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By C. Scala, F. Bernstein

General expressions are derived relating the streaming potential to the electrochemical potential of a permeable junction separating two electrolytic solutions. By the methods of irreversible thermodynamics, these potentials are related to the flux of mobile species arising from differences in pressure and chemical composition. The flux is conveniently expressed as a transference number, the equivalents of a mobile species transferred through the junction per faraday of electricity. At constant pressure, the flux equation gives the electrochemical potential across the permeable junction. The streaming potential across the junction is obtained by imposing the condition of constant chemical composition. The familiar expression for the borehole SP is shown to be a particular solution of the general flux equation. The derived exprersions were verified in experiments with seven shale samples. Both the theory and experimental data provide additional proof of the presence and magniturle of the streaming potential in shales. INTRODUCTION In the interpretation of electric logs, there has been much concern over the correction to be made to the recorded SP for an electrokinetic or streaming potential originating from the difference in pressure between the mud column and the formation. It was long recognized by M. R. J. Wyllie that such an electrokinetic potential source mas in the mud cake opposite permeable sands.' More recently, M. Gondouin and C. Scala2 showed an additional but opposing electrokinetic source of potential to be in shale beds. Their work was supported by the very extensive and excellent work of Hill and Anderson.3 n the present contribution, the general relationships between the electrochemical potential and the corresponding streaming or electrokinetic potential are examined and clarified. A notable advance towards understanding the relation between the various sources of electrical potential in a borehole has been the application of the now common methods of irreversible thermodynamics to describe the passage of ions and water through charged electrolyte junctions such as in shales.' The advantages of these methods are twofold. Unlike classical thermodynamics, they can be applied to non-equilibrium sys- tems, i.e., systems in which flow takes place, provided the flow processes have reached a steady state. They also permit conventional thermodynamic properties to be related to the steady-state dynamic properties. These methods arc used here specifically to derive expressions relating the potential behavior of shales in terms of ion and water transport. Experimental verification of the expression for the shale streaming potential is given which in turn offers theoretical support to the existence of a streaming potential in shales.' Importantly, the relative contribution of transport parameters to the SP is clarified. The theoretical treatment is essentially that of Lori-mer, Boterenbrood and Hermans3 ut has been extended to derive the general expression for the SP.

By Harold B. Janzen, M. A. Nobles

Campbell and Martin1 described a new exploratory logging tool which was developed as the result of a new concept of the phenomenon that takes place when a porous formation containing mobile hydrocarbons and irreducible interstitial water is invaded by mud filtrate. They postulate that invasion will create a bank of interstitial water between the flushed zone and the undisturbed formation. This postulate was confirmed by Wyllie.' Thus, if R,, is greater than R, and mobile hydrocarbons are present, the resistivity profile with increasing radii from the borehole is high:low:high, as shown in Fig. 1. If mobile hydrocarbons are absent, the bank of interstitial water is indistinguishable from the undisturbed formation and the resistivity profile, when R is greater than R,, is high:low, as shown in Fig. 2. The zone containing the bank of interstitial water and residual oil was named the "low zone". Campbell and Martin conducted extensive electrolytic wedge and resistivity analog model studies to select a logging system that would reflect the presence or absence of the low zone. The system they selected was that of a family of lateral curves. The model studies indicated that the presence of a low zone could be detected in approximately 60 to 70 per cent of the producing sand reservoirs with a family of lateral curves provided that the sonde effectively occupies 90 per cent of the diameter of the hole. These studies were conducted with D, = 5d. The presence or absence of the low zone is best determined by a plot of the apparent resistivity vs spacing as shown in Figs. 3 and 4. A dip or inflection in the profile that repeats at successive depth intervals is considered diagnostic. To reduce the shunting effect of the hole and mud, a sonde of insulating material that is surrounded by a series of flexible rubber bumpers was designed. The bumper efficiency is considered to be the ratio of the bumper diameter to the hole diameter. Field tests conducted by Campbell and Martin on 10 wells with a tool that provided for station readings indicated that dips or inflections could be detected in 19 out of 25 known producing zones. INSTRUMENT DEVELOPMENT Halliburton developed a logging system to simultaneously and continuously record eight lateral curves. The instrumentation is described in the literature.' The original logs made

By C. D. Stahl, S. M. Farouq Ali

This investigation attempts to describe and simulate the alcohol displacement process by means of a cell model, as employed in chemical engineering practice. The proposed model is more simple than previously proposed models. and utilizes parameters chosen on a theoretical basis. The model successfully reproduced the formation of the stabilized bank and the breakthrough of alcohol, the latter depending on one of the model parameters, which may be correlated with the length of the porous medium. Moreover, the effects of the phase behavior of the liquid system involved, as observed in experimental studies, were reproduced. Several variations of the basic model were devised and tested on a digital computer. These included the cases in which: (1) the actual value of fractional flow was used in cell-to-cell computations; (2) the number of cells was varied within the same run; and (3) incomplete rather than complete phase equilibrium was assumed within each cell. The proposed cell model clarifies the basic mechanism of the process. Detailed concentration profiles obtained for each cell, for instance, showed the mechanism of bank formation in relation to the phase behavior characteristics. The results obtained indicated a varying degree of phase equilibrium concommitant with changes in the velocities of the phases in an actual alcohol displacement. This condition was approximated by changing the number of cells during the simulation. Interesting information was obtained on the influence of path length on the efficiency of alcohol displacement, which has been the subject of some controversy. Certain limitations preclude the use of the proposed model as a substitute for experimental studies. The results obtained were, nevertheless, of value in interpreting the experimental results. INTRODUCTION During recent years considerable effort has been directed toward an understanding of alcohol displacement, the process whereby oil and water are recovered from a porous medium by the continuous injection of a solvent. The complex nature of the physical process involved has so far defied a complete mathematical treatment. Other methods of approach, amounting to an overall material balance, have been proposed, yielding useful information on certain aspects of the process.1-3 Taber et al, in particular, defined the displacement mechanism in terms of the phase behavior of the alcohol-oil-brine system invoIved.2 Wachmann reported a mathematical treatment of alcohol displacement subject to certain simplifying assumptions? Donohue proposed the use of a "cell model" for simulating alcohol disPlacement.5 The nature of the assumptions involved limited the utility of the model. The present work attempts to examine the variables involved in the simulation of alcohol displacement and discusses several possible versions of the basic cell model. Under certain conditions the model results are similar to the experimental results. In particular, the spontaneous formation of of the stabilized bank and the effects of the system phase behavior were successfully reproduced. PREVIOUS WORK ON CELL MODELS Cell models and the theoretical plate concept are often used in solving chemical engineering problems in which an explicit mathematical solution may be difficult or impossible to obtain. Examples of such applications occur in distillation, gas-liquid chroma tography,6 reactor technology, absorption, etc. In petroleum engineering, such a model was used by Attra7 to describe non-equilibrium gas drive, and by Higgins and Leigh ton8 to calculate sweep efficiency in water flooding. Aris and Amundsen pointed out the equivalence between the diffusion model and perfectly mixed cells connected in series.9 Deans proposed a three-parameter cell model to simulate two-component

Jan 1, 1966

By A. H. Heim

A method and apparatus for measuring gas porosities of rocks are described. The apparatus can be assembled from commercially available components. In principle, measurements are made by volume substitution at constant pressure. The maximum error is not more than 0.3 porosity per cent. Typical results are given. INTRODUCTION Determining the porosity of rock samples is one of the most important and yet most varied types of measurement in core analysis. Among the many techniques devised are the so-called "gas porosity" methods. An old and well known example is the Washburn-Bunting method.' The U. S. Bureau of Mines2-' described and later improved the apparatus for a now widely used method generally known as the "Boyle's law" method. In the present form of the Washburn-Bunting method,' the volume of air in the pores of a rock sample at atmospheric pressure is extracted and then collected in a graduated burette at atmospheric pressure. The volume of air is read directly as the pore volume of the sample. The absolute error in reading the collected volume of gas is independent of the total volume; thus, the relative error is larger when the volume is small, as it is for rocks of low porosity. In addition, the sample after measurement contains mercury, which limits its use for other analyses. The Bureau of Mines (or Boyle's law) method measures directly the solids volume of a sample from which the pore volume and porosity are derived, using a separate measurement of the bulk volume. Gas at a few atmospheres pressure is introduced into a sample chamber of known volume containing the rock sample. The pressure is accurately measured. Following, the gas is expanded into a burette at 1 atm, and the gas volume is read directly. From the initial pressure p, and the final pressure p2 and volume v,, the initial gas volume v1 is calculated using Boyle's law; that is, p1v1 = p2v2. Volume v, minus the volume of the empty sample chamber is the solids volume of the sample. The accuracy of the method is limited, unless corrections are made, by deviations of the gas from the "ideal" gas-law behavior assumed in the simple form of Boyle's law. The purpose of the present paper is to describe a method for measuring the gas porosity of a rock which avoids many of these difficulties. Gas volumes are measured directly with the same accuracy as the bulk volumes. Pressures of at least an order of magnitude larger than those of previous methods are employed to insure rapid penetration of the gas into the sample. While special equipment may be built to apply the method, the porosimeter may be constructed as well from commercially available components. For simplicity, the apparatus described will be referred to as the "Constant-Pressure gas porosimeter". THE CONSTANT-PRESSURE METHOD Fig. 1 shows schematically the arrangement of components comprising the present Constant-Pressure porosimeter. Briefly, the method is one of volume substitution and may be considered a null measurement. Omitting (for the present) some of the operational details, the method of measurement consists of the following three steps. 1. After evacuation, the volume of the measuring system (a ballast chamber, a manifold, two gauges and their connections) up to the sample chamber is filled with gas to a high pressure (- 1,000 psi). A sample of the gas at this pressure is trapped in one side of a sensitive differential pressure gauge to serve as the reference pressure for subsequent steps. 2. The evacuated sample chamber containing the rock sample is opened to the measuring system. As the gas expands into the chamber, the resulting decrease in pressure unbalances the differential pressure gauge. 3. The pressure is restored by means of a mercury volumetric pump. The volume of mercury injected exactly equals the free or void volume of the sample chamber (volume of empty chamber minus the solids volume of the rock within). From the injected volume and the known empty chamber volume, the solids volume is obtained and the porosity calculated. The pressure and the volume occupied by the gas are the same before and after opening the sample chamber. Expansion and compression of the gas are incidental operations and do not enter into the calculation of porosity. By the pressure balancing or nulling, the free volume of the sample chamber is merely substituted by an equal and measured volume of mercury. Since the measurements are at constant pressure, there are no compressibility corrections necessary for the sample chamber.

By C. S. Matthews, M. J. Fischer

Distortions caused by dip on five-spot sweep pattern were determined by visual experiments on a model composed of parallel plates of glass with holes drilled at the well positions. This model war designed to simulate a reservoir depleted of its primary oil and containing a gas phase. Prior to interference, it was found that the injected fluid took a nearly circular shape about the injection well for the cases studied. However, these circles moved down-dip considerably under the influence of gravity. It was concluded that pattern flooding should be practical for many dipping reservoirs. A quantitative measure is given of how much the well position should be altered to allow for dip. Experimental deternzinarions of injectivity showed this to be about the same in a dipping reservoir as in a horizontal one. INTRODUCTION This investigation was undertaken at the suggestion of the Pacific Coast area, Shell Oil Co., where a pilot water flood in a dipping reservoir was being carried out at the time this paper was in preparation. Prior to the undertaking of this flood there had been considerable speculation as to how gravity would distort a flood pattern in a dipping reservoir. The thought was expressed in some quarters that a five-spot flood would not be applicable to a dipping reservoir because the flood pattern would elongate in the direction of dip and would never form a "seal" around each five-spot. If the flood pattern elongated sufficiently, both the oil and the water might flow largely downdip rather than into the pattern producers. In view of the importance of the California water floods, it was deemed important to determine the effects of gravity on a five-spot pattern in a dipping reservoir. Previous work had also been carried out in this laboratory,1 but only for the single-fluid case. The present investigation extends these results to multifluid cases. Several sets of pictures of flood fronts are presented so that the process can be visualized. From these pictures, curves are obtained which show the correct drilling position for achieving equal times of oil-bank breakthrough. Several injectivity curves are also presented which may aid in prediction of injection rate. ASSUMPTIONS The present results were obtained with a model composed of two parallel plates of glass spaced about 1 mm apart. When the inertial forces are small, as in these experiments, the flow between the plates follows Darcy's law and is analogous to flow through porous media. In using these plates, it is not possible to scale effects of capillarity correctly as regards a transition zone between injected and displaced fluid. Calculations show, however, that effects of such capillarity should not be important in determining the gross flooding pattern in the field. In fact, in this model the capillary forces tend to circularize the pattern to a greater degree than in the field. To preserve a large ratio between the viscous forces and the circularizing capillary forces in the laboratory model, viswus glycerine was used as the flooding liquid and the rate of flooding was very high. The circularizing force is believed negligible in these experiments. In addition to the effect of the surface tension y, which tends to circularize the pattern, another capillary force must be considered. This is the adhesion tension y cos, which acts at right angles to the circularizing term y. If the plates in the model are "wetted" the same all around the periphery, then the adhesion tension,

Jan 1, 1957

By R. E. Collins

The existence of fluid migration across fixed boundaries in oil and gas reservoirs has been known for many years. Several techniques have been developed in the past for estimating The rate of migration across fixed boundaries as an aid in planninng field development and in the valuation of oilfied properties. The principal de/errc7nt to the use of these techniques lies in the rather extensive reservoir and field data required for ca1cula1ion.s of fluid migration. For this reason, a new, simplified procedure has been developed which makes possible the calculation of fluid migration with a minimum of field and resrvoir data. This new technique is based on certain. solutions of the differential equations describing flour in the reservoir which assume, for the portion of the reservoir of interest, that the formation can be approximated by a homogeneous rock of uniforrn thickness, that only a Single mobile fluid phase exists, and {hut fluid production at the well is solely a result of expun.rioti of rile resersoir fluds. The results of the, present work are compiled in a set of curves. These curves can he used to calculate both rule and cummulative fluid migration when the abovc assumptions ore ju.r/ifiril. The only data required for such calcu1ations are the production hisrories of all wells in the field, the permeability and porosity of the reservoir., the, compressibility and viscosity of the fluid. and the shape of the reservoir, INTRODUCTION The migration of fluids across fixed boundaries in oil and gas reservoirs has long been recognized as an engineering and economic problem, but no methods for estimating the extent of migration during the development stage of a reservoir have been published. The investigation reported here has been directed toward providing a simple technique for estimating the extent of such migration and has therefore been restricted to the simple case of a homogeneous reservoir of uniform thickncss containing compressible liquids. It is assumed that only a single mobile fluid phase exists and that fluid production at the wells is solely by expansion of the reservoir fluids. Obviously, few, if any. reservoirs conform to these assumptions during their entire productive life. On the other hand, most reservoirs approximate fluid expansion reservoirs during their initial stages of primary production. Consequently, the present work should be applicable during the field development of most reservoirs, regardless of whether or not they ultimately are gas cap, dissolved-gas, or water-drive reservoirs. The mathematical analysis of this problem has been outlined in the Appendix, while the results of the investigation are presented in the following discussion. DISCUSSION OF CALCULATION PROCEDURE SLNGLE Well, Ini-iimite Reservoir The procedure for calculating fluid migrations is based on solutions of the partial differentia1 equations describing the flow of a homogeneous compressible liquid in a homogeneous reservoir of uniform thickness. This differential equation has the form of the well-known diffusion equation. The equation, as applicable to our problem, is where p is fluid density, is porosity, EL is viscosity, c is compressibility,* and k is permeability; .r, y and t are the space coordinates and time. Derivation of this equation is given in the Appendix. The analytical solution of this equation for a single well of negligible radius producing at a constant rate. q, from a reservoir of infinite areal extent for a time, 1. can he readily obtained. From this solution the rate of flow and also the cumulative flow across a straight line segment of length, 1, at a perpendicular distance. 6. from the well can be computed. One end of the line must be on the perpendicular joining the well to the

By J. M. McDowell

A study has been made to deter~r~irre how the behavior of a water drivee reservoir changes as a function of the permeability of the formation and as a function of the size of the aquifer. The effect of pressure maintenance programs on the rote of nat11rn1 water influx is also studied, (IS well as the influence on the influx of the position of water injection we1ls with reference to the oil-water contact. The results are determined by the use of an electrical reservoir analyzer and are plotted as families of curves of pressure, natural water influx, and water injection vs time or cumulalive production. Two size aquifers are studided. One is limited with a 36-inile radius, and the, other approximates an infinite reservoir. The field remains the same in all respects for the completed. study except that the permeability is changed to agree with that assumed for the aquifer. 1 N TR ODUCTI ON this study was undertaken in -dcr to determine how the pres- sure decline and water influx of a water drive field would change as a function of the rate of fluid withdrawals, as a function of the permeability of the formation, and as a function of the size of the aquifer. The effect of pressure maintenance programs on the rate of natural water influx was also studied as was the position of water injection wells with reference to the oil-water contact. The information has been determined by the use of an electrical reservoir analyzer1, sometimes referred to as a Carter type analyzer. This instrument is an analog type computer which can be set up to represent a water-drive oil field. It can reproduce the past performance of the field in terms of analogous electrical quantities when the proper circuit analog of resistors and condensers has been determined, also, it can determine the future behavior for any method of production which may be selected, or permit the comparison among several different methods. Although most applications of the reservoir analyzer"" have been to thc study of specific field performance, it is intended to use the analyzer here to help understand how various reservoir characteristics effect the final performance of the field and to sec under what conditions water injection will be of benefit and to what extent. Within the oil industry the performance of water-drive reservoirs was first described analytically by LV. Hurst' and M. Muskat2 for certain types of boundary conditions. The solutions of the equations developed involved products of exponential functions and Bessel functions. Later publications by Hurst4 and van Ever-clingen" and the hooks of Muskat6 gave further information on reservoirs with other boundary conditions. They also presented rather extensive curves and tables to help evaluate the analytical expressions involved. A rccent series of publications by Chatas. bascd on the work of Hurst and van Everdingen, gives quite complete charts and tables of data to evaluate the pressure decline or water influx in infinite or limited radial reservoirs. and in linear reservoirs. The curves in all of these publications are given in dimensionless form

Jan 1, 1956

By C. R. Johnson, R. A. Greenkorn, E. G. Woods

A new method of reservoir evaluation called pulse-testins has been developed for describing formation properties between wells. Pulse-testing utilizes a sensitive differential-pressure gauge at a responding well to measure and record the response generated by a series of flow rate changes (pulses) at an adjacent or pulsing well. Since the pulse-test instruments have a sensitivity of about 0.001 psi, pulses of several hours or less in duration will generate a measurable response in most reservoirs. For this reason, many well pairs can be tested in a short period of time with little interference in field operations. Comparison of pulse-test results to conventional testing methods shows that the pulses obey unsteady-state, compressible-flow theory and thus provide a measure of both transmissibility (kh/µ) and storage (Fch). In additian, the method can be used qualitatively to describe communication across faults and between zones, and direction and tnagnitude of fracture trends. INTRODUCTION To obtain maximum usefulness of new reservoir analysis methods as well as to aid the field engineer in understanding local reservoir anomalies, improved methods are needed to describe heterogeneities in specific reservoirs. Optimum well spacing, equipment specifications, well completion design and especially the economics of pressure maintenance and secondary recovery programs depend on the extent and location of the heterogeneities that affect flow behavior in the field. One method for obtaining a measure of areal reservoir heterogeneity is an interference test. In the simplest interference test. constant-rate production or injection is initiated at one well and the effect of this flow is measured as pressure vs time at another well. Thus, during interference tests the flow characteristics of the formation are determined in situ. Despite their obvious usefulness in reservoir delineation. interference tests have not enjoyed frequent usage. Reasons for this have been the weeks or months required in many fields to obtain a pressure response measurable with conventional gauges, and the interruption of routine field operation during the field-wide shut-in normally required during these long times. Also, only average properties of the relatively large volume of the reservoir disturbed during these long times can normally be obtained — variations in properties between individual wells cannot be delineated. To eliminate these drawbacks of conventional tests, a special reservoir description technique called pulse testing' has been developed. Pulse-testing has several advantages that make it more convenient and practical for field use than interference tests. Valid tests are more certain because use of a series of flow disturbances gives rise to a diagnostic pressure response that can more readily be distinguished from unknown trends in reservoir pressure and other "noise." The time required is only a fraction of that for an interference test because a special, high-sensitivity differential pressure gauge can detect a much smaller pressure change. Variations in properties between individual wells can be delineated because of the relatively small volume of the reservoir disturbed during these short times. Finally, wells other than the pulsing and responding wells do not have to be shut in or carefully regulated.

Jan 1, 1967

By J. P. Heller, H. B. Bradley, A. S. Odeh

It is an accepted procedure to represent oil field displacement problems in which the pressure is relatively constant by a potentiometric model. Variations in reservoir permeability thickness product are represented in the model by proportional variations in electrolyte depth. It is shown that significant errors are present wherever these variations are of such magnitude that the consequent differences in the electrolyte depth in the model or any regions therein are appreciable in comparison with their respective horizontal dimensions. To demonstrate the order of magnitude of this error, a hypothetical linear oil field in which permeability varies linearly with distance was treated both analytic.ally and by potentiometric model. The results obtained from the model are compared with the exact solution for various configurations of linear geometry. Deviations in flow times up to 11 per cent were observed. It is confirmed that these discrepancies can be reduced by increasing the areal scale and/or decreasing the vertical scale of the model. It is the purpose of this work to assist in the design of potentiometric models by indicating the relationship between this error and the chosen scale factors. INTRODUCTION The fundamental geometrical criterion which determines whether the pressure distribution in an oil or gas reservoir may be simulated in a conventional potentiometric model is the requirement that all reservoir parameters be constant along any vertical line between its upper and lower boundaries. This situation never obtains exactly; nevertheless, it is usually assumed that there is no permeability stratification within the section of interest, that all wells completely penetrate the sand, that there are no gravitational effects, and that variations in thickness of the sand are small compared to the horizontal extent of the field. In these ways the reservoir problem is reduced to two dimensions in which there is no variation of pressure in the vertical direction within the limits of the sand. Variations in sand thickness and/or horizontal pesmeability may be taken into account as described by Muskat, by considering the fluid conductivity of the reservoir in its horizontal plane to be proportional to the product of permeability and sand thickness. If the variations in sand thickness are small compared to the horizontal dimensions of the reservoir, then it is a good approximation to consider the reservoir problem to be essentially two dimensional. The same approximation, however, is made in the simulation of the assumed two-dimensional reservoir by the electrolytic model. The model, if it is to take into account variations of permeability-thickness product, must of necessity be made in three dimensions, that is, with variable depth. Most potentiometric model studies and theoretical treatments reported in the literature to date2,3,4,5,6,7,8 are those in which either an average permeability was used or a constant permeability was assumed. This is understandable since detailed permeability data are not usually available for the majority of reservoirs. Many investigators have recognized that the failure to incorporate permeability variations within a reservoir in model studies leads to serious error in predictions made

Jan 1, 1957

By L. A. Rapoport

This paper is intended as an aid in the perfornzance and interpretation of experimental studies of multi-phase flow in porous tnedia. The mathenmatical formulation of incompressible, two-phase flow phenomena in a three-dimensional porous medium is presented. The resulting equations which account for the effects of gravitational and capillary forces are used to derive general scaling laws permitting accurate representation of water-oil displacement processes by means of experimental flow models. The significance of the scaling laws is examined in the light of the physical concepts of micro-behavior of fluids in porous media. Formulations of scaling con(litions obtained by application of dimensional analysis are reviewed. The principles of application and the limitations of flow model studies are discussed. INTRODUCTION Most of the problems facing a reservoir engineer involve three-dimensional, two-phase flow systems. influenced by the effects of gravitational and capillary forces. Present computational procedures generally do not permit the evaluation of such systems without introducing greatly simplifying assumptions. The actual reservoir configuration is usually replaced by one or a combination of several "equivalent" linear systems, and the effects of capillarity and gravity are considered to be negligible. While these assumptions may in many instances be justified, there are a number of important practical problems for which the results obtained on the basis of such simplifications remain open to question. An alternate approach to the problems that are not amenable to analytical treatment consists of seeking their solution by experimental laboratory investigations. The main difficulty of such investigations lies in the fact that multiphase flow phenomena subjected to gravity and capillary forces proceed differently at different rates and in systems of different dimensions. Therefore, unless great discrimination is exercised in selecting the conditions and interpreting the results of laboratory tests, such tests may be entirely misleading in regard to field applications. The purpose of this paper is to present a theoretical background for the performance of laboratory tests that are more nearly representative of field behavior. It will be shown that simulation of field behavior can be achieved by conducting laboratory tests under conditions selected in accordance with definite scaling laws. The formulation of conditions required for the scaling of two-phase flow experiments in porous media has been presented in the literature on several occasions.12,3,4 One of these formulations advanced by Leverett1, cannot be considered applicable to most practical cases inasmuch as it does not cover generally the regime of laminar flow. An adequate treatment, based on the consideration of dimensional analysis, was presented by Engelberts and Klinkenberg.2 Their treatment, however, is concerned primarily with the scaling of linear flow systems, and hence somewhat limited in scope. Accordingly, the formulation of more general scaling laws, applicable to three-dimensional systems, appeared to be warranted. It appeared, furthermore. desirable to establish this formulation on the basis of a mathematical treatment, rather than by application of general principles of dimensional analysis, to permit more explicit

Jan 1, 1956

By K. Leutwyler, H. L. Bigelow

Jan 1, 1966

By H. E. Ellis, O. L. Patterson, Granville Dutton

A large-scale water-injection program for the Holt Bryant reservoir of the Delhi field has been established by means of the new automatic multi-pool analyzer. The objectives of this program were to increase reservoir pressure and eliminate a cross-dip gradient of 200 psi. Pressure increases during the first 15 months of operation have allowed for increased oil production rates of 2,600 BID or 25 per cent. To determine the required water injection it was necessary to predict natural water influx ram. Elimination of the pressure gradients required the water-injection program to account for the fluid migration and pressure interaction between segments of the reservoir. An integral part of the ana1ysis included the determination of in-place volumes of oil and gas. The operation of the automatic multi-pool analyzer in determining the water-injection program is described. New features of the analyzer inc1ude an automatic maferial balance analog and a leart-squares computer for determination of unknown reservoir properties. Operational procedures are sufficiently simple to allow direct use by reservoir engineers on a routine basis. The rapidity and accuracy with which problems may be solved permit repeated ana1yses as a check upon reservoir performance. INTRODUCTION This paper describes the use of a new analog com- puter in determining the optimum injection program for a complex reservoir. The Holt-Bryant (Fig. 1) is the major reservoir in the Delhi field of Louisiana. Production is from the Tuscaloosa at subsea depths of 2,970 to 3,286 ft. The sand dips to the south and southwest from three to five degrees. The reservoir is approximately 12 miles long and varies in width from 1/2 to 2 miles. Various calculations have placed the initial volume of oil in-place in the wide range of 275 to 355 million STB. Two small gas caps were initially present in the reservoir. The Holt-Bryant reservoir was discovered in Dec.. 1944, and was fully developed with 188 producing wells by 1948. After development was completed, an east-west pressure gradient began to establish itself. This

Jan 1, 1957

By B. C. Craft, Murray F. Hawkins, John R. Rayne

A reservoir mechanism of sulfur recovery by the Frasch process is presented. Improving the economic, of recovery appears to be largely a well, rather than a reservoir problem. A most important factor is the limitation of the lateral extraction of .sulfur about the wells due to the almost vertical .flow of the injected hot water. Model studies are described which confirin the mechanism of sulfur recovery by presently used methods. Studies on a five-spot system indicate improved econo171ics regarding recovery, recovery rate, therinal efficiency, and well spacing. INTRODUCTION Sulfur mining by the Frasch process is one of the most important methods of sulfur production. Developed by Herman Frasch in 1894, it enjoyed a rapid growth and presently supplies 40 per cent of the world's sulfur and about 80 per cent of the U. S. sulfur." The Frasch process of mining the underground native sulfur deposits involves melting the sulfur in place by introducing super-heated water into the sulfur bearing formations, and producing the sulfur as a liquid. Many problems are encountered in mining these native sulfur deposits, and sulfur production depends chiefly upon the nature of the sulfur deposit and the underground thermal efficiency of the heat transfer system. THE RECOVERY MECHANISM Sulfur is typically found in salt dome cap rock as crystal aggregates occurring within the cavities, seams, and pores of limestone and gypsum formations.' Sulfur-bearing limestone is typically cavernous, vugular, and fractured. Porosity averages about 20 per cent and sulfur saturation averages approximately 25 per cent by weight.1,15 Vhe sulfur bearing formations are very heterogeneous in nature and physical characteristics vary widely within small distances. Total cap rock thickness may vary from 50 to more than 1,000 ft, and net productive thicknesses from a few feet to a few hundred feet. In most cases the upper portion of the limestone formation is barren of sulfur, varying from 5 to 200 ft in thickness.1,15,33 The immediately overlying sediments and the underlying dense anhydrite formation provide permeability barriers to the flow of water from the cap rock section.'" Sulfur wells are drilled with conventional oilfield drilling rigs. Fig. 1 illustrates the casing program of a completed sulfur well. The cap rock sections are drilled with core bits and continuous core logs are recorded. After coring, the cap rock section is underreamed to a diameter of 9 7/8 in. The bottom of the 8-in. casing is welded closed and rests on the bottom of the drill hole, which extends through the cap rock to a few feet above the bottom of the sulfur-bearing formation. The 8-in. pipe contains 5 ft of 3/4-in. perforations on 6-in. centers and 15 ft of 1-in. perforations on 1-ft centers. The 20-ft perforated section is set opposite the lower 20 ft of the sulfur-bearing formation. The 4-in. sulfur pipe is set inside the 8-in. water line on a seat ring at the top of the 5-ft section of 3/4-in. perforations. The mining water, heated to about 330°F under a pressure of 250 psi, is forced down the annulus between the 8-in. and 441-1. pipes and through the 1-in. perforations to the sulfur-bearing formation. Compressed air, at 400 psi pressure, is supplied through a 1-in. air line near the bottom of the well to provide air-lift of the liquid

Jan 1, 1957

By A. Lubinski, C. R. Stewart, K. A. Blenkarn

Storage porosity has been considered one of the important pore geometry characteristics of heterogeneous-porosity limestones. Storage pores are only containers for fluids, in contrast to flow channel pores which both contain fluids and provide continuity for fluid flow. The concept of another geometry characteristic, "poro-striction", is presented as a second pertinent variable in describing limestone pore space. In simple terms, poro-striction is a measure of the flow resistance between storage and flow channel pores. Alternating-flow core-testing theory provides flow relationships which can be used to divide the pore space of heterogeneous-porosity media into flow channel and storage pores and to measure the "porostriction" of the latter. Experimental application of this theory to naturally occurring heterogeneous limestones shows that "porostriction" and the ratio of storage to flow channel pores can be estimated. Porostriction and porosity ratio are microscopic characteristics which should influence oil-recovery efficiencies during certain types of displacement processes. A knowledge of pore geometry should be valuable in designing or selecting the most effective oil-recovery process for heterogeneous limestones containing storage porosity. INTRODUCTION The primary and secondary recovery of oil from limestone (carbonate) reservoirs has been and will continue to be a major source of the world's supply of petroleum. Oil recovery from this type of formation is strongly influenced by the size, shape and arrangement of the pore spaces which hold the formation fluids. Therefore, it is expected that better knowledge of pore geometry would lead to design of more effective recovery processes. The concept of storage and flow channel pores has been presented as an important characteristic of limestone porosity.1 In this concept, storage pores are only containers for fluids, whereas flow channel pores are those pores which both contain and transmit the fluids. The work reported herein involves the concept of an additional pore characteristic called "porostriction". This characteristic can be visualized as the flow resistance between the storage and flow channel pores. This resistance is an important pore geometry characteristic because it determines how easily gas and oil can replenish fluids removed from the flow channel pores. Fatt and co-workers at the U. of California have studied the effect of storage porosity and a factor similar to porostriction on transient-flow response of models of porous media during a pressure build-up or drawdown test.55 These models consisted of porous sandstone cores with artificially simulated storage pores. It is the purpose of this paper to describe tests which permit the storage porosity and porostriction of actual reservoir core samples to be estimated. The tests used for this purpose involved not transient tests but, rather, an alternating-flow technique in which pressure and flow are varied according to a sine-wave relationship. This technique may offer advantages over transient methods. THE POROSTRICTION CONCEPT Before discussing alternating-flow (A.F.) tests in core samples, it is necessary to define the porostriction parameter. For this purpose attention is focused upon an elemental volume v representative of a porous medium having both flow channel pores and restricted-access storage pores. For such an elemental volume, it is considered that the storage porosity is combined to form a single pore and that the access into the storage porosity may be represented by a single capillary tube connecting the single pore to the flow channels of the elemental volume. Let the radius of the capillary tube be designated as Rc and its length as L,. When fluid flows into or out of the pore, the following equation (Poiseuille's law) relates flow rate q to pressure drop p, and the physical characteristics of the tube and the fluid of viscosity jx.

By G. A. Croes, N. Schwarz

The dimensionless groups, to which the variables that govern the displacement of oil from reservoirs by liquids can be combined, are derived. Three types of displacement are considered, viz. cold-water drive, hot-water drive and solvent injection. The derivation of the dimerzsionless groups is carried out by means of the relevant basic equations (in-spectional analysis). The resulting sets of groups are afterwards completed by means of dimensional analysis. The form of the groups is given in such a way that they can be adapted to suit the various boundary conditions that are encountered in practice. The physical meaning of the groups is discussed. They have all been brought together on a chart, from which their mutual relation and their corresporzdence to related dimensionless groups in common use in other engineering sciences can be read off. The limitations of dimensionally scaled model experiments as a useful tool for studying liquid flow in porous media, as occurring in oil reservoirs, are discussed. The use of models to study fluid mechanics has an appeal for everyone endowed with natural curiosity. What active boy has not played with ship and airplane models, or crude models of dam and drainage systems Even in the most advanced technical engineering, such models play a fundamental and indispensable role. "And yet in few departments of the physical sciences is there a wider gap between theory and practice, between scientific knowledge and the state of art, than in the use of models to study hydrodynamic phenomena." Garrett Birkhoff1 INTRODUCTION Laboratory displacement experiments are extensively used to investigate, directly or indirectly, the production behavior of petroleum reservoirs. Such experiments are representative of the reservoirs as a whole, if they are carried out with models that are "properly scaled." The performance of oil reservoirs is governed by the values of a number of variables, which can be combined to dimensionless groups. Two general methods are available for the derivation of these groups. In the first of these, dimensional analysis, combination of the variables is done essentially by trial and error. Its only premise is knowledge of the complete set of relevant variables. In the second method, inspectional analysis, the dimensional homogeneity of the equations describing the behavior of the system to be studied is used. The resulting dimensionless groups can be divided into: (a) independent (variable) groups, such as dimensionless length, time, etc.; (b) dependent groups, giving the dimensionless form of the variables, such as recovery and pressure, that can, at least in principle, be measured during an experiment; (c) similarity groups, which are independent constant groups, such as ratio of length to height of the reservoir and ratio of

Jan 1, 1957

By O. C. Baptist

The Umiat oil field is in Naval Petroleum Reserve No. 4 between the Brooks Range and Arctic Ocean in far-northern Alaska. The Umiat anticline has been tested by 11 wells, six of which produced oil ; however, [lie productive capacity and recoverable reserves of the field are subject to considerable speculation because of unusual reservoir conditions and because several wells appear to have been .seriously damaged during drilling and completion. Oil is produced at depths of 275 to 1,100 ft; the depth to the bottom of the permanently frozen zone varies from about 800 to 1,100 ft, .so that most of the oil reserves are in the permafrost Reservoir pressures are estimated to range from 50 to 350 psi, increasing with depth, and the small amount of gas dissolved in the oil is the major source of energy for production. Laboratory tests were made on cores under simulated permafrost conditions to estimate oil recoverable by solution-gas expansion from low saturation pressures. The cores were also tested for clay content and susceptibility to productivity impaiment by swelling clays and increased water. content if exposed to fresh water. The results indicate that oil can be produced fronz reservoir rocks in the permafrost and that substantial amounts of oil can be produced from depletion-drive reservoirs by a pre.s.r~lrr drop of as little as 100 psi below the saturation pressure. Freezing of formation water reduces oil productivity much more than that due to increased oil viscosity: Failure of we1ls drilled with rtuter-base mud to produce is attributed to freezing of water in the urea immediately surrounding the wellbore. Swelling clays apparently contributed very little to the plugging of the wells. INTRODUCTION Naval Petroleum Reserve No. 4 lies between the Brooks Range and the Arctic Ocean in northern Alaska. The Umiat oil field is located in the southeastern part of the Reserve and is about 180 miles southeast of Point Barrow (the only permanent settlement in the Reserve and the primary supply point for drilling of the wells at Umiat). Eleven wells were drilled for the U. S. Department of the Navy, Office of Naval Petroleum and Oil Shale Reserves, between 1944 and 1953 to test the oil and gas possibilities of the Umiat anticline. Six of these wells produced oil in varying quantities and the best one pumped about 400 B/D.' Estimates of recoverable oil range from 30 to 100 million bbl. The main oil-producing zones are two marine sandstone beds in the Grandstand formation of Cretaceous age: these are referred to as the upper and lower sands. Good oil shows were found throughout the sand settions in the first three wells drilled on the structure, but the highest rate of oil production obtained on any 01 the many tests was about 24 BOPD. These first wells were drilled with conventional rotary methods using water-base mud; later wells were drilled either with cablc tools using brine or rotary tools using oil or oil-base mud. These experiments were successful as is shown by comparing the oil production from Well No. 2 with that from No. 5. These two wells are only 200 ft apart and are located at about the same elevation on the structure. Well No. 2. drilled with a rotary rig using water-base mud, was abandoned as a dry hole after all formation tests were negative. Well No. 5. drilled with cable tools and reamed with a rotary using oil, pumped 400 BOPD which was the maximum capacity of the pump and less than the capacity of the well. These field results indicated that the producing sands were extremely "water sensitive" and it was assumed that the cause of this sensitivity was the presence of swelling clays in the sands. Because of the very unusual reservoir conditions and the difficulties encountered in completing oil wells in the permafrost. the Navy asked the U. S. Bureau of Mines to make laboratory studies under simulated permafrost conditions to assist them in estimating the production potential of the field and the recoverable reserves. These tests were designed to determine the cause of the plugging of wells in the permafrost and to test oil recovery from frozen sand by solution-gas expansion with the oil gas-saturated at very low pressures. EXPERIMENTAL METHODS AND PROCEDURES Samples Analyzed Core samples were analyzed that represent the lower sand in Umiat Well No. 7, the upper sand in No. 3. and both the upper and lower sands in No. 9. These sands should be productive in all of the wells because of their location on the structure. Core samples from

By Walter J. Karplus

By means of a finite difference expansion a fluid flow field in cylindrical coordinates with axial symmetry, is simulated by a network of electrical resistors. A series of DC analog computing units, connected to the network bounrlary corresponding to the free surface effectively adjust the network parameterr to satisfy simultaneously Laplace's equation and the borrndary conditions. The system automatica1ly "re1axes" to the correct boundary shape in a fraction of a second. A detailed application of the method to the water coning problem in oil wells is presented. INTRODUCTION The water coning problem in petroleum production is a prime example of a class of problems known as free surface problems. A free surface arises in fluid flow regions when the shape of the boundary of a single phase flow region is affected in some manner by the pressure or potential gradients within the region. When there exists an interface of two fluids of different densities, the shape of this interface will be determined by the relative velocities of the two fluids and will therefore comprise a free surface. Other free surface problems of interest to petroleum engineers include the gas coning, water injection, and gravity drainage probIcms. In each case it is desired to predict the shape of the boundary of the oil zone as a function of the flow rate, the well penetration and the physical parameters and dimensions of the system. Such problems do not lend themselves to the classical analytical treatments of boundary value problems because in this case the boundary of the potential field and the potential distribution within the field are interdependent and must be calculated simultaneously. A further difficulty arises from geometric considerations. Since the fluid flow is of the radial type in most cases, such effective methods as conformal transformation and the hodograph are not applicable. It is the purpose of this paper to present a method whereby free surface problems of this type may be solved rapidly to the desired order of accuracy by means of a combination of a network of electrical resistors and conventional analog computer units. The water coning problem is treated in some detail. Then the procedure for extending the technique to other free surface problems is indicated. FORMULATION Most commercial oil reservoirs consist of an oil-saturated sand, bounded above by an impermeable strata and below by a water-saturated sand. Gravimetric separation results in a horizontal oil-water interface. When an oil well is drilled so that it partially penetrates the oil zone, and production is initiated at a constant rate, the oil-water interface will assume a new shape as a result of the pressure gradients in the oil zone. It should be emphasized that this is a steady phenomenon. After an initial transient period the water phase will be completely static and only the oil will be in motion. The new steady-state shape of the interface, as shown in Fig. 1, depends upon the depth of penetration, d, the thickness of the oil zone, I, the production rate, q, the hydraulic permeability zone of the sand, k, the viscosity of the oil, µ, and the densities of the oil and water, p.,. and pw.. The problem is to determine this relationship. Darcy's law + = - (P f g p z).....(1) it.

Jan 1, 1957

By Morris Muskat, J. M. McDowell

In both production and refining operations of the oil industry many processes are controlled by the gas-liquid phase relationships of the hydrocarbon mixtures of interest. The quantitative behavior of fractionating or distillation columns and stills depends on the changes in the equilibrium distributions of the hydrocarbon components between the gas and liquid phases as the pressure or temperature conditions may vary along the length of the column or as the gross operating parameters and compositions are varied. In oil producing operations the nature and amount of gas phase developed within an underground reservoir as it is being depleted and its pressure declines also involve the basic equilibrium gas-liquid phase interactions. The influence of surface conditions of temperature and pressure and various separation procedures on the nature and amount of stock tank oil and natural gas recovered from a given well stream is likewise determined by the same basic phase equilibrium characteristics of the hydrocarbon systems. Such equilibrium separation of the gas and liquid phases in a well stream is of importance both in establishing optimum separator conditions for obtaining maximum stock tank oil yields and in the general problem of crude stabilization. The phenomena involved in these problems and their quantitative aspects can, of course, be established in each individual instance by appropriate laboratory experimentation. It has been found, however, that by associating with the individual hydrocarbon components functional characteristics de- scribing their individual gas - liquid phase equilibrium behavior, the phase properties of the composite mixtures can be predicted by mathematical computation. These characteristics have been termed "equilibrium constants", or "equilibrium ratios"," and are defined by the general equation: where K, is the equilibrium ratio for the jth component, y, is the rnol fraction of that component in the gas phase, and xj is the corresponding mol fraction in the coexisting liquid phase. Considering the Kj as representing available and measurable properties of the individual components of a mixture, and as defined by Eq. (l), it is readily shown that in a coexisting gas-liquid system the rnol fraction concentrations of the individual components in the gas phase, y,, and those in the liquid phase, x,, are given by the expressions: where the nj are the corresponding rnol fraction concentrations in the composite system, and ng is the mol fraction of the whole which is in the gas phase. The basic unknown in Eq. (2) is n,: which gives the gross separation between the gas and liquid phases. This may be determined in principle by imposing the requirement that the sums of the mol fractions in both the gas and liquid phases are unity, i.e. by the equations: The solution of the equivalent Eqs. (3) lying between 0 and 1, when inserted into Eqs. (2), provides a complete description of the compositions of the coexisting gas and liquid phases. While the Eqs. (3) are essentially equivalent to polynomial equations, such transformations are of little value when dealing with multicomponent sys-

Jan 1, 1949

By J. M. McDowell, Morris Muskat

In both production and refining operations of the oil industry many processes are controlled by the gas-liquid phase relationships of the hydrocarbon mixtures of interest. The quantitative behavior of fractionating or distillation columns and stills depends on the changes in the equilibrium distributions of the hydrocarbon components between the gas and liquid phases as the pressure or temperature conditions may vary along the length of the column or as the gross operating parameters and compositions are varied. In oil producing operations the nature and amount of gas phase developed within an underground reservoir as it is being depleted and its pressure declines also involve the basic equilibrium gas-liquid phase interactions. The influence of surface conditions of temperature and pressure and various separation procedures on the nature and amount of stock tank oil and natural gas recovered from a given well stream is likewise determined by the same basic phase equilibrium characteristics of the hydrocarbon systems. Such equilibrium separation of the gas and liquid phases in a well stream is of importance both in establishing optimum separator conditions for obtaining maximum stock tank oil yields and in the general problem of crude stabilization. The phenomena involved in these problems and their quantitative aspects can, of course, be established in each individual instance by appropriate laboratory experimentation. It has been found, however, that by associating with the individual hydrocarbon components functional characteristics de- scribing their individual gas - liquid phase equilibrium behavior, the phase properties of the composite mixtures can be predicted by mathematical computation. These characteristics have been termed "equilibrium constants", or "equilibrium ratios"," and are defined by the general equation: where K, is the equilibrium ratio for the jth component, y, is the rnol fraction of that component in the gas phase, and xj is the corresponding mol fraction in the coexisting liquid phase. Considering the Kj as representing available and measurable properties of the individual components of a mixture, and as defined by Eq. (l), it is readily shown that in a coexisting gas-liquid system the rnol fraction concentrations of the individual components in the gas phase, y,, and those in the liquid phase, x,, are given by the expressions: where the nj are the corresponding rnol fraction concentrations in the composite system, and ng is the mol fraction of the whole which is in the gas phase. The basic unknown in Eq. (2) is n,: which gives the gross separation between the gas and liquid phases. This may be determined in principle by imposing the requirement that the sums of the mol fractions in both the gas and liquid phases are unity, i.e. by the equations: The solution of the equivalent Eqs. (3) lying between 0 and 1, when inserted into Eqs. (2), provides a complete description of the compositions of the coexisting gas and liquid phases. While the Eqs. (3) are essentially equivalent to polynomial equations, such transformations are of little value when dealing with multicomponent sys-

Jan 1, 1949

By Harold L. Overton, Leonard B. Lipson

The first paper in this series1 outlined practical methods for applying the theory of steady-state flow of an ideal Bingham plastic liquid through a circular pipe and axially through a stationary concentric annulus to the enginekring analysis of friction loss problems. In the present paper the properties of another useful rheological model, the pseudoplastic generalized Newtonian liquid, are considered. The terminology applied to the rheological models is derived from the following classification of rheological models. The term "generalized Newtonian" is a class designation applied to models which do not have yield points but which exhibit a dependence on shear rate. The term "pseudoplastic" applies to a sub-group of models within this class which exhibit a decrease in "viscosity" with increasing shear rate. The term "power model" refers to a mathematical model proposed for describing the behavior of the pseudoplastic liquid. This model is of interest since certain of the salt saturated, oil emulsion, and "low solids" drilling fluids, inverted emulsion- and oil-base drilling fluids, aqueous gels, gelled crudes and blocking agents employed in hydraulic fracturing operations are of this type. The present treatment considers the equations describing steady-state Poiseuille flow through a circular pipe and a stationary concentric annul us, and Couette flow between concentric rotating cylinders for a particular pseudoplastic model. As in the previous paper' it is demonstrated how these equations can be applied to engineering calculations of friction losses. Graphical procedures for recovering the various flow curves and cxample calculations are included. A method for simplifying turbulent-flow correlations is also proposed. PROPERTIES OF PSEUDOPLASTIC GENERALIZED NEWTONIAN LIQUID In contrast to the Bingham plastic the generalized Newtonian liquid is yield point-free. However, in the flow region the relation between stress and rate of shear is not linear as it is with ordinary Newtonian fluids. If the stress increases with shear rate at a less than linear rate the system is called pseudoplastic. While there is no yield point in this system, the Bingham advocates' coined the term pseudoplastic to distinguish this kind of behavior from Bingham plastic behavior. For the pseudoplastic a plot of flow rate vs friction loss or rpm vs torque, from pipe flow or rotational viscometer measurements, respectively, generally has three qualities: (1) it can be clearly extrapolated toward the origin; (2) there is very little slope at the origin. This accounts for the high apparent viscosities at low mixing and pumping rates; and (3) the flow curve possesses a continuous curvature which gradually diminishes as the shear increases. Rheological measurements on a pseudoplastic liquid can be easily misinterpreted as indicating Bingham plastic behavior. A typical case in point is illustrated in Fig. 1. These data were obtained from measurements on an oil-emulsion mud using a commercially available rotational-type viscometer (Model 35 Fann V-G meter). Applying the "two-point"* method3 gives a plastic viscosity of 17 cp and a yield point of 24 lb/100 ft2; at the same time weighting material settles very easily in this system. This illustrates that it is not possible to distinguish between the pseudoplastic and the Bingham plastic from only two data points. It is possible to determine qualitatively which model is more appropriate by a combination of dynamic and static measurements. Experience indicates that the simplest criterion for the Bingham plastic liquid is that the calculated dynamic yield point should be of the same magnitude as the measured gel strengths. For the oil-emulsion drilling fluid illustrated in Fig. 1, a value of 4 lb/100 ft2 was obtained for gel strengths measured