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

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

By L. L. Handy

Solvent floods using slugs of solvent have been found to show continuity in behavior from the vapor pressure of the solvent to the critical pressure for the two-component driving gas-solvent system. In the pressure region between the solvent vapor pressure and the critical pressure for the gas-solvent system, the gar and solvent are only partially miscible. Although complete miscibility cannot be obtained at these pressures, complete oil recovery is possible in principle. In two-phase solvent floods the solvent is propagated tllrough the reservoir, primarily, in the vapor phase. The carrier gas requirements constitute a significant factor in the economics of the process. A qualitative theory is proposed for estimating the amount of dry gas required to move the solvent through the reservoir. The theory shows that for two-phase solvent floods the total gas needed is a minimum at the vapor pressure of the solvent and at the critical pressure for the gas-solvent system, and is a maximum at some intermediate pressure. The predictions of the theory are supported by experimental studies using methane, butane and decane or methane, propane and decane in a natural sandstone core. INTRODUCTION Previously, solvent slug processes have been found effective for oil recovery in two pressure ranges. First, conventional miscible displacements are possible at pressures greater than the critical pressure for the gas-solvent system. Second, Jenks, et al,' have shown that, at pressures slightly in excess of the vapor pressure of the solvent, a solvent slug can be propagated through a reservoir by a gas essentially insoluble in the liquid solvent. The solvent bank displaces the oil ahead of it. Both of these processes, at least ideally, are capable of recovering all of the oil in the swept regions. Slug processes for which the gas and solvent are partially miscible have not been considered; that is, those systems for which the solvent and driving gas form two equilibrium phases in which the vapor phase contains a significant amount of solvent and the liquid phase an appreciable amount of the driving gas. Welge and Johnson' have shown that the gas needed to movc a solvent slug through the reservoir increases with increasing pressure above the vapor pressure of the solvent. It will be shown that solvent slug processes can, theoretically, recover all of the oil at any pressure greater than the vapor pressure of the solvent. But the amount of gas required to move the solvent through the reservoir depends very much on the pressure and temperature. In the present study a maximum in the gas requirements was both predicted theoretically and observed experimentally. This result has not been reported previously, and would not have been predicted from the Welge and Johnson model. The gas requirements are a minimum at the pressures corresponding to the vapor pressure of the solvent and again at the critical pressure for the gas-solvent system, and are a maximum at some intermediate pressure. AN APPROXIMATE THEORY FOR TWO-PHASE SOLVENT FLOODING The differences and similarities between conventional solvent floods and two-phase solvent floods are best understood by referring to concepts developed for miscible displacement in which miscibility is generated in the reservoir. In Fig. 1(A), a ternary diagram is shown for a hypothetical gas-solvent-oil system. To be rigorous the three components should each consist of a single molecular species. The pressure for Fig. 1(A) is greater than the critical pressure for the binary gas-solvent system at the specified temperature. Diagrams of this type are the ones most frequently referred to in discussions of enriched-gas drive and miscible displacement. A limiting tie line is shown tangent to the two-phase envelope and intersecting the gas-solvent line at Point A. To obtain generated miscibility with this type system, others have shown that, for an oil of Composition D, a mixture of gas and solvent must be injected which is richer in solvent than that composition indicated by A. An oil repeatedly contacted with a gas phase richer than A changes toward a composition which would be at equilibrium with the injected mixture, that is, a composition lying on a tie line which passes through the injected-gas composition. Since no such tie line exists, the oil is enriched to the point at which it becomes directly miscible with the injected mixture. At pressures lower than the critical pressure for the gas-solvent system, other types of phase diagrams are observed. The ones of interest in this paper are for pressures greater than the vapor pressure of the solvent, but less than the critical pressure of the gas-solvent system. Such a ternary diagram is shown on Fig. 1(B). In this case, two-phase behavior is observed not only for gas-oil mixtures, but also for certain compositions in the gas-solvent system. If a gas of Composition A (a dew-point vapor) is injected, once again the original oil is enriched by successive

By F. Selig, A. S. Odeh

A second-order approximation to the exact solution of the diffusivity equation corresponding to the pressure build-up of a well producing at a variable rate is derived. This approximation is applicable when the well's shut-in time is larger than the total time elapsed since the well was first produced. The resulting equations are compact in form and easy to use. Thus, the need for Horner's' theoretically precise but rather laborious solution to the above problem is eliminated. In addition, these equations apply where the use of Horner's widely known approximate method is questionable. From a practical point of view, the reported method is best suited for analysis of drill-stem tests and short production tests conducted on new wells. INTRODUCTION The utility of drill-stem and short production tests in reservoir studies has long been recognized by the reservoir engineer. If interpreted correctly they could lead to a wealth of information upon which may depend the success or failure of reservoirs' analyses. Initial reservoir pressure and the average flow capacity are two quantities that are normally sought from a drill-stem and/or a short production test analysis. Pressures are the most valuable and useful data in reservoir engineering. Directly or indirectly, they enter into all phases of reservoir engineering calculations. Therefore, their accurate determination is of utmost importance. The flow capacity kh of the reservoir is indicative of its commercial capability. In addition, it can indicate the presence of a darnaged zone around the wellbore and, thus, the necessity for remedial measures. Of the several methods used to analyze drill-stem and short production tests, Horner's' method is by and large the most common. It applies to an infinite reservoir and or a limited reservoir where the effect of production has not been felt by the boundary. Horner's method makes use of the so-called "point-source" solution of the diffusivity equation. The point-source solution is approximated by a logarithmic function and the superposition theorem is utilized to give the familiar pressure build-up equation where is the shut-in time, q is in reservoir barrels per day and the rest of the symbols conform with AIME nomenclature. Eq. 1 was derived for a well which produced at a constant rate q from time zero to time t and was then shut in. In actuality, such a constant rate of production does not normally obtain. Therefore, a correction must be applied to Eq. 1 to account for the varying rates of production. Horner suggested two methods. The first, which results in a theoretically accurate solution, is rather lengthy and laborious and, thus, it is not suited for routine analysis. The second which has been termed a "good working approximation" is the one used by the majority of the reservoir engineers. In the second method, Eq. 1 is modified by simply introducing a corrected time t, and writing where q is the last established production rate prior to shut-in, and t, is obtained by dividing the total cumulative production by the last established rate. Horner's original paper does not give any indication that this method of correction is based on any theoretical justification. In addition, there is a question as to what constitutes the last established rate. In case of a drill-stem test some engineers use the average rate obtained by dividing the total fluid produced by the total flow time, while others calculate the average rate by dividing the total fluid produced by the last flow-period time. Obviously, different results obtain for the different flow rates used. Because of this, a simple method to the varying-rate case was developed which is theoretically sound and which defines clearly the flow rate and its associated time to be used in the calculations. The final equation arrived at is where q* and t* are a modified rate and time, respectively, and can be easily calculated. In addition, it is shown theoretically that Horner's approximate method, if used for a variable-rate case, gives the correct pressure but would not be expected to give the correct flow capacity. MATHEMATICAL ANALYSIS The general equation governing the flow of slightly compressible fluid in porous media may be written as The elementary solution to Eq. 4, representing an instantaneous withdrawal of Q units volume of fluid at the origin at t = 0, is known as the instantaneous sink

By D. Havlena, A. S. Odeh

The material balance equation used by reservoir engineers is arranged algebraically, resulting in an equation of a straight line. The straight line method of analysis imposes an additional necessary condition that a successful solution of the material balance equatiott should meet. In addition, this algebraic arrangement attaches a dynamic ineuning to the otherwise static material balance equation. The straight line method requires the plotting of one variable group vs mother variable group. The sequence of the plotted points as well as the general shape of the resulting plot is of utmost importance. Therefore, one cannot progrm the method entirely on a digital computer ar is usually done in the routine solution of the material balance equation. If this method is applied, then plotting and anaIysis are asential. Only the appropriate equations and the method of analysis and interpremtion with comments and discussion are presented in this paper. Illustrative field examples for the various cases treated are deferred to a subsequent writing. INTRODUCTION One of the fundamental principles utilized in engineering work is the law of conservation of matter. The application of this principle to hydrocarbon reservoirs for the purpose of quantitative deductions and prediction is termed "the material balance method of reservoir analysis". While the construction of the material balance equation (MBE) and the computations that go with its application are not difficult tasks, the criteria that a successful solution of the MBE should fulfill have always been a problem facing the reservoir engineer. True and complete criteria should embody necessary and dcient conditions. The criteria which the reservoir engineer uses possess a few necessary but no sufficient conditions. Because of this, the answers obtained from the MBB are always open to question. However, the degree of their acceptability should increase with the increase in the number of the necessary conditions that they should satisfy. Generally, the necessary conditions commonly used are (1) an unspecified consistency of the results and (2) the agreement between the MBE results and those determined volumetrically. This second criterion is usually overemphasized. Actually, the volumetrically determined results are based on geological and petrophysical data of unknown accuracy. In addition, the oil-in-place obtained by the MBE is that oil which contributes to the pressure-production history,' while the volumetrically calculated oil-in-place refers to the total oil, part of which may not contribute to said history. Because of this difference, the disagreement between the two answers might be of paramount importance, and the concordance between them should not be overemphasized as the measure of correctness of either one. In this paper, a third necessary condition of mathematical as well as physical significance is discussed. It is not subject to any geological or petrophysical interpretation, and as such, it is probably the most important necessary condition. It consists essentially of rearranging the MBE to result in an equation of a straight line. This straight line method of the MBE solution has invalidated a few long time accepted concepts. For instance, it has always been advocated that if a water drive exists, but one neglects to take it into account in the MBE, the calculated oil-in-place should increase with time. The straight line method shows that in some cases, depending on the size of the neglected aquifer, the calculated oil-in-place might decrease with time. The straight line method requires the plotting of a variable group vs another variable group, with the variable group selection depending on the mechanism of production under which the reservoir is producing. The most important aspect of this method of solution is that it attaches a significance to the sequence of the plotted points, the direction in which they plot, and to the shape of the resulting plot. Thus, a dynamic meaning has been introduced into the picture in arriving at the final answer. Since the emphasis of this method is placed on the interpretation of the sequence of the points and the shape of the plot, one cannot completely automate the whole sequence to obtain "the best value" as normally done in the routine application of the MBE. If one uses the straight line method, then plotting and analysis are musts. The straight line method was first recognized by van Everdingen, et al,2 but for some reason it was never fully exploited. The advantages and the elegance of this method can be more appreciated after a few cases are carefully treated and worked out by it. SOLUTION OF THE MATERIAL BALANCE EQUATION SATURATED RESERVOIRS The MBE for saturated reservoirs written in AIME symbols is

By R. G. Hawthorne

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

By D. J. Pye

Certain high molecular weight synthetic polymers in very dilute solutions decrease water mobility in porous media 5 to 20 times more than would be expected from the solution viscosity. This indicates that increasing water viscosity to reduce adverse mobility ratios in water floods is economically feasible in many situations. Laboratory results are consistent with accepted theory with respect to improved areal sweep, permeability distribution and displacement efficiencies as would be predicted from the solution mobility measured in the core. The laboratory results have been confirmed by field pilot floods which recovered 80 per cent and 100 per cent more oil than comparative water-flood pilots in fields containing 16 cp and 130 cp oil respectively. Economic analyses based on pilot data indicate the method to be profitable. INTRODUCTION The desirability of improving the water-oil mobility ratio in waterflood-ing operations has long been appreciated, especially for high viscosity crudes. Methods of reducing the oil viscosity by application of heat, gas resaturation and miscible drives have met with some success. Conversely, it has been generally conceded that increasing the water viscosity would also be effective, but the amount of thickening agent required to effect an appreciable increase in viscosity is discouragingly high. The use of such materials as glycerin, sugar, or glycols is completely out of the question economically, so that considerable attention has been given to the use of the much more efficient synthetic water-soluble polymers as viscosity improvers; but even these materials are not economically attractive. If increased water viscosity could be economically realized, marked improvements in areal sweep efficiency, as discussed by Caudle and Wittel and others, would be realized in field operations. Similarly, the improved mobility ratio would bring about increased recoveries by correcting permeability distribution problems as discussed by Stiles2 and Dykstra and Parsons,' and improvements in displacement efficiency would be obtained as recognized by Buckley and Leverett4 and Welge.5 LABORATORY STUDIES THE RESISTANCE EFFECT In the course of research to develop a more efficient water-soluble polymer viscosity builder, we discovered that a very few types of water-soluble polymers exhibited a most unusual and very interesting property not previously observed. Among the earliest materials showing such activity were polymers containing acry-lamide. Fig. 1 shows a typical plot of the solution viscosity vs concentration for one of these polymer solutions as determined in an Ostwald viscometer. As usual, such a plot is an approximate straight line on semi-logarithmic paper. Now, an inspection of Darcy's law, shows that a viscosity value can also be determined in a formation sample if the pressure, flow rate and permeability are known. If the core permeability is previously determined with wa- ter or brine, then the viscosity of another aqueous fluid may be determined from the equation. The viscosities so determined for glycerin (glycol or polyvinyl alcohol solutions, for example) agree with the viscometer determinations. However, in the case of the water-soluble polymer solutions in which we are interested, the viscosities measured in the formation sample depart very markedly from the viscometer values. A typical relationship for a polymer solution is also shown in Fig. 1. This unusual departure from the expected response is referred to as a resistance property of the polymer and is quantified as "resistance factor". Since the resistance factor R can best be determined in a core under study, the term has been defined on the basis of the ratio of the brine mobility to the polymer solution mobility at residual oil saturation so that:

Jan 1, 1965

By T. K. Perkins, R. N. Hoffman, O. C. Johnston

A simplified theory of viscous fingering in miscible systems has been developed. It predicts the correct functional relationship between pertinent variables and permits the calculation of the order of magnitude of fingering behavior for simple systems, such as linear or radial. The theory is based on the following four observations: (1) cross-flow takes place only near the ends of fingers; (2) the relative finger width is about 0.5; (3) fingers can be suppressed by transverse dispersion, the suppression being quantitatively described by a critical ratio of dispersion times; and (4) the widths of extending fingers are close to the minimum size finger that can grow at any stage of displacement. Fingering is studied in two-dimensional linear and diverging radial systems, both theoretically and experimentally. For linear systems, the length of the fingered region is proportional to mean displacement; the finger width is proportional to square root of mean displacement; and there is a small initial region void of fingers because of suppression by longitudinal dispersion. For the radial system, two limiting conditions are recognized. If the mean displacement is small compared with the wellbore radius, the length of the fingered region is proportional to the mean displacement. The width is proportional to the square root of mean displacement. If the mean displacement is large compared with wellbore radius, length of the fingered region is proportional to mean displacement, but the number of fingers approaches a constant value. Also, in the radial case there is a small initial region void of fingers because of longitudinal dispersion. INTRODUCTION The behavior of viscous fingers in miscible systems has been of interest to the oil industry for many years. Previous studies have clearly shown the existence and nature of fingers in small models.1,2,3,4 Engineering techniques for extrapolating to reservoir situations have been proposed. 5,6 Still, because of the lack of a real understanding of the mechanics of fingering, there remains uncertainty and disagreement as to the best way to scale or calculate fingering behavior in the reservoir. In this paper we discuss a study of the mechanics of fingering in miscible systems (i.e., why do fingers behave as they do?). A simplified theory is developed which we believe will (1) predict the correct functional relationship between pertinent variables, (2) permit us to calculate the order of magnitude of fingering behavior for simple systems such as linear and radial, and (3) give further insight into the problem of scaling or otherwise extending the results to more complicated reservoir conditions. The paper includes the following sections: (1) brief summary of the mixing behavior of miscible fluids in linear and radial systems; (2) four fundamental observations of fluid flow (under fingering conditions); (3) based on these four observations, a simplified theory of fingering in linear and radial systems is developed; and (4) the theoretical equations are compared with experimental fingering data measured in laboratory models. A REVIEW OF DIFFUSION AND DISPERSION As will be shown later, the behavior of viscous fingers in miscible systems is controlled in large degree by the mixing between the two fluids. A quantitative description of fingers will first require a quantitative description of mixing. Fortunately, much work has been done to clarify this subject; a fairly comprehensive review has been presented in a previous paper. 7 For present purposes let us (1) briefly summarize the quantitative description of diffusion and dispersion (both longitudinal and transverse) within a differential element; (2) present a simplification of the "width of mixed zone"; and (3) describe the effect of geometry on width of mixed zone. If one fluid is displaced from a porous medium by another miscible fluid (in the absence of fingers)

Jan 1, 1966

By R. Wiesenthal

A method is developed for improving the low recovery efficiency which results when viscous oils are flooded by water. Viscous oil has been diluted with a lighter liquid miscible in it in any ratio which produces a mixture possessing a reduced viscosity and which is then displaced by water. Light gasoline was used in these tests to reduce the viscosity of the oil in place. Oils possessing viscosities in the range of 1.28 to 324 cp at the test temperature of 30C were displaced from a cylindrical unconsolidated sand core, with a length of 2 m (6.6 ft) and a diameter of 10 cm (3.9 in.) at a flood of 6.27 cc/sq cm/hr. In two test series up to I PV of light gasoline was injected into the core before water flooding. In the first series the recovery of a low-viscosity oil (1.28 cp) increased as the volume of light gasoline was increased. However, only the oil which could not be recovered by water flooding alone could be replaced by the light gasoline. In the other series, the recovery of a more viscous oil (13.5 cp) was not increased even if large quantities of light gasoline were injected. An oil bank built up ahead of the water front so that the flood water met with equivalent saturation conditions, as would be the case in flooding solely with water. However, the recovery of more viscous oils could be increased substantially when the core was first flooded with water, followed by a slug of light gasoline, and then flooded with water again. Furthermore, an oil bank ,built up ahead of the second water front. The formation of this oil bank is particularly favorable for the displacement process; the greater part of the light gasoline injected was pushed out of the core ahead of the oil and recovered. Using a light oil of low viscosity, it was possible to replace only the residual oil that could not be recovered by water flooding by the light gasoline. As the basic study had shown that highly viscous oils could be recovered by flooding first with water, then with light gasoline, followed by a second injection of water, the study was extended by tests made with reservoir fluids and cores taken from the low-gravity oil reservoir of the Wietze field, Germany. A great increase in recovery over flooding with water alone was obtained in these tests. The results may show a way to recover high-viscosity oils by a comparatively simple flooding procedure. INTRODUCTION In fields where oil recovery has not benefited from natural water drive, increased recovery may be obtained by the injection of water into the reservoir, especially in the case of fields which produce high-gravity, low-viscosity oil. There is a rather close relation between the viscosity of the reservoir fluid and the recovery that can be obtained by water flooding. As the viscosity increases the oil recovery decreases proportionately, so that an oil viscosity about 30 times that of water is generally considered to be the upper limit for an economically successful water flood.' This limitation has been recognized theoretically by Buckley and Leverett,' and has been verified by numerous laboratory flooding tests, such as by Croes and Schwarz,V n the viscosity ratio range of oil to water from 1 to 500. The reason for the inferior recovery of viscous oils by water flooding can best be understood by considering the mechanism of flood tests. In such tests water is introduced into one end of a sand-packed tube which is saturated with oil of the viscosity to be tested, and oil is produced from, the other end. Oil, with a viscosity lower than that of water, is pushed ahead of the advancing water with considerable uniformity, and irregularities in the displacement process tend to be eliminated as the oil saturation decreases and the water saturation increases. However, oil recovery is not complete, for the water forms interfaces with the oil and traps residual oil droplets in certain of the pore spaces. These residual oil droplets are distributed rather uniformly through the sand, and they are not susceptible to recovery regardless of the amount of water passed through the test cylinder. In the case of an oil possessing a viscosity greater than that of water, the injected water tends to move irregularly through the sand in the test cylinder. At the place in the sand where initial water movement takes place, resistance to the movement of water is lowered and the water finger so formed will grow perceptibly; it will extend to the outlet end of the test cylinder so that a breakthrough of water takes place long before much of the oil-saturated volume of the sand has come in contact with water. Once the breakthrough of water takes place, the water-oil ratio is gradually approaching the economic limit, even though much of the oil in the test cylinder remains unaffected by water. Disconnected drops of oil are trapped in the portion of the sand that has been invaded by water, just as when an oil of low viscosity is flooded. The poor recovery of the more viscous oils, therefore, is due to the irregularity of water movement through the test cylinder, and the rate of water movement through restricted sections of the oil-saturated sand increases with the more viscous oils. There are three ways whereby the recovery of viscous oils by water flooding can be improved:' (1) adding materials which will increase the vis-

Jan 1, 1965

By N. T. Cotman, C. H. Pickering, P. B. Crawford

A beat-conduction model has been developed to study the flow of fluids in a stratified oil reservoir which is being subjected to unsteady-state depletion. To simulate stratification, plates of different metals were joined together so that cross flow could occur between strata. A description of the model, plate construction and necessary instruments is included. The data indicate that stratified oil reservoirs experiencing cross flow may be treated as a single, homogeneous, producing sand having properties intermediate between those of the layers making up the system. A technique for averaging core-analysis data to arrive at the proper average value of rock permeability and porosity is presented INTRODUCTION Although oil has been produced from reservoirs for 100 years and reservoir injection processes have been undertaken for 40 years, the industry is still inadequately informed on the problem of reservoir inhomogeneities. Stratification is perhaps the best-known type of reservoir heterogeneity. Urenl discussed the possible significance of this nonuniformity in his 1927 paper on the theoretical aspects of water flooding. Since this time the problem of vertical permeability variations has been the subject of much discussion and numerous papers. Levorsen2 simply describes the origin of layered beds in his Geology of Petroleum. Some beds are field-wide and may be traced from well to well, while others apparently pinch-out and become discontinuous. In addition, the physical characteristics of a layer may vary from place to place. For these reasons, a stratified system is not easily defined. In calculations involving oil reservoirs in which permeabi1ities vary, methods presently used contain many simplifying assumptions. The reservoir frequently is considered as consisting of permeable layers, each of uniform vertical and lateral permeability and each behaving as if separated from adjacent layers by thin impermeable layers. Other assumptions frequently made in reservoir calculations are steady-state flow conditions, unit mobility ratio, linear geometry, average or homogeneous porosity, and negligible capillary-pressure and gravity effects. Muskat3 has. developed solutions to the stratification problem for mobility ratios other than unity for both linear and exponential permeability distributions. He assumes lateral uniformity and continuity of all productive strata and does not consider porosity. This study will take into account transient effects, cross flow and varying porosity, as well as the variations in permeability, by means of a heat analogy. Muskat5 has shown the similarity of the equations which describe the flow of fluids in a porous medium to the flow of heat through a solid. Landrum, et al,6 made an analogy between the flow of fluid in an unsymmetrical reservoir and the conduction of heat in a similarly shaped metal plate. This analogy is summarized briefly as follows. If fluid is produced from an oil reservoir above the bubble point, the resulting pressure distribution will be identical to the temperature which would result in a metal or heat-conducting plate if heat were similarly removed. This means that a study of heat flow in stratified thermal models will behave like fluid flow in stratified oil reservoirs. For mathematical correspondencies, set- Ref. 6. The fluid mobility k/p is proportional to the thermal conductivity, and the porosity-times-compressibility product is proportional to the density times specific heat of the thermal plate. EXPERIMENTAL EQUIPMENT AND PROCEDURE The stratified reservoir model used in this study is illustrated schematically in Fig. 1. A Leeds and Northrup Speedomax Type G temperature recorder

By L. G. Jones

An approximate method of calculation is developed in this paper which allows duplication of radial unsteady-state gas flow computer results where Darcy's law applies, such as those reported by Aronofsky and Jenkins and Bruce, Peaceman, Rachford, and Rice. Moreover, the new calculation method can be used to obtain results for radial unsteady-state gas flow obeying the quadratic flow law proposed by Duwez and Green. Means are discussed for predicting well behavior at single or superimposed flow rates in finite or infinite reservoirs, determining reservoir rock properties from well-test data, reproducing and interpreting back-pressure test data, and determining the radial extent and reserves of gas reservoirs from well-test data. Example calculations are presented for gas flow following both Darcy's law and the quadratic flow law. INTRODUCTION Since the publishing of U. S. Bureau of Mines Monograph 7,1 most gas-well testing methods have been based on the equation q = y (pl2 - pw2)a, where q = production rate, pl and pw, are formation pressure and sandface pressure, respectively, and y and a are constants to be obtained from test data. These methods, used for predicting both deliverability and "open-flow" capacity of gas wells, have been useful and accurate in many cases but unsatisfactory in others. Even at best, however, they do not supply information about the formation or lead to an understanding of nonsteady-state gas flow in porous media. Many theoretically based studies2-4 of gas flow obeying Darcy's law have been made.' Since the partial differential equations which result from combining Darcy's flow law with the continuity equation are nonlinear, most of the published research consists of either numerical solutions or analytical solutions for linear approximating equations. Such solutions have been of limited value in field work due to their unhandy form and their failure to correlate most field data. There is evidence5,6 which indicates that Darcy's law is inadequate to describe gas flow at some flow rates of practical interest. A quadratic flow law: which reduces to Darcy's law at low rates, is more successful in accounting for experimentally observed behavior. This flow law has been applied successfully to a few hypothetical reservoir cases in work which has recently been published.10 However, these numerical solutions of the equations involved have been successful only on a special analog computer. Routine applications to field cases would be awkward and have not been attempted. The present paper describes an approximate method for computing nonsteady-state gas flow solutions which has been completely successful in predicting the results for both Darcy flow and quadratic flow obtained by elaborate numerical methods. The new calculation method allows determination of the observable variables in gas-well testing at constant rates. It is similar to the scheme of using a succession of steady states suggested by Muskat7 in that it makes use of steady-state and material-balance equations. It also is similar to the work of Aronofsky and Jenkins2 in that the new method includes Aronofsky and Jenkins' stabilized flow equation as a special case. It improves upon both of these calculation schemes in that it accurately describes all portions of reservoir history and suggests means of determining reservoir rock properties from well-test data. This paper deals only with production from a reservoir, in which case the rate is defined as being negative. The reservoir studied here is a homogeneous, disk-shaped porous body of uniform thickness, with all boundaries sealed except the inner radial boundary of the well.

By R. A. Greenkorn

This project originated in a practical problem—we needed five tracers that could be used together to locate flow paths in a pilot flood. While tracers for subsurface liquids have been used since the turn of the century,1-15 none of those reported in the literature appeared to be either consistent or quantitative enough for our purposes. Most were used in field systems without controlled experiments to determine the accuracy and precision of analysis, and many were tracers that could not be used collectively. The ideal tracer, of course, would follow the fiuid of interest exactly, traveling at the same velocity as the fluid front. But the ideal is impractical to attain because adsorption-desorption effects cause the tracer to lag behind the front; these effects, plus diffusion-dispersion effects, cause the tracer front to spread more than the fluid front. Thus, our objective was not to locate a tracer that would be ideal for all circumstances but rather, to find one that would approximately follow the fluid, or one that under controlled conditions could be corrected to calculate the movement of the fluid front. We considered tracers satisfactory—(1) if they were easy to analyze; (2) if their breakthrough-elution curves were not too different from those for the chloride ion, a tracer believed to follow the fluid front closely; and (3) if we could calculate from the curves a material balance, at 1.25-PV (pore volume) injected, within 5 per cent of that calculated from chloride curves. Of a possible 35 materials, we selected 13 tracers that could be quickly and easily identified and whose analysis was claimed to be accurate within 5 per cent. Only one of these, tritiated water, was a radioactive tracer. Radioactive tracers are easy to detect even in small quantities, but they require special handling and special equipment. Also, those that can be used together are limited because special equipment is required to separate emissions from the various tracers. Two of the original 13 tracers were eliminated in static tests to determine how accurately and precisely they could be analyzed, and to check on gross adsorption. The remaining 11 were flowed through a 9-ft linear sandstone model, and breakthrough-elution curves were obtained. Finally, three tracers were field-tested as breakthrough tracers. These tests are described in the following sections. The 13 tracers considered in these experiments were EDTA (ethylene diamine tetra acetic acid), fluo-rescein, picric acid, salicylic acid and ammonium, boron (as borate), bromide, dichromate, iodide, nitrate and thiocyanate ions, plus chloride ion and tritiated water. All but the chloride ion and tritiated water were subjected to static sand tests to eliminate the tracers that could not be analyzed quickly and accurately and to eliminate those that showed excessive adsorption. All but two of the tracers, EDTA and salicylic acid, qualified for flow tests on this basis. The procedures used in the static tests (see Appendix A for analytical details) were as follows. Each tracer (in the amount shown in Table 1) was dissolved in 1,000 ml of water. Then, 800 ml of this solution was added to 500 gm of sand, and the remaining 200 ml was reserved as a control standard. The sand-tracer mixture was agitated once a day over a 10-week period, and during this period we analyzed five duplicate samples from this mixture, along with five duplicate samples from the control solution. From the control-sample analyses, we constructed a control chart on which the sand-tracer analysis results were plotted. The control chart, and its use to determine accuracy and precision of analytical results, as well as the amount of adsorption, is described in Appendix B. The results of the static tests are summarized in Table 1, which lists the concentration c (concentration at time t) divided by c, (initial concentration). If there was no adsorption, c/c, = 1. The acceptable deviation from this value varied from tracer to tracer, depending upon the limits of analytical precision established for the different tracers. The best results were obtained with boron, bromide and dichromate, with all values falling within limits of accuracy and precision. Ammonium, iodide, nitrate and picric acid also were satisfactory, although the ammonium results were erratic. Flourescein and thiocyanate were unsatisfactory ini-

By J. E. Warren

To clarify the specific questions raised by J. E. Warren's discussion, the following remarks are offered. 1. It is correct that stratification is assumed in the application of the statistical zonation technique, but this does not constitute a limitation. If, in fact, the reservoir is not stratified the zonation technique will show it. The reservoir is then considered as one zone with a particular distribution of permeabilities. 2. The statistical zonation technique was intended as wells may not be an adequate measure of the lateral permeability changes; e.g., if the data presented in Table A-4 of the subject paper are evaluated on the basis of two crossed classifications with interaction, the variation between zones is significant with a probability greater than .99; however, the variation betwen wells is also significant with a probability greater than .87. It is apparent that the technique suggested by the author can be generalized to provide a powerful method for describing petroleum reservoirs. This generalization can be achieved by treating the core data as a two-way classification to determine the between-zones and the between-wells variations; then, the build-up permeabilities can be analyzed, perhaps on at least-squares basis, to establish the probable distribution of the permeabilities within the zones associated with the individual wells. an additional tool for the reservoir engineer to use together with other techniques and all available information. If pressure build-up permeabilities are available, they should certainly be included. Pressure build-up tests were not available for the reservoir considered in the paper. 3. The proposed correlation of zones between adjacent wells was the best of the techniques that I examined. It is not so much a "measure" as an indication of lateral permeability changes, regardless of whether they are seemingly random or form an obvious trend.

By J. J. Taber, I. S. K. Kamath, R. L. Reed

Alcohol floods of consolidated sandstone cores have shown the process to be strongly dependent on the phase behavior of the particular alcobol-oil-water system used. This means that in many cases the mechanism does not conform to the idea of a piston-like displacement. Instead, it is found that by changing the alcohol it is possible to change the relative velocities of the oil and water and, in fact, the entire mechanism of the process. The effects of rate, viscosity, initial saturation, distance travelled and hysteresis of relative permeability on the alcohol flooding mechanism are discussed. INTRODUCTION Reasons for interest in the use of alcohol to miscibly displace oil and water from a porous medium appear in the existing literature.l-7 The mechanism of the displacement has been considered and the apparent implications formulated into a theory which presumably would enable one to predict the essential features of the process.6 Unfortunately, most of the reported experiments have been performed with unconsolidated or artificially consolidated sands. With these systems some of the noteworthy facets of the process are obscured and resulting data appear uncertain. It is the purpose of this paper to show how the use of consolidated sandstones has led to revision of the mechanism and, hence, the theory of alcohol flooding. The practical result is increased pessimism toward the possibilities of commercial application of the simplest form of the alcohol-slug process. However, elucidation of the mechanism has made it possible to define the essential characteristics of a system of slugs which will behave in a nearly piston-like fashion and, thus, yield the best possible result. EQUILIBRIUM PHASE BEHAVIOR Fig. 1 is a diagram of the ternary system isopropyl alcohol (IPA)-Soltrol-calcium chloride brine. Brine was used to prevent plugging of the core and calcium chloride was used because sodium-chloride brine exhibits a solid phase with Soltrol and IPA. If alcohol is added in increments to the immiscible mixture of water and oil represented by Point A, the path followed by the successively equilibrated samples will be on the Line ABC and pass from the immiscible region to the miscible region by crossing the binodal curve at B. Consider the intersection D of this path with the tie Line EF. The quantity of oleic phase is proportional to the Segment ED and the quantity of the aqueous phase is proportional to DF. Compositions of the two phases are specified by Points E and F. It is clear that in the case shown the oleic phase is diminishing and entirely disappears when miscibility is achieved. Such will

By F. R. Van Horn

WALLACE MCKEEHAN, Douglas, Ariz. (communication to the Secretary*).-In going over this report, which I have done very carefully, I find that the summary as composed deals with the various problems almost entirely from the employer's standpoint and that practically no consideration is given to the employee's point of view. An experience of 3 ½ years as safety inspector, and almost 30 years as workman-and official, has convinced me that any plan for the betterment of working conditions which will achieve any degree of success must give the workmen their share of responsibility and allow them full participation. The company may finance and direct the movement, but its ultimate success depends upon the support given by the workmen. The cooperation of the men doing the actual work is so essential that all writers insist that this must be secured, but they do not agree as to how this can best be done.

Jan 4, 1917

By D. R. McCord

Data of a known statistical quality has been successfully used in a system of conventional fluid mechanics equations essentially free of empirical "conformance factors to prediet early detailed field 03 in a reservoir where gravity drainage is important and oil and rock properties vary con. siderably. The system is useful where comparative recovery predictions for one or more cases of natural depletion and gas cap pressure maintenance are needed and the reservoir is not too heterogeneous. INTRODUCTION Many authors, have examined various facet* of the gravity drainage-gas cap pressure maintenance problem over the last several years. Two of them. Burtchaell and Elkins, have published practical studies. In the case described in the present paper the best development and operating plan for one of the important reservoirs of the Bolivar Coastal Field in Western Venezuela was urgently needed. It was requested that sound comparative economics be presented along with an engineering evaluation of no additional drilling, maximum drilling. several cases of gas pressure maintenance, of water flooding and combination cases. A rapid analysis of the effect on recovery of the range of crude and rock properties indicated that none of the published methods using average properties would yield adequate comparative predictions. This paper covers the predictive system developed to deal with natural depletion and gas cap pressure maintenance. The large amount of data available for this field permitted statistically reliable correlations to be made of most rock and oil properties, and the evident smooth areal variations of these properties encouraged the construction of a lineal mathematical model of material balance and fluid flow equations, which could be manipulated by a modified iterative procedure. The manipulation is much more adaptable to computing machines than manually operated calculators and subsequent studies have been prepared for the IBM Card Program Calculator. A matrix as opposed to linear model has been tentatively worked out by other members of our group to deal with more heterogeneous reservoirs. It is more adaptable to computers such as IBM-701 or the Remington-Rand Univac. In 1950, when the system reported below was developed, no rapid computing machines were available to the author. Their existence and promise encouraged an initial approach by hand. Some details of oil and rock Property correlation are included because there are a surprisingly large number of medium-sized reservoirs in the world that show systematic property variations which are significant for a rigorous analycon. All large reservoirs (over half a billion barrels recovery) with which the author is familiar contain at minimum a significant temperature variation. Using the rock and oil property correlations, past pressure and production, the gas-oil ratio history of the reservoir was computed and found to be in close agreement with field measurements. The future performance under cases of natural depletion, gas injection of 100 MMcf/day, and of 130 MMcf/day, was then computed; a 70 MMcf/day case was also carried far enough to provide data for comparative economics. The 130 MMcf/day case proved to be the most attractive. A 50,000 horsepower gas turbine driven centrifugal compressor plant to take this quantity of adjacent field gas from 30 to 1,950 psia and inject it into the reservoir is under construction. The total cost of the project will exceed 20 million dollars.

Jan 1, 1953

By J. N. Chew, C. A. Connally

A correlation is presented for predicting the viscosity of gas-saturated crude oils under reservoir conditions. It is based on the dead oil viscosity and the solution GOR. The correlation was developed from a study which showed that at a fixed solution GOR, the relation between the gas-saturated oil viscosities and the correrponding dead oil viscosities is a straight line on logarithmic coordinates. Data from 457 crude oil samples from the major producing areas of the U. S., Canada and South America were used. The best straight lines through the data were fitted by the method of least squares with a digital computer. The correlation is presented in the form of an equation and also in a convenient gra-phical form. INTRODUCTION The viscosity of gas-saturated crude oils under reservoir conditions is an important physical property used in reservoir engineering calculations. Many difficulties are inherent in the sampling of gas-saturated oils and in the laboratory measurement of their viscosities. Therefore, it is frequently neither possible nor convenient to obtain measured values, so the viscosity must be estimated from other, more readily available data. This paper presents a correlation for predicting the viscosity of gas-saturated crude oils. The correlation is based on the amount of gas in solution and the viscosity of the dead oil at the temperature of interest. A previous correlation of these variables was presented by Beal. In comparison, the correlation presented here extends to a higher GOR and to a lower dead oil viscosity. Over the range of variables covered by both, this correlation is based on approximately 11 times as many oil samples. DEVELOPMENT OF CORRELATION A study was made of the variables or parameters which could be used to relate the viscosity of a gas-saturated oil to its dead oil viscosity. Preference was given to those variables which are easily and commonly measured in the field or in the laboratory. It was found that either the differential liquid formation volume factor or the solution GOR of the gas-saturated oil could be used as a correlating parameter. The solution GOR was se- lected for the correlation because of its simplicity and availability. In an attempt to refine the correlation by introducing a second parameter, the following additional variables were tested: differential liquid formation volume factor at the reservoir bubble point, the formation volume factor at the saturation pressure for the GOR of interest, reservoir temperature, viscosity of C+ fraction at 100°F and atmospheric pressure, weighted average gravity of gas in solution, oil gravity, reservoir bubble point pressure, saturation pressure at the selected GOR and reservoir temperature, and gas mol fraction. None of these improved the correlation to any significant extent. It was found that at any fixed solution GOR, the relation between the gas-saturated oil viscosities and the corresponding dead oil viscosities is a straight line on logarithmic coordinates, for the range of variables investigated. This straight-line relationship was used for the correlation. Viscosity data on a total of 457 crude oil samples were used. The data were obtained with rolling ball-type viscometers. The samples came from all of the important producing areas of the U. S., as well as from Canada and South America. Their geographical distribution is shown in Table 1. The ranges of properties represented by these samples are as follows: For each crude oil sample the viscosity of the liquid phase from each step of a differential liberation was plotted against its solution GOR on semi-logarithmic paper. A smooth curve was drawn through the points and values of viscosity in centipoises were read from

By C. C. Miller, A. B. Dyes

The cost of finding and developing new reserves is continually rising. We must meet these rising costs with more economical operations. This can he accomplished if we revise our ideas of proper well spacing and well allowable to consider the concept of optimum well spacing. According to this concept, the optimum spacing is the one which leads to the maximum present worth for a reservoir when ail factors affecting total cost and total revenue are considered and when the wells are produced in the most eficient manner. Application of this principle efficiently utilizes available well potential and properly considers the recovery eficiency in addition to fixing the spacing on the basis of the amount and value of the oil to he recovered. This Study presents an analysis of one producing zone containing low gravity crude to illustrate the effect of these factors on the present worth and on the optimum economic spacing under two production drives—-evolved gas and water drive. The maximum present worth occurs when the optimum number of wells for open-flow operation is employed. Frequently, this optimum development cal1s for very wide spacing and the ideal field rates are not unreasonable. Under other circumstance. where proration is necessary, an optimum combination of well spacing and well allowable exists which permits production at relatively high rates. The optimum well density in a field depends on the recovery efficiency and the valule of the oil. In solution gas-driven reservoirs this optimum spacing for operation at high producing rates can vary from extremely wide spacing to handle viscous low gravity oil in thin formations to relatively close spacing in thick sands where good recoveries are expected. Because of the better recovery from water-driven fields, the optimum spacing in these fields is closer than in solution gas-driven fields. Also, the water encroachment pattern is dependent upon the well spacing, and an adequate numher of wells is needed to assure a good sweep eficiency. The economic optimum well density in a water-driven field is high enough for this purpose. INTRODUCTION From year to year domestic oil becomes more difficult to find. The fields we do locate are frequently smaller and deeper than older fields and the costs for men, material and equipment are continually rising. To replenish our reserves we must continue to search for and develop new fields, but experience has shown we cannot expect prices to rise in proportion to costs. Consequently, we must meet increasing costs by more economical operations. A vigorous effort is being made within the industry to improve exploration methods, to cope with the problems of deeper drilling, and to obtain a secondary yield from older fields. Many significant contributions have resulted from these efforts. The economic operation of new fields can be further improved by developing these fields on optimum spacing and producing the wells at higher rates. This would avoid the drilling of unnecessary wells and provide additional capital for seeking new oil. While the benefits of these practices are obvious, the problem of defining the optimum development of a field for natural depletion can become very complex. A study of the effect of well spacing and several reservoir variables on economic worth of a specific field is reported here to illustrate the problem and to show the magnitude of the benefits to be realized. This study is necessarily limited to the field conditions selected and is not intended as a general solution of the well spacing problem. It does, however, indicate factors to be considered, the trends to be expected, and the direction in which we should proceed in developing new fields. NO consideration is given to land and legal considerations which might arise. METHOD OF ANALYSIS The optimum method of developing and producing a field is to use the combination of spacing and prora-tion which gives the maximum return. In addition, we do not want to lose recovery. These considerations of maximum return and maximum recovery present no serious conflict. The value of each method of operation is conveniently expressed in terms of its present worth. According to the present worth method of evaluation,' an acceptable annual percentage return is assigned to the operation, and all incomes, capital costs, and expenses are discounted at this rate to the start of the operation. This net value is the present worth. If we apply the same discount rate to several alternate methods of operation, the one yielding the greatest present worth is the best method. If we express net income in terms of price per barrel of oil produced, drilling and equipment costs on a well basis, and expenses as cost per well year, this evalu-

By W. F. Hower, W. Brown

Large-scale sand consolidation tests were conducted in an effort to determine the reasons for the successes and failures of this method of sand control. Several different consolidating materials were used in treating both clean and bentonitic sands that were packed in a chamber having a capacity of 3.3 cu ft. The results were essentially the same for all of the different consolidating materials, The data show that low-viscosity consolidating materials pumped at a relatively slow rate gave the best results. Where the formation has produced sand, the treating fluids can compress the formation, thus permitting the channeling of fluids to another horizon. Pressure-packing these zones before attempting to consolidate is recommended. Sands containing more than 4 per cent of water-swelling clays are not good candidates for consolidation. It is indicated that loose sand, particularly when it is bentonitic, can be fractured during the placement of the treating fluids. INTRODUCTION Sand production in oil and gas wells has plagued the industry for many years, and numerous cures for this problem have been suggested. Most methods have been successful to a certain degree, but the great variety of well conditions that exist in the different areas has magni- fied the problem and limited the successful use of the various systems. Four review papers1-4 present a wealth of information concerning the degrees of success that have been obtained by the different sand-control methods. The bridging of sand grains by the use of gravel packs and screens has been quite successful. However, these methods do not leave the casing clear for all types of multiple completions, and the cure does not last for the production life of the well in some instance:;. The control of loose sands by sand consolidation with resins has never been as successful as desired. It has always been hoped that such a treatment would eliminate all sand problems for the life of the well, but. initial applications, starting in the middle 1940's, were only moderately successful. Lott, et a1,3 reported a success ratio of approximately 50 per cent and made the following conclusions. The highest percentage of successes were obtained where: a. Consolidation of a zone was made at the time of initial completion or prior to the production of sand. b. The interval treated was less than 12 ft in length. c. Between 30 and 50 gill plastic/ft of producing interval was displaced through the perforations. REASONS FOR SAND CONSOLIDATION FAILURES Our own experiences in the field of sand consolidation point toward the following conditions as the major reasons for the failure of sand consolidation attempts. 1. Mud-plugged perforations and mud invasion of the formation. 2. Sand in the casing covering all or part of the perforations. This sand could be either formation sand or one of the coarser sands used as propping agents in hydraulic fracturing. 3. Holes in the casing. 4. Channels behind the casing. 5. Attempting to treat too long a perforated section. 6. Too high a percentage of water-swelling clays in the formation. 7. Formations that have produced sand. Recent attempts were made to treat perforated sections ranging from 10 to 30 ft, in wells that have produced sand, by using a straddle packer that was raised and lowered through the perforations as the consolidating material was being pumped. In most instances, the pressure required to pump fluid into the formation varied considerably as the tool was raised and lowered. This suggested the possibility that significant differences in permeability were present or that only part of the formation had produced sand. There were times when a sudden break in pressure indicated that a fracture was being formed. Research conducted several years ago concerning the problem of the control of water in air and gas drilling indicated that shale sections could be fractured quite easily. In addition, it was determined that it was easier to pump fluids into shale bodies by fracturing the shale itself, or the interface between the shale and sand, than to pump into a fluid-saturated formation. Formations that produce sand are usually adjacent to shale bodies and frequently have shale streaks of various thicknesses inter-bedded in the sand. Therefore, where shale is exposed to fluid pressure it

By V. J. Sikora, T. S. Hutchinson

This paper presents a method for making a water-rlrive ana1gsis without prior knowledge of aquifer geometry and uniformity using a standard desk calculator. Although it is necessary to know the initial oil in place to use this method, this is a minor limitation to the scope of (he method since in most reservoirs it is possible to obtain reasonably accurate volumetric oil-in-place estimates. An equation is developed which relates pressure at the water-oil contact to the water influx rate as a function of time by a factor called the resistance function. The resistance function introduces the composite effect of the aquifer geometry and flow resistance disrribution. It is pointed out that the characteristic shape of this function makes it possible to start with on approximation of the function and successively improve the approximation until the correct resistance function curve is obtained. In this fashion water-drive estimates can be made without the limitation of assuming simplifed aquifer shapes and flow distributions. This is the novel feature of this development. Methods are given for extrapolating the final curve to calculate future aquifer behavior. Equations are developed for adjusting the pressures and water influx rates where it appears that possible errors in these quantities make it difficult or impossible to obtain a useuble resistance function curve without this adjustment. Application of the pressure build-up analysis techiiique to estimate. some of the aquifer properties is also presented INTRODUCTION A great many oil pools are the result of oil accumulation in some type of trap in an otherwise large and continuous porous stratum. The void space of this stratum outside of the oil pool itself is filled with water or brine. In analyzing performance of the oil pool surrounded by this water aquifer, it is quite necessary. in most cases, to include behavior of the water. When the pressure at any point in a fluid system is lowered, such as by opening a well in an oil sand, fluids in the immediate neighborhood of this point will begin flowing towards this lower pressure sink. As pressure in this area drops due to flow towards the sink, fluids from farther out will start to flow towards the lower pressure. As more and more fluid is removed from the system, the distance from the sink or well within which flow is occurring will continually increase; that is, thc region of disturbance will grow. If some rigid boundary such as a fault is reached by the disturbance, this area will cease to grow; but if some movable boundary is reached, such as a water-oil contact, the area will grow on out into the water, although rate of growth may and almost always does change. The relation between amount of pressure drop and amount of fluid flowing at any point and at any time in an aquifer depends on such factors as compressibility and viscosity of the fluids, the porosity and permeability of the rock, geometry of the whole system and withdrawal rate or pressure drop. With these factors known, it is theoretically possible to calculate the pressure-flow behavior of the system, However, in practice true solutions to the problem are next to impossible due to complexity of reservoir systems. A number of approximate methods of solution have been developed based on various simplifying assumptions. One frequent assumption is that the water-oil contact can be located and equations defining oil reservoir and aquifer solved separately by assigning values to various parameters so as to match past pressure and production history. Usually the properties of the aquifer are not known, since few, if any, wells are drilled through the aquifer; but the water influx and the pressure at the reservoir boundary are known over some time. If it can now be assumed that the aquifer is circular, pie-shaped, or linear, and that it has uniform properties, it is possible to fit a theoretical dimensionless curve to the past aquifer performance history and therefore calculate future water drive. These theoretical curves are available in the literature,1,2,3 Of course, the assumption of uniform aquifer properties is almost always somewhat in error. Fortunately, however, moderate variations from the average have little effect on behavior of the system; and, hence, the best fit of a theoretical curve is frequently satisfactory. In other cases variations in aquifer properties and geometry are large enough that none of the available theoretical curves will give an acceptable fit of the data. In these cases, methods of fitting the data with various electrical analyzers have been developed.1,2' It is the purpose of this paper to present an approach

By L. L. Handy

The development of a theory for miscible liquid displacement requires evaluation of the variables which affect growth of the mixing zone between solvent and displaced oil. Factors which appear to be important are individual fluid viscosities, viscosity ratios, flood rate, fluid densities, flow characteristics of the porous medium and molecular diffusion coefficients of the fluid components. The primary purpose of this paper is to evaluate diffusion effects. Theoretical treatments to date have been limited to floods for which the viscosity ratio is one. Two principal theories have been proposed. Von Rosenberg adapted for porous media a theory derived for capillary tubes by G. Taylor.' ,' In this theory molecular diffusion perpendicular to the direction of flow is a primary factor in sharpening the flood front. Slow floods give sharper fronts for a given distance traveled than fast floods. An alternative theory considers miscible liquid displacement as a statistical problem.3,4,5 Diffusion is not an important factor in this theory, but it leads to the same general type of equation as von Rosenberg's. Both theories predict S-shaped concentration profiles with the same dependence on distance traveled. The statistical or "dispersion" theory predicts rate independence, however. To supplement rate studies a direct measurement of a diffusion effect would be helpful in evaluating which of the two proposed mechanisms best describes miscible liquid displacement for one-to-one viscosity ratio systems. No quantitative theory has been proposed for floods in which a low-viscosity fluid displaces a high-viscosity fluid. It might be anticipated, however, that the extensive fingering observed in floods with adverse viscosity ratios would increase opportunities for an exchange of components between displaced and displacing liquids by a diffusional process. Even if molecular diffusion were not an important mixing mechanism for one-to-one viscosity ratio systems, it could be significant in systems with adverse viscosity ratios. METHOD FOR EVALUATING DIFFUSIONAL MIXING In solvent flooding the displacing liquid can become mixed with the displaced liquid by a number of processes; in general, either mechanical or diffusional in character. To evaluate the contribution of diffusion, a method was sought that would distinguish diffusional mixing from mechanical mixing. Suppose two soluble tracers are added to the displacing liquid and that one tracer has a diffusion coefficient much greater than the other. Then, if diffusion is important during miscible liquid displacement and if diffusional transport is primarily transverse to the direction of flow, substances with high diffusion coefficients will have sharper concentration profiles than those with comparatively low molecular diffusion coefficients. In order to have material balance the profiles for the two tracers must intersect. For one-to-one viscosity ratios the intersection would occur at the 50 per cent concentration point. It is generally assumed that diffusion in the direction of flow is negligible. This assumption appears reasonable because flow velocities are ordinarily orders of magnitude greater than diffusional velocities. If this is not a valid assumption, however, the shapes of the fronts will be further degraded by longitudinal diffusion. The effects of rate and molecular diffusion coefficients are the opposite for longitudinal diffusion to those for lateral diffusion. Higher rates or lower diffusion coefficients would tend to give sharper fronts. The double tracer method offers several unique advantages in evaluating effect of molecular diffusion. First, the method gives a direct measure of the effect of diffusion on mixing zone size. Second, the effect of a difference in diffusion coefficients is determined from a single experiment. Comparison of several experiments at different rates is not required. Third, the method is applicable for any viscosity ratio. Fourth, the method is unaffected by density differences between the two fluids which might result in rate dependent gravity effects. EXPERIMENTAL PROCEDURE Suitable tracers are substances whose diffusion coefficients differ as widely as possible, but whose adsorptions on rock surfaces is negligible. The two tracers selected were methanol and sucrose in water solutions. The approximate diffusion coefficients for methanol and sucrose, as given in the International Critical Tables, are 1.3 X 10-5 and 0.3 X 10-5 m2/sec, respectively. Concentrations in the water solutions were 200 gm/liter of solution for both methanol and sucrose. These high