillary forces eliminates the triple-valued Buckley-Leverett saturation profiles. In the present paper, the one-dimensional displacement equation for a homogeneous permeable medium, including the effects of capillary and gravity forces, is solved. The method of solution resembles that described in the paper by Douglas, et al, since it involves a similar transformation of variable. However, it extends their work in that the effect of gravity is included and, in addition, pressure profiles may be calculated. Also, the functions of saturation [kro(S), krw(S) and Pc (S) ] required are entered in tabular form rather than as low-order polynomials. This simplifies data preparation and permits the use of a greater range of function types. Unlike the method of Fayers and Sheldon, the corresponding elapsed time required for the development of each saturation profile is calculated and, also, saturation profiles may be calculated after breakthrough. THEORETICAL CONSIDERATIONS The dimensionless form of the one-dimensional displacement equation for a homogeneous porous medium as obtained from Darcy's law and material balance relations can be written as follows.

By R. J. Blackwell, A. L. Pozzi

Scaled laboratory-model studies provide a powerful method for evaluation of a proposed oil-recovery process. In recent years, models have been used extensively to evaluate processes in which solvents displace oil, both for general cases and for specific reservoir conditions. Since the performance of a miscible flood in a horizontal reservoir can be significantly affected by transverse mixing between solvent and oil, this displacement mechanism must be accurately simulated in the scaled model studies. Unfortunately, precise scaling of transverse dispersion coupled with the requirement of geometric similarity requires impractically large laboratory models and long times for experiments. If scaling requirements for miscible displacements could be relaxed while accurate simulation of essential displacement mechanisms is maintained, the utility of model studies would be greatly enhanced. The purpose of the work reported herein was to evaluate the relative importance of various mechanisms affecting miscible displacement and to ascer tain whether the essential features of the displacement process can be simulated even though some scaling groups are not satisfied. These studies were performed with completely miscible systems in linear, horizontal models packed with unconsolidated media. From the experimental results, a set of relaxed scaling criteria was formulated which allows the requirements of geometric similarity and equality of the ratio of viscous to gravity forces to be omitted for specified conditions. The relaxed criteria are valid whether transverse mixing is by molecular diffusion or by convective dispersion. Correlations which permit prediction of vertical sweep efficiencies in linear, horizontal reservoirs were developed from the experimental data when transverse mixing is by molecular diffusion. These same correlations may be used when transverse mixing is by convective dispersion if an empirically defined, effective, transverse dispersion coefficient is used in the description of the mixing process. The effective transverse dispersion coefficient correlation essentially duplicates the dispersion coefficient correlation for equal-viscosity, equal-density fluid systems. Experimental values for the effective transverse dispersion coefficient can be measured readily. INTRODUCTION One of the most effective methods for evaluation of miscible-displacement oil-recovery processes is that of displacements in laboratory models scaled to simulate reservoir conditions. For these laboratory studies to be meaningful, however, the essential displacement mechanisms affecting reservoir performance must be accurately simulated. Since the performance of a miscible flood in horizontal reservoirs, or in dipping reservoirs at high rates, can be significantly affected by transverse mixing of solvent and oil, this mechanism must be considered in the design of laboratory experiments. Unfortunately, precise scaling of transverse dispersion coupled with the requirement of geometric similarity requires impractically large laboratory models and long experiment times. This difficulty seriously limits the utility of laboratory model studies. Craig,et al,1 demonstrated that geometric similarity is not required when mixing is unimportant. Their experimental data indicate, for the cases studied, that the displacement is sufficiently characterized by scaling the ratio of viscous-to-gravitational forces. This work suggests that relaxation of the requirement of geometric similarity and, possibly, other criteria might also be permissible when mixing is important, provided suitable groups describing the mixing process are scaled. The purpose of the work reported here was to evaluate the relative importance of various mechanisms affecting miscible displacement and to ascertain whether the essential features of the displacement process can be simulated even though some scaling groups are not satisfied. Consequently, laboratory investigations were made of the dependence of recovery on dispersion and other variables,

By Christoph Sagadin

Corrosion mechanisms between high melting synthetic ferronickel slags and refractory were investigated. The used slags were prepared by mixing and melting of specific oxides. Substrates of the applied refractory material and specimens of FeNi slags were heated in a hot stage microscope up to 1650 °C. The experiments were performed under a defined gas atmosphere of 60% CO and 40% CO2. A further examination of the formed phases between slag and refractory occurred by scanning electron microscope. The investigations indicate that the slag penetrates between magnesia grains and partly dissolves magnesia. Spot analyses show that iron diffuses into the magnesia grains, which transform to magnesiawustite, meanwhile SiO2 forms different types of olivine like forsterite and monticellite. Thermodynamic calculations confirm the formation of these phases. The combination of practical lab scale experiments and thermodynamic calculations should finally contribute to an improvement of the refractory lifetime and performance.

COLLOQUIUM PRACTICAL APPLICATIONS OF INSTRUMENTATION IN THE MINING AND METALLURGICAL INDUSTRIES Rapporteur: J. Laschinger A colloquium and general meeting of the Institute held jointly with the Instrument and Control Society of Southern Africa at Kelvin House, Johannesburg on 18th October, 1972, had as its theme "Practical Applications of Instrumentation in the Mining and Metallurgical Industries" Dr J. P. Hugo (President) was in the chair and about 200 members and visitors were present.

By P. D. White, J. T. Moss, J. S. McNiel

Results of experimental work on the in situ combustion process were first published in this country in 1953' when Kuhn and Koch described results of a three-well test in Jefferson County, Okla. Shortly there-after Grant and Szasz2 described field studies in Nowata County, Okla., in which several modifications of the basic combustion process were investigated by injection of various oxidizing gas mixtures other than air. In the past five years speculation concerning this method of oil recovery has continued, with a number of interesting papers on various aspects of in situ combustion appearing in the literature3,4,5,6,7. However, only one of these' provided actual engineering information on pattern-type field experiments. The primary objectives of the Oklahoma three-well test were to check operation of specially designed electrical ignition equipment and evaluate problems associated with initiation of combustion in a natural oil sand reservoir. Immediately following successful completion of this test in the Pontotoc sand, plans were activated for a second field experiment to be conducted in the same sand a short distance away. The objective of this second field test1 as to establish a relationship between field observations and information derived from laboratory experimentation and computations. Secondary objectives of the test were to develop operating tech- niques and define operating problems, and to obtain information that would assist in the economic and engineering appraisals of the process. Careful planning was required since this test was designed with the hope of providing basic information on the characteristics and operation of a combustion drive. It was necessary to include test facilities which would not only permit precise control and measurement of the process variables but would also provide sufficient flexibility of control to insure that variables in the combustion process could be studied during the course of the test under a wide range of conditions. In order to accomplish these objectives elaborate instrumentation and several control wells were included in the design of the test installation. This type installation was felt to be well suited for an experimental study, but was in no respect intended as a counterpart of a commercial-scale operation. PRELIMINARY LABORATORY WORK An important objective of the field experiment, as stated previously, has been to obtain reliable data on operating variables which might be compared with similar data obtained in laboratory combustion tests. The value of successful correlations between quantities that can be measured in the laboratory and those obtained in a full-scale commercial venture has been well established for other industrial processes. The difficulty of following a process underground should make such correlations even more essential here. Also. approaching the field test phase of development of in situ combustion, it was considered desirable to precede design of test equipment with laboratory combustion studies under a variety of conditions to establish likely ranges of variables. These included temperature, air injection rate, frontal advance rate and produced gas composition.

By Thomas Egleston

DURING the year 1873, my attention was called to the frequent accidents, resulting from the breaking of rails, on the different railroads in this' country, and I was requested to investigate the subject. The plan of investigation I proposed was to ascertain, in as many cases as possible, the exact hist* of broken rails, both in this country and in Europe; to collect information in Europe relating to the tests which rails used there were required to undergo, and to experiment upon the pieces of rails broken on the road, with a view to ascertain whether their fracture was due to their chemical constitution, bad manufacture, to a reduction in strength owing to temperature, or to physical changes in the constitution of the rail. The investigation was commenced by the collection of statistics, and for this purpose I spent several months in Europe, collecting information on the subject of broken rails, as well as the life of iron and steel rails generally, and inquiring into the systems of purchasing and of testing them employed by the railroads there, intending, on my return, to make a series of analyses of rails broken, both in this country and in Europe; to ascertain, if possible, how far great cold had an influence on the fracture; to examine whether the rail broken, either on the road or afterwards, underwent, at or about the fracture, any physical change, and to make a series of experiments on the strength of rails manufactured in the United States or sold in the American market. Unfortunately, the panic of 1873 came on, just before my return from Europe, and prevented the carrying out of the larger part of the plan proposed; as, however, I had collected in the meantime, a large number of statistics of different kinds from all parts of Europe, and had made a number of interesting experiments and investigations, I have thought it would be worth while to communicate some

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

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

By E. C. Pechin

The attention of the members of the Institute of Mining Engineers is asked to a description of the minerals of Southwestern Pennsylvania, as representing the minerals of an enormous area, stretching continuotisly, with local variations, along the western slopes, and at the base of the Alleghany Mountains, through Western Pennsylvania, West Virginia, and Kentucky, and destined to play no mean part in the future of the iron trade of the country. The particular locality is at and near Donbar, in Fayette County, where the most careful examinations and extensive developments have been made. The deep gorges cut by the Youghiogheny and Monongahela Rivers, in their passage through the mountains and lower lands, give admirable facilities for observing the measures lying between the sandstones of No. 11 and the upper coal measures. Apart from its varied mineral and agricultural resources this section of country has great historic interest attaching to it. It was down the Monongahela that Braddock led his ill-fated army to its defeat under the wails of Port Duquesne, and it was up the Youghiogheny River and Dunbar Creek that the shattered remnants hastily fled to gain the camp of Colonel Dunbar, the second in command, who was entrenched upon a high knob, at the head of the creek which bears his name, and which is still known and visited as Dunbar's Camp. Numerous streams, runs, and localities preserve the names of the guides and pioneers who accompanied, in early colonial times, young George Washington, who was pushing his surveys into the trackless and hostile wilderness, and many of them taking up large tracts of

By J. E. Warren

The mathematical problem considered by the author1 can be given another physical interpretation which is of some practical significance. The alternative physical problem involves the approximate behavior of a single well which is producing at a constant rate through an axially symmetric, horizontal fracture of infinite flow capacity which is located at the center, top or bottom of a uniformly thick, horizontal, homogeneous, anisotropic reservoir of infinite lateral extent which contains a single, slightly compressible fluid. Using Coats' asymptotic results, the wellbore pressure in the fractured system is given by the following equation.

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 Clayton A. Signor

Abstract: Expansive soils cause more damage to structures annually than a combination of other major natural disasters. Because of the cost to our society, all means and methods need to be fully explored to mitigate the problems associated with expansive soils. This study will present a foundation design approach that is underutilized in this application, driven piles. The main objective of the study is to present pile test results and analysis from four driven pile project sites in three types of expansive soils found in central Texas: Del Rio formation, Taylor/Navarro formation, and expansive alluvium. High strain dynamic pile tests were conducted on each of the four studies with rigorous signal matching analysis from the CAse Pile Wave Analysis Program (CAPWAP). Ultimate pile capacities ranged from 73 to 311 kips with an average of 61% of the total capacity coming from the pile shaft and were two to six times the structural capacity needed. Allowable loads calculated from Modified Gates dynamic formula best modeled allowable test results. Average unit skin frictions ranged from 0.50 to 4.71 ksf. Restrike pile tests of 1 to 17 days after initial driving reported 30 to 100% shaft capacity gain. All open-ended pipe piles driven produced soil plugs ranging from 4 to 14 feet thick. Small diameter, thick-walled, open-ended pipe piles reached penetration of twice the depth of designated zone of seasonal moisture change without problem. The observed production rate of the driven piles was on average 8 minutes, which implied daily production of 15 to 40 piles. Predrills or augered holes should be specified for underground obstructions found in soil investigation.

By J. Davies

This paper describes the successful management and construction of the 5.18 m (17 ft) diameter, segmentally lined, North Dorchester Bay CSO storage tunnel in Boston, MA. This 3,300 m (10,830 ft) long tunnel was mined through variable soft ground with only 6 m (20 ft) of cover, using an earth pressure balance tunnel boring machine. These extensive data collected from the projects unique automated and internet accessible geotechnical and TBM monitoring system, will be examined and used to explore the relationships between geological conditions, TBM operating parameters and measured settlements. The paper will also discuss the use of screw revolutions as an indicator of muck volume.

By Youn-Bae Kang

Evaporation mechanism of S from liquid Fe–C–S alloys at 1873 K was proposed by analyzing available experimental data. It has been known that increasing C content in liquid alloy increases activity coefficient of S ðfSÞ, and it could raise driving force for the evaporation reaction S = S(g). However, experimental data of the evaporation of S in the Fe–C–S alloys could not be accounted for only by considering the increases of fS. In the present study, formation of carbosulfides, CS(g) and CS2(g), was additionally taken into account in order to explain role of C for the accelerated S evaporation. Surface adsorption of S was also taken into account, which retards the evaporation rate of S. An evaporation model equation was formulated. It can be applied to calculate the evaporation rate of S over wider C content (from zero to its saturation to liquid alloy).

By B. Manana

In heap leaching, the oxidative dissolution of value minerals encapsulated deep within large particles cannot be fully understood on the basis of the bulk solution conditions relative to variable solution conditions near the mineral surface. In the present study, the diffusion of the oxidising species through inner particle pores is simulated in a model apparatus that separates an inert platinum electrode under controlled reduction potential from the bulk solution through narrow pores of varying length. The Cu-NH3 system was selected as a model system and Cu(II)/Cu (I) as the redox couple of interest. Platinum probes inserted in the pores allowed the measurement of varying redox potentials along the pore length. Results showed that in a 10 mm pore, there exists a narrow (0.5 mm) zone near the reduction surface that is depleted of the oxidant and in which its supply is strongly diffusion limited, even if it is in abundance in the bulk solution. The consistent presence of this narrow diffusion region across all experimental results indicated that the transportation of the reduced species away from the reaction surface had the most significant influence on the potential variations within the pore. This was further supported by the minimal effect of varying concentrations of both Cu(II) and dissolved oxygen on the solution potential within this region.

By C. R. Boyd

Without attempting to do more than give a preliminary or skeleton report upon the geology and minerals of Southwestern Virginia at this time, I am led to hope that the great commercial importance of the sixty miles of cross-section here crudely treated of

By E. R. Spencer, E. Reeves

National Institute for Occupational Safety and Health (NIOSH) researchers conducted an investigation to quantify sound levels and to determine the amount of sound reduction provided by engineering noise controls installed in a talc processing plant. Baseline sound level and sound intensity measurements were performed at the plant and the measurement locations were recorded for comparison to post-control measurements. Follow-up measurements were then made at the same locations after the initial noise controls were installed. The plant subsequently decided to implement additional noise controls and the researchers returned to conduct measurements for a final analysis of all noise controls. The most significant results showed a sound level reduction in the main mill area from a range of 93-104 dB(A) down to 90-94 dB(A), and a total sound power level reduction of 21 dB(A) for air classifying mill 3.

By A. A. Hazell

Mine management by objectives by A. A. Hazel The procedure of setting production targets is illustrated in Appendix I of the paper. This procedure is intended only as a guide although the standards on which it is based were established by work studies and found subsequently to be well within the limits of a trained crew.

By Albert L. Brautigam, Michael R. Yenchek, Jeffrey Shawn Peterson, James C. Cawley

The National Institute for Occupational Safety and Health investigated how the starting of induction motors may cause nuisance tripping of short-circuit protection on coal mine power systems. This research had a dual purpose: 1) to identify how motor-start waveforms differ from those for short circuits and 2) to devise a method to provide short-circuit protection without intentional time delays to account for motor starts. This technology will help ensure that surface temperatures of energized electrical apparatus will not exceed gas or dust ignition thresholds when short circuits occur.

By R. G. Hawthorne

This report is concerned with fluid displacement in porous media, in those cases where viscous and gravitational forces control the displacement. Such a system would usually be found in a sand body of large physical dimensions such as an oil reservoir, although it is possible to create such a system in the laboratory. It is shown that the position of the fluid interface can be predicted by numerical calculations using a basic idea presented by Dietz. Fluid flow is considered in a vertical plane in a homogeneous, porous medium of sufficient thickness that the capillary transition zone is small in comparison with the total reservoir. A theory developed by Dietz' is used to make numerical calculations of the position of the fluid interface. The results for several conditions are compared with scaled model experiments. The results show that, for gas drive in a reservoir of steep dip, a relatively low flow rate can displace large volumes of oil before gas breakthrough. On the other hand, water injection at favorable mobility ratio and low dip may show best performance at high rates. Water tends to underride the oil and, given sufficient rime, will break through without much oil displacement. For certain conditions, which include relatively low flow rate, the interface is a straight line and its behavior is simple to calculate. At higher flow rates, the interface is unstable, and a numerical solution was programed for an automatic computer. In general, good agreement is shown between the fluid model and the computed results so long as gravitational forces have control. For a water drive at very unfavorable mobility ratio, many small water fingers appear. These viscous fingers are not controlled by the relatively small gravitational forces. When viscous fingering becomes the controlling factor, the mathematical model is oversimplified, and results do not check the fluid flow model. INTRODUCTION Present methods of reservoir analysis depend upon certain simplifying assumptions to obtain mathematical descriptions of practical use. Material-balance methods (Muskat2 or Tarner2) assume uniform fluid saturations in the entire reservoir, or in a few subdivisions of the reservoir. An unsteady-state flow calculation by West, er at considered pressure and saturation changes in flow to a well during solution gas drive, and neglected gravity effects. Results showed only a 4 per cent difference for ultimate oil recovery by the Muskat method, even though the case chosen for study was one in which unsteady-state effects should be high. The Buckley-Leverett5 method commonly assumes a one-dimensional flow system. It is applicable at high flow rates where viscous forces predominate over gravity forces. Simultaneous, parallel flow of the two fluids is assumed, and the concept of a fluid interface is not introduced. Permeabilities to each fluid for a given saturation must be known. The method is not applicable for a two-dimensional system where cross flow becomes possible. Less well known is the displacement equation derived by Dietz. This method is designed for two-dimensional flow systems and assumes a definable fluid interface within the porous medium. Dietz showed that, for a range of low flow rates, the interface would be stable. straight and at an angle of inclination which could be simply calculated. At a certain critical flow rate, the calculated interface tilt would equal the formation dip. For higher flow rates, a finger of displacing fluid would invade the displaced fluid. Dietz indicated that his method applied only to macroscopic reservoir behavior, while the Buckley-Leverett method applied to the small transition zone at the fluid interface. The examples worked out in this report are based on the fluid-displacement theory of Dietz. It is shown that the Dietz theory may be used to derive equations analogous to the Buckley-Leverett equations. In contrast to the Buckley-Leverett method, flow is considered in a plane rather than being limited to a line. Rather than a frontal advance, the movement of a fluid interface is followed. For flow rates substantially exceeding the critical rate and for high viscosity ratio, many fingers of invading fluid occur-—rather than the single finger assumed by Dietz. On the other hand, so long as some gravitational influence remains, the flow is not entirely parallel to the bedding planes as assumed by Buckley and Leverett; therefore, both methods fail to give an adequate descrip-