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By J. W. Whalen

The design, operation and evaluation of an instrumental method for the determination of fluid saturation in porous materials during multiphase flow studies is described. The presence of a magnetic tracer comprising approximately 20 per cent by weight of the aqueous phase is necessary in the utilization of this method. The results of two-phase relative permeability studies on two samples of porous media are presented. The realizable accuracy of the magnetic susceptibility apparatus in fluid saturation determination is estimated to be 2 saturation per cent. The response characteristics of the instrument, together with its simplicity of operation and maintenance are sufficient to recommend its use in routine laboratory studies. INTRODUCTION The problem of fluid saturation determination in laboratory multiphase flow studies characterizing oil recovery processes is one which has been given exhaustive study. Briefly the requirements which are considered essential for a method sewing this purpose are: (1) that the measurement be made external to the porous rock sample under study with no interruption of the fluid flow pattern, (2) the saturation indication must be independent of fluid distribution in the volume scanned by the measuring device, (3) the sensing element should have a small field of definition enabling several independent measurements to be made along the length of a core sample, and (4) the properties of the fluid phases should not be too greatly influenced by required tracer substances. Several methods of saturation determination have found acceptance as meeting most of the above requirements. These include resistivity,1 X-Ray absorption,2,3,4 radiotracers,5,6,7 and neutron8 diffraction The merits of these techniques have been discussed elsewhere in detail and will not be considered here. It may, however. be pointed out that the elaborate instrumentation required by the methods mentioned above, with the exception of the resistivity method (which is undesirable from other standpoints) encourages the development of methods of saturation determination which, while meeting the requirements listed, are simple to operate and maintain. It is the purpose of this paper to describe the instrumentation and evaluation of a technique for the determination of fluid saturation which combines relative simplicity with an accuracy and applicability comparable with methods now in practice. DESCRIPTION OF THE METHOD If the magnetic susceptibility of one of the fluid phases contained within a porous material is sufficiently greater than that of the remaining phases, the fluid content of the material with respect to that phase may be determined by an induction measurement. This may be accomplished by measuring the induced voltage in the secondary of a transformer which contains the porous sample and its interstitial fluids as a portion of the magnetic flux path. Since the fluids used in multiphase flow studies involving petroleum reservoir samples possess uniformly low magnetic susceptibilities, the use of a magnetic tracer in one of the fluid phases is required. In view of the relatively high paramagnetic susceptibility of salts of the transition elements, together with their high water solubility. these compounds become logical choices as tracer materials. Cobaltous chloride, having a specific magnetic susceptibility of 17.5 x 106 cgs electromagnetic units, was used as the tracer material throughout this work, in a concentration of approximately 20 per cent by weight in the aqueous phase. MAGNETIC SUSCEPTIBILITY APPARATUS Basically, the magnetic susceptibility apparatus consists of a transformer serving as the detecting head, a

Jan 1, 1955

By C. H. Stewart, R. A. Fagin

A two-dimensional, three-phasereservoir simulator was programmed for a large memory digital computer. It was designed to provide a practical solution to describing the complex physical relation between the natural forces and the physical properties of a heterogeneous reservoir when subjected to a specific set of conditions. A reservoir study is briefly described to illustrate application of the model. A full volumetric account of three phases (oil, gas and water) is performed simultaneously throughout an integration net representing the reservoir. Absence of one or two of the phases is treated as a special case of the more general situation. Expansion (or contraction) of all phases, including rock expansion, is performed so that the pressure calculation is the general unsteady-state case. To account for the large variations of subsurface elevation encountered in some reseruoirs, and to allow for segregation of the various phases, a gravity head term is included in the basic drive potential. Appropriate fluid and rock properties are used in polynominal surface form (functions of pressure and/or depth) or they can be entered as space variables at each position of the integration net. An unsteady-state water influx calculation, based on the method of van Everdingen and Hurst, l was connected to the boundary of the matrix to simulate aquifers of various sizes. In addition to reservoir calculations, three-phase fluid flow from the producing depth to the wellhead, including provisions for gas lift, was incorporated in the simulator. A workover routine was also built which can automatically switch to a different set of production relations when a gas-oil ratio or water fraction reaches a limit; or it can shut-in the well if prescribed. INTRODUCTION This paper describes a reservoir engineering mathematical simulator used to represent the complex interaction of natural forces and physical properties of a reservoir during natural depletion or with various injection schemes. The simulator, which was programmed for a large memory digital computer, is a two-dimensional calculation which handles three mobile fluid phases simultaneously (oil, gas and water). Basic requisites for the method are individual well production and pressure data, hydrocarbon fluid properties, geological data (producing depth and net sand), capillary pressure data, relative permeability data and permeability and porosity information. Matching the past performance of a combination drive reservoir often has yielded information concerning continuity and the validity of basic data. Detailed predictions of future performance can be made for continuation of current depletion methods (natural depletion) as well as for various types of recovery by gas or water injection. Combination injection cases and pattern studies can also be performed. Workover programs, gas lift and different types of artificial lift programs have been investigated using a technique similar to that described by Kern and Nicholson2 except that conditions of pressure and saturation at the block within which the well is located are used rather than average reservoir conditions. Drilling additional wells to optimize profit was explored, both as to number and location, by placing wells at different spots within the reservoir matrix. Special depletion processes can be examined, such as upstructure drainage and lateral (or strike) waterfloods in thin oil columns. In one case the mathematics of the simulator were modified to calculate the displacement in the vertical plane rather than in the horizontal plane. In this manner specific reservoir problems can be studied, such as coning of gas and/or water around production points, fingering along permeable stringers or, more generally, frontal advances in a heterogeneous section.

Jan 1, 1967

By B. P. Nutter, M. Lebourg, J. A. McAlister

The Multi-Flow Evaluator (MFE) is a new system of tools providing an original approach in drill-stem testing. It improves control during the testing operation and gives a more accurate evaluation of the fluid recovered while providing additional pressure information for reservoir analysis. The tools are operated entirely by up and down motion of the drill stem. This up and down operation provides a positive means of control and offers easily observed surface indications of tool operating position. An unlimited number of shut-in and flow periods may be taken with this tool while in the hole. The key to the success of the MFE system of tools is the safety seal packer. Until this development, operation of the tools by up and down motion only had proven unreliable. The success ratio of the MFE now exceeds the ratio achieved by conventional tools. The equipment includes a 2,750-cc chamber in which a representative sample of the flowing formation fluid is trapped at the end of the last flow period and brought to the surface under pressure. The sample can be evaluated at the wellsite or transferred under pressure for laboratory analysis. The sampling feature of the tool allows a sample to be obtained from a reservoir which has suffered minimum influence from production. A representative sample can be obtained for laboratory or empirical analysis by employing a testing technique to minimize drawdown during flow periods. Interpretation methods to take advantage of this additional information are presented and supported by actual field examples. INTRODUCTlON The Multi-Flow Evaluator (MFE) represents the most recent technical advance in formation testing. Drill-stem testing techniques prior to the MFE have been limited in three areas—(1) operaiion, (2) recovery analysis, and (3) reservoir analysis. The tools in the MFE system allow maximum control during testing and provide test information superior to that obtainable with conventional tools. The system of tools and the operation procedures are described to illustrate the mechanical advantages over present tools. Methods of pressure build-up and fluid

Jan 1, 1966

By Mohamed Mortada

A practical analytical method is developed in this paper which provides the practicing reservoir engineer with a handy method for analyzing oilfield interference problems. The procedure employed entails the principle of superposition of individual eflects. A set of charts has been developed and can be used to evaluate the individual effects. The charts represent non-steady state pressure distribution in an extensive aquifer due to a constant rate of production from an oilfield which can be approximated by circular geometry; they are constructed from solutions to the diffusivity equation considering a circular sink as distinguished from a point sink. The solutions are obtained partially from mathenlatical considerations and partially from analog and digital computations. By virtue of the method of images, the charts can be used also to treat the case of production near a fault. In addition, provisions are made for treating oil fields, which are produced by partial water-drive. Treating an actual field case is deferred to a subyequent writing; however, illustrative example is presented to demonstrate the method of application. The pressure distribution in linear and limited radial aquifers is currently under study. INTRODUCTION With the increasing difficulty of discovering new oil reserves, attention is necessarily focused on the efficient development and production of existing reservoirs. Efficient production depends on the availability of reliable means for predicting the pressure-production behavior of oil fields. The prediction is usually performed by analytical methods or computers. Analytical methods are often limited to systems which can be approximated by simple geometry. However, analytical methods have the distinct advantage of being readily performed without special equipment. Analog and digital computers lend themselves to a variety of problems, but are not always locally available and consequently their use is limited. One particular situation in which analytical techniques are needed involves studies of multiple, water-drive oil fields sharing a common aquifer. Such oil fields are in hydrodynamic communication. Therefore, production from any of the oil fields is accompanied by a pressure decline which is transmitted through the aquifer to the other oil fields and manifested as pressure interference. The rate of propagation of the pressure decline is such that the pressure may be significantly reduced many miles away from a producing pool. An example of pressure interference is found in the Woodbine formation in East Texas. There. a dozen or more pools, including several of major size, draw on the Woodbine formation for water drive. The pressure in the water-bearing portion of the formation has been significantly reduced over an area of several thousand square miles. This reduction of pressure has been such that when the Hawkins pool was discovered in 1940, the pressure in it had already been reduced 280 psi below the initial value in the aquifer. This resulted from prior production from other Woodbine sand pools, including, in particular, the East Texas field some 20 miles to the east of Hawkins.' The above example indicates that the interference effect can be significant and should be taken into con-

Jan 1, 1956

By R. N. Hoffman

A new technique for determining capillary pressure curves has been developed and tested. The technique differs from previously reported centrifuge techniques in that the centrifuge is slowly accelerated from zero to the maximum desired speed rather than being held constant at particular, progressively higher speeds. An important advantage of this technique over other methods of determining capillary pressure curves is the short time required to obtain the desired amount of data over the chosen pressure range. For example, to obtain 30 data points between 1.2 and 104 psig with a 1.55-in. long, 3/4-in. diameter core using a brine-air system, 6.6 hours were required with this technique. An equally important development of this paper is an analytic method for the conversion of the data from the centrifuge experiment to capillary pressure curve data. Previously there has been only an approximate conversion available. Although the capillary pressure curves determined by this technique appear to be as accurate as those determined by other techniques, the accuracy could be improved if certain variables, not treated in this experiment, were investigated. Among these are the dynamic distortion of the centrifuge equipment and imperfect initial saturation of the cores. INTRODUCTION Pirson 1 defines capillary pressure in a porous medium as "the differential pressure that exists between two fluid phases at their interfaces when they are distributed under static equilibrium within a porous material". Capillary pressure in rocks is known to be a function of fluid saturation, among other things, and a capillary pressure curve is defined for the purposes of this paper as a plot of the capillary pressure-wetting phase saturation relationship for a particular rock sample. Several methods are used for determining capillary pressure curves for small rock cores. Prominent among these are the semi-permeable barrier, mercury injection, and, to a lesser extent, centrifuge methods. The semi-permeable barrier method is currently the most popular. It features simplicity in both execution and the mathematical conversion of the experimental data into a capillary pressure curve. The main disadvantages of the semi-permeable barrier method are the time required — as long as two months — to obtain several points for the curve, and a fairly low maximum pressure before breakthrough of the non-wetting phase into the barrier occurs, for example, about 32 psig for a brine-air system. It is for these reasons that other methods such as the centrifuge method have been introduced. High accelerations and the absence of a barrier result in quicker attainment of saturation equilibrium at a given pressure. However, the centrifuge method involves much more expensive equipment and more difficult procedures and calculations than does the barrier method. The purpose of this investigation has been to improve the equipment and procedures of the centrifuge method and to develop an analytic method for the conversion of the experimental data into a capillary pressure curve. Hassler and Brunner did the original work in the determination of capillary pressure using a centrifuge. In their work the centrifuge speed was increased in a step-wise manner, each speed being held constant until saturation equilibrium was reached in each core. Saturation equilibrium was indicated when the volume of liquid collected in the graduated pipette of the core holder remained constant. According to Hassler and Brunner, equilibrium was reached in "a few minutes to one-half hour or more". In the centrifuge method, as opposed to the barrier method, the fluid saturation of the core is not a constant throughout the length of the core, but varies with the radius of centrifugation. Also, the capillary pressure cannot be read directly but must be calculated from a knowledge of the centrifuge speed and other parameters. For these calculations, Hassler and Brunner derived the following equations:

By D. L. Luffel

This paper presents two main developments: (1) a short, constantly accelerated centrifuge method of measuring drainage capillary pressure properties of cores, and (2) a new analytic method of treatment of centrifuge measurements to obtain saturation-capillary pressure relations. Toward the first objective, a clever experimental method is presented which is certainly more rapid than previous centrifuge methods. Toward the second objective, a new analytic method is derived to eliminate the approximate method of Hassler and runner. The derivation of this new analytic method, however, appears to be in error. The new analytic solution is developed as Eq. 5 in Appendix A. At the beginning of this derivation, the average wetting phase saturation of a core is expressed as, where is the centrifuge angular velocity. Eq. D1.2 makes explicit the dependence of S and S upon The second equation shown in Appendix A expresses the relation between the three variables Since for a given porous medium the saturation is uniquely related to capillary pressure, the functional dependence of S upon P is trivial, i.e., Pc.ri does not actually appear. Thus, the function 5 (r, ) may be expressed as a new function, S*, of PC, , alone The author transforms the variable of integration in Eq. Dl through use of Eq. D2 and writes

Jan 1, 1965

By A. M. Khan, I. Fatt

A new type of steady-state thermal conductivity apparatus is presented and described. Results of measurements on Berea sandstone, Solehofen limestone, Bandera sandstone and Bakelite are presented. The measurements were made in a two-hour period at temperatures up to 80C and axial stress to 125 bars. Results show that the thermoelectric heat pump and tbermistor-controlled electric heater can be used in a steady-state apparatus for measuring thermal conductivity of mck. Steady-state is reached within two hours. With Urethane foam insulation, not more than 3 per cent of heat is lost by radial flaw. The apparatus can be used as a comparator or for absolute thermal conductivity determinations. INTRODUCTION Calculations of heat flow through the earth, both for scientific purposes and engineering design, require thermal-conductivity data on rock. However, the high variability of rock often makes handbook thermal-conductivity data of little value. For engineering purposes in particular, it is desirable to have measured data on samples of the rock mass under study. Laboratory methods for measuring thermal conductivity of rock use either time-dependent1-3 or steady - state temperatures .4,5 The time -dependent (transient) techniques give thermal diffusivity from which thermal conductivity can be calculated if the specific heat and density of the sample is known. Most transient techniques require lengthy and involved sample preparation. Also, the accuracy tends to be low. Steady-state techniques are more exact and are the most commonly used. The steady-state methods, however, also have their difficulties. A long time is often required to reach steady state, troubles have been encountered in establishing good thermal contact between the heating and cooling elements and the rock sample, and unaccounted-for heat losses are difficult to prevent. In this paper we present a description of a new type of steady-state thermal conductivity apparatus. With it, measurements have been made in two hours at temperatures up to 80C and axial stress to 125 bars. Sample preparation is simple and no thermal conductivity standard is needed.6 APPARATUS Our steady-state apparatus is an improvement . over the conduction pile used by Zierfuss and van der vliet4 and Somerton.5 A linear heat flux is established through a conduction pile consisting of the rock sample and a standard material of known thermal conductivity in series. The thermal conductivity of the sample is calculated from the ratio of the temperature drop across the sample

Jan 1, 1966

By R. F. Krueger, L. C. Vogel

A design is presented for an electrical analog computer which can solve non-steady state heat transfer problems in an extensive radial forrnation containing a moving cylindrical source. The computer is used to simulate a .simplified thermal oil recovery process in which heat transfer from a moving, constant temperature source takes place radially by conduction only. Temperature distribution curves are .shown for several different assumed modes of travel for the heat source. The data are used to estimate the residual fuel requirements necessary to maintain a self-propagating isotherma1 front for the particular system being studied. Although the computer is designed to represent an unique system. conversion factors can he adjusted to show the effects of changing the assumed values of system constants, such as thermal conductivity of the formation and temperature of the source isotherm. Examples are given to illustrate the relative influence the assumed values of these variab1es will have upon the quantity of fuel required to maintain the source. The maximum effect of heat transfer by the flowing air stream is estimated mathematically, and full requirements for this system are shown to he markedly reduced. Inasmuch as the examples discussrd refer to a simplified process, the results are not directly applicable to a practical field operation. It is believed, however, that the data il1ustr-ate some general trends which are important in thermal recovery processes. Data from laboratory and field experimentation can he used to modify the computer to take into account the influence of heat transfer by various mechanisms, for example, by injected air and by fluids ahead of the hot zone. With .such refinements it would be possible to estimate more accurately the limiting values for the rate of travel of the high temperature front and for the required air- injection rates. Use of this computer should aid in evaluating the economic feasibility of oil recovery by various thermal processes. It is hoped this paper will stimulate further work by others to help accomplish this objective. As the costs of exploration for and development of new oil fields rise. increasing the recovery of oil from established fields becomes more and more important. In fields containing low gravity crude oil, ultimate recoveries by primary means are sometimes less than 10 per cent and often total only 10 to 20 per cent of the oil in place. Conventional secondary recovery methods are not always effective in increasing this total. Methods which have been proposed for increasing the recovery of viscous oils include thermal processes for increasing the mobility of the oil. One method is to provide a moving heat source generated within the formation to heat the oil and a flowing gaseous medium (for example, air) to sweep the oil to the producing wells. The possibility of such a process has intrigued engineers for many years.'. '." but little or no information has been available in the literature relative to the technical or economic feasibility of this method. Recently, however, interest has been revived by publication of laboratory and field test results indicating the feasibility of maintaining

Jan 1, 1956

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