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By T. K. Perkins, L. R. Kern, R. E. Wyant

Many factors influence the results of hydraulic fracturing to stimulate well productivity. Most of these factors have been studied and their effects discussed in the literature. However, the movement of sand through the crack during fracturing has received little consideration. Exactly where the sand goes in the fracture, how it gets there, and how this might influence results have not been studied, judging from published reports. To get an answer to some of these questions we have studied sand movement through a simulated crack segment in the laboratory. Although the work is preliminary, we have answered some questions about sand movement in a semiquantitative sense. We are reporting our results at this time in the hope of stimulating interest and further study in this important aspect of fracturing. LABORATORY EXPERIMENTS Fig. 1 is a schematic diagram of equipment used in this study. A segment of a vertical crack 0.25-in. wide was formed between two Plexiglas plates, and movement of sand in fluid through the crack was observed under various conditions. Sand and fluid were injected into the crack at constant rates. The top and bottom of the crack were closed so that the fluid flowed horizontally. Because of gravity most of the sand settled to the bottom before moving very far through the crack. If the fluid velocity exceeded a certain critical value, then all the sand injected was washed on through the crack even though it had settled to the bottom. When the fluid velocity was less than this value when sand injection was started, most of the sand which settled in the crack remained in place until enough of the area open to fluid flow was blocked by the settled sand to increase the fluid velocity to the critical value.. At this point all injected sand moved on through the crack even though it settled to the lower portion of the moving fluid section. All sand previously settled remained in place. When fluid velocity was increased at this paint by increasing fluid injection rate (without also increasing sand injection rate), some of the previously settled sand was washed out until enough area was open to flow that the velocity dropped to the critical value. Conversely, when the fluid velocity was decreased by decreasing the fluid injection rate (without changing the rate of sand injection), additional sand would settle until the critical velocity was again attained. In short, the critical velocity is an equilibrium value which the system will automatically seek, if possible. The equilibrium (or critical) velocity was measured under several conditions. The results of these measurements are shown in Pig. 2 where the equilibrium velocity is shown as a function of the amount of sand injected per minute. The points falling in the shaded band shown in Fig. 2 were obtained with water. The higher viscosities were obtained by adding CMC (carboxymethylcellu-lose). Data indicate that the equilibrium velocity is not very sensitive to sand size or transport fluid viscosity, or to sand injection rate (except at very low rates). Points connected by the solid CUrve in Fig. 2 were obtained with water and 20- to 40-mesh steel shot. The equilibrium velocity is apparently fairly sensitive to difference in density between particle and fluid. Only one run was made with a gelled fluid and that was gelled water at zero sand injection rate as shown in Fig. 2. (Measurements of equilibrium velocity at zero sand injection rate were made by filling the crack

By H. T. Kennedy, S. G. Dardaganian, J. L. Clanton

This paper presents equilibrium ratios measured on oil and gas samples taken from the Kelly-Snyder field at the reservoir temperature of 128°F. From these data smoothed curves were constructed for the reservoir fluid up to the bubble-point, 1,874 psia. A material balance calculation, based on the use of equilibrium ratios instead of the usual solubility and shrinkage meayr~rements is made on the field and compared to the con\.entional method. DETERMINATION OF EQUILIBRIUM RATIOS The equilibrium ratios, or K values were obtained by recombining separator samples of oil and gas taken when the reservoir was above its bubble point, and thus represented the reservoir fluid before any change had occurred due to the evolution of gas. The well sampled had a gas-oil ratio of 617 cu ft of high pressure sepalator gas per barrel of separator oil. and a bottom-hole temperature of 128°F. Samples were brought to equilibrium by agitating for 24 hours in a thermostat, maintained at the proper temperature within 0.2ºF, mercury being added when necessary to maintain the pressure constant. Samples of oil and gas were then taken by constant pressure displacement, and analyzed by low temperature frac-tionation. Equilibrium ratios were determined at 128°F and 500, 800, 1,000 and 1,400 psig. At the latter pressure, there were indications that the gas samples were contaminated with small quantities of oil, and only data taken at the first three pressures were included in Fig. I. This figure has been extended to higher and lower pressures by the use of the Fluor charts for 3,000 psi convergence pressure, on which the experimental points fit well. MATERIAL BALANCE EQUATIONS The possible errors involved in conventional material balance equations have been analyzed by Muskat1. Where the errors occur in field measurements, or in obtaining effective averages from variables such as gas-oil ratio and pressure, they will evidently be involved in any attempt to estimate the content of the reservoir. On the other hand, substantial errors may be introduced in conventional material balance calculations by: (1) the neglect of the amount of oil dissolved in gas at high pressures and temperatures in the reservoir, but which condenses under separator conditions; (2) the assumption that the separation of oil and gas in the reservoir may be duplicated by a differential liberation in the

By M. E. Hassan, J. C. Calhoun, R. F. Nielson

An apparatus was constructed for the measurement of inter-facial tensions over a range of temperatures and pressures. This apparatus utilized the pendent drop method, and resembles in construction similar apparatus recently described in the literature and in use in some petroleum research laboratories. The interfacial tensions of benzene, propane, n-pentane, n-hexane. n-octane. and iso-octane against water were measured at temperatures ranging from 26" to 82" C and at pressures ranging from 1 to 204 atm. Values of interfacial tensions for the henzene-water system and their variations with temperature and pressure are generally in good agreement with value? of previous investigations. The data in all cases showed a slight decrease of interfacial tension with pressure at constant temperature in the range studied. The effect of pressure became less as the pressure was increased, with an indication of a reversal of the effect at higher pressures. There was a decrease of interfacial tension with temperature at constant pressure in all cases, as would normally be expected. This rate of decrease became greater the higher the temperature. A general equation is presented for the interfacial tensions as a function of presPurp and temperature over the range studied, and the constants calculated for each system. A definite trend was found in the effect of molecular weight on the interfacial tension at a given temperature and pressure, for the homologous series from propane to n-octane. Data for n-decane from the literature fitted well into this trend. INTRODUCTION It has been recognized for many years that surface forces play an active part in the production history of an oil reservoir as well as in determining the amount of unrecoverable oil. The magnitudes of these forces are governed by the values of the interfacial tensions. These vary greatly with composition, pressure, and temperature. The effect of composition is greater than that of pressure and temperature over the ranges normally encountered. In fact, because of the variety of compounds present in a crude, no attempt has yet been made to correlate crude composition with its surface tension or with its interfacial tension against water. The amount of dissolved methane greatly affects the surface properties of an oil, and there are "surface-active" compounds in crudes which have a much greater effect on boundary tensions than do equal amounts of hydrocarbons. The purpose of the present work was partly to contribute data that would eventually be used in correlations applicable

Jan 1, 1953

By A. S. Odeh

During the last few years, several authors1-6 have advanced mathematical solutions, both exact and approximate, to the pressure behavior of a well producing from a region bounded by a circular discontinuity. The most recent solutions were reported by Carter, and by Bixel and van Poollen. Varter solved the appropriate equations analytically. Bixel and van Poollen reported a numerical solution and provided correlation charts for the interpretation of flow tests; their method uses an overlay technique and is valid for wide ranges of mobility and storage capacity ratios. The mobility ratio is defined as the ratio between k/p of the two zones, and the capacity ratio is defined as the ratio between $c of the two zones. We have developed independently an analytic solution, programmed for the CDC 1604, that agreed with Carter's.'1 The same differential equation, assumptions, boundary, and initial conditions were used. We ran 30 drawdowns and 30 buildups of a unit capacity ratio and of varied mobility ratios. We also varied the radial distances to the discontinuity, and the radii of drainage. We examined the results and obtained correlations that were simple to apply and useful in flow test interpretation. The results and the correlations reported here in graphical form allow the determination of the mobility of the two zones and the radial distance to the discontinuity. They eliminate the need to fit field data to a curve as is required in the overlay technique.6 Thus, the advantages of the present correlations are that they require less effort and they provide a unique answer. They also provide guidelines for selecting the parts of the flow test plot to be used in the interpretations. Moreover, the results obtained from the programmed analytic solution define the time intervals where the approximate solution reported by Hurstl and discussed by Larkin2 are applicable. Normal buildup and drawdown plots resulted in an early straight line, a long transition zone and a late (second) straight line. However, the second straight line will not occur if the drainage radius is not at least 10 times the radial distance to the discontinuity. The mobility of the imer and outer zones can be calculated from the slopes of the first and second straight lines, respectively, by the normal methods of the flow test analysis. We found that the radial distance to the discontinuity may be determined from the abscissa value for the point of intersection of the extensions of the two straight lines. Results similar to some of these were reported by Adams et aL7 The approximate solution compared favorably with the exact one during the first and second straight-line periods. However, during the transition zone, deviation occurred between the two solutions. Discussion of Figures Figs. 1 through 7 summarize the results of this investigation. On Figs. 3 through 7 there appears the equation of each respective plot. In this paper the mobility ratio M is defined as klp2/kzpl, where subscripts 1 and 2 refer respectively to the inner and outer zones, and dimensionless time = t, = 6.32 X In the investigation, M ranged from 0.25 to 50;

Jan 1, 1970

By R. F. Jones, N. E. Truitt, M. Prats

In most published discussions and theories of in situ combustion, the combustion fronts are assumed to be vertical. However, evidence from field tests leaves no doubt that combustion fronts often advance more rapidly along the top than near the bottom of a formation as a result of difference in density between injected air and formation liquids. The approximation proposed in this paper to determine the movement of the resultant tilted combustion surfaces states that the vertical rate of movement of combustion surfaces is proportional to the horizontal oxygen flux. Where self-ignition is possible, the proposed method demands that a secondary combustion surface exist around production wells which produce some oxygen. These secondary combustion surfaces may be formed long before the primary combustion surface can advance from injection to production wells. Heat liberated near production wells at these secondary combustion surfaces can contribute to an early increase in production rate. Results indicate that significant oil recoveries cannot be obtained from the usual flood patterns (five-spots, seven-spots, etc.) without producing large volumes of unused oxygen. Ideally, to increase oxygen- consumption efficiency, well patterns should allow oil production from a first line of production wells and gas production from more distant lines of producers. However, it may be desirable to produce some gas at all wells to support (and benefit from) active secondary combustion surfaces. Results indicate that the well spacing through which combustion can be advanced is larger than that predicted by other methods. A large number of production wells may still be desirable to take quick advantage of gravity drainage. From a comparison with results at South Belridge field, California, it appears that this method adequately describes oxygen concentration and temperature histories and combustion-front shapes. However, this method does not accurately locate the most advanced point of the combustion surface. There is some field evidence to substantiate the actual presence of secondary combustion surfaces at South Belridge. Use of the proposed method appears warranted at this time when lay-over of the combustion surface can be anticipated. INTRODUCTION The assumption of vertical combustion fronts has been embodied in all previous publicationsl-6 which use the movement of combustion fronts away from injection wells to determine the temperature and fluid distributions in the reservoir. The only paper concerned with a mathematical model of the combustion process in which a nonvertical combustion front is used was written by Gottfried.11 Actually, nonvertical combustion fronts have been observed in most in situ combustion field tests for which adequate data are available.7, 8, 6 In practice, the typical vertical extent of the burned zone decreases with distance from the injection well, and this burned zone is at or near the top of the sand body. In some cases, such as at South Belridge field near Taft, Calif., the combustion surface is almost horizontal over a very large area.8 Thus, for some years an obvious and serious gap has existed between theory (vertical fronts) and practice (tilted fronts). This is indicated in Fig. 1. Tilted combustion fronts such as observed at South Belridge8 sometimes result from the natural tendency of injected gases to rise to the top of an oil sand. An initially present gas cap, even if thin, would make apparent the effect of gravity segregation from the first, particularly since formations considered for in situ combustion field operations are generally selected for their

Jan 1, 1969

By L. F. Elkins

In 1946 Littlefield, Gray and Godbold published a thorough geologic description of the West Edmond Hunton Lime reservoir, located in Central Oklahoma, and discussion of its early performance.l They stressed implications of the nature of the reservoir rock on performance to be expected with production both by natural means and by pressure maintenance with gas injection. The essence of their analysis was that presence of an extensive interconnected system of fractures and solution channels constituting about 10 percent of the total reservoir void space in otherwise very low permeability reservoir rock would result in severe channeling of naturally encroaching water or injected gas through the fractures with little or no benefit to ultimate recovery of oil. Considerations were in progress to unitize the reservoir for pressure maintenance by gas injection. Some other engineers involved in the project had considerably different opinions at the time regarding continuity of fractures and regarding effects of such fractures on performance. The field was unitized Oct. 1, 1947. Subsequently, pilot gas injection tests were conducted in four areas of the field; large scale waterflooding was conducted in two parts of the field in the Bois d'Arc or Upper Hunton; and pilot waterflooding was conducted in the Chimney Hill or Lower Hunton. Full scale water-flooding is now in progress in the Chimney Hill section where it is oil productive. The primary purpose of this paper is to divine some of the secrets of this reservoir through analysis of all aspects of its performance, with the hope that it will be of help to engineers and operators in developing and operating other low permeability fractured reservoirs. In part, this considers variance of actual performance from some widely used conventional reservoir engineering assumptions. For brevity, other aspects of unit operation and its benefits to the owners are not discussed. Oil and Gas in Place For unit participation, net pay in the Bois #Arc defined by the gross SP anomaly on electric logs totalled 2,100,000 acre-ft in the 30,000-acre reservoir. Development of West Edmond predated modem logging techniques that now would be used to determine "net" or "productive" pay. In Fig. 1 this log pick is compared with detailed core analysis and core descripfion by Littlefield et a1.l for one well identified as A on the index map in Fig. 2. Not all of the gross Bois #Arc section had oil staining, but all cores with measureable intergranular permeability were stained. (The committee selection of the Bois d'Arc SP interval also includes the Frisco section as identified from cores by Littlefield.) The Bois d'Arc is oil productive throughout the entire 600-ft monoclinal accumulation from the eastern upstructure erosional pinchout to the west-side oil-water contact. (See Littlefield et al.l and McGee and Jenkins for geologic detail.)

Jan 1, 1970

By E. L. Anders

The Mile Six pool is located on the La Brea-Parinas Cullcession of International Petroleum Co., Ltd., in northwestern Peru on the west coast of South America. The reservoir pressure in this pool has been maintained within 200 psi of its initial value throughout its history, and gravity drainage has played an important role in the production behavior. It has now produced 95 per cent of its estimated ultimate recovery. It is estimated that this interesting oil pool will ultimately produce 67 per cent of the initial oil in place and that the resulting residual oil saturation may be as low as 19 per cent of the pore volume (29 per cent of the hydrocarbon pore volume). An evaluation of reservoir rock and fluid characteristics and ultimate oil recovery is presented. INTRODUCTION This study of Mile Six pool was made to evaluate its performance according to latest available information. The production performance of this pool has been discussed in various articles in the past. and the reported behavior has been used as an example for application of computation procedures for gravity drainage depletion' and as an illustration of field behavior under gravity drainage or expanding gas cap drive. There have been wide variations in reported values of initial oil in place, reservoir oil volume factor. connate-water sauration, volume of effective sand, and ultimate recovery because of the paucity of reliable basic data. These various factors have been determined as accurately as practicable with the latest available information, and this evaluation is presented herein. The production history of Mile Six is an excellent example of gravity drainage depleion with effective pressure maintenance by gar injection. GENERAL Mile Six pool was discovered by cable-tool drilling in November. 1927. when well 1996 was completed in the Parinas sand. After slow development with cable tools and sporadic production. the pool was opened to continuous pro(iuction in November. 1933. and develpment was completed with rotary rigs. Pressure maintenance was started in December, 1933. by returning gas to upstructure wells. Most of the development was 'completed by 1937, but some additional wells were drilled in the period 1939-1947. and several old wells were deepened. A total of 46 oil and gas wells and 4 dry holes were drilled on approximately 7-acre spacing. Of the producers. 21 are now flowing. 2 are pumping. 44 are gas input wells. 3 are abandoned. I is a gas well shut in. and 15 are shut in because of non-commercial production or high gas-oil ratio. The locations of all wells are shown on the map of Fig. I. Total oil production on Dec. 31. 1952, was 30,867,373 bbl: cumulative gas production was 22,023,777 Mcf; and 26,410,946 Mcf of gas had been returned to the reservoir. These figures do not include oil and gas lost ill a blowout in January. 1940. GEOLOGICAL DESCRIPTION Mile Six pool is located on the northern end of a structural spur projecting from the La Brea-Negritos uplift.' The spur is probably a reflection of a basement structure. It plunges gently to the north, i broken into a complex series of fault blocks. and contain.; the Verdun Alto. Section Sixteen. and Mile Six pools. The Parinas handstone (lower Eocene!. which is the producing formation in Mile Six. occurs at an average depth of 2,200 ft in the pool and dips north and east at from 15' to 20". The pool covers an area of approximately 350 acres. Mile Six is downfaulted about 600 ft from Section Sixteen pool to the soutb. and a major fault forms its western boundarv. The north and east boundaries are formed by the intersection of the sand top with the water-oil contact which occurs at approximately 2.440 ft subsea. An original gas-oil contact probably existed at about 1.875 ft subsea. Fig. 1 presents the latest structural interpretation of the pool. and Fig. 2 is an isopach map showing thickness of the total Parinas formation above the original water-oil contact. The heavy lints of Fig. 1 are contours on the sand top, and the fine lines are contours on the fault planes. This type of straight-line structural map was developed 1)) International's geologists to reflect structural conditions where the bedding planes dip and have no curvature. 'The La Brea-Parinas Concession is highly faulted by normal fault.. The beds are flat wherever exposed. The Parinas formation is approximately 635 ft thick. and it is etimated that 62 percent II the formation is effective sand. The original oil zone was about 565 ft thick. Fig. 211 presents an electric log showing typical Parinas sand development ill Mile Six pool. The Parinas band in Mile Six i. a well-sorted. medium- to-coal-se-grained. cross-bedded sand with minor lenses of shale and small lenses and pockets of pebble conglomerate. The sand grains are subangular to rounded and consist chiefly of quartz with feldspars. biotite hornblende, and augite as accessory minerals. Because of faulting of the Parinas- formation to the east and north of the pool. there is probably l possibility of a significant, natural water drive in Mile six. The faults within the pool. as indicated in Figs. 1 and 2. are of smaller disI,laceInent and seem to act only , partial barriers to fluid movement within the reservoir. RESERVOIR CHARACTERISTICS Core analysis data are available from five wells. The data were obtained from three wells (Nos. 3401. 3586 and 3719) at the time of their completion and from well 1996 when the original liner was sidetracked and the well was deepened ill 1946. Data from well 2779 were obtained in 1943 from old cores taken when the well was deepened in 1934. From these core analyses. the average porosity was estimated to be 22.6 per cent. and the average permeability to dry ail. was estimated to be 780 and Measured productility indices varied from 3.1 to 71.4 B/D per psi differential. Specific productivity indices varied approximately from 0.1 to 0.3 B./D per psi per ft of sand.

Jan 1, 1953

By H. S. Price, J. C. Cavendish, R. S. Varga

A numerical formulation of bigh-order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. INTRODUCTION The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusionconvection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation:

Jan 1, 1969

By R. W. Snyder, W. H. Guilinger, B. S. Gottfried

Two numerical methods are presented JOT solving the equations Jor one-dimensional, multiphase /low in porous media. The case oF variable physical properties is included in the Formulation, although gravity and capillarity are ignored. Both methods are analyzed mathematically, resulting in upper and lower bounds for the ratio of time step to mesh spacing. The methods are applied to two- and three-phase waterflooding problems in laboratory-size cores, and resulting saturation and pressure distributions and production histories are presented graphically. Results oJ the tuo-phase flow problem are in agreement with the predictions of the Buckley-Leuerett theory. Several three-phase flow problems are presented which consider variations in the water injection rate and changes in the initial oil-and ujater-saturation distributions. The results are different physically from the two- phase case; however, it is shown that the Buckley-Lez~erett theory can accurately predict fluid interlace velocities and displacing-fluid frontal snturations for three-phase flow, providing the correct assumptions are made. The above solutions are used as a basis JOT et~aluuting the numerical methods with respect to machine time requirements and allowable time step for a fixed mesh spacing. INTRODUCTION Considerable progress has been made in recent years in obtaining numerical solutions of the equations for two-phase flow in porous media. Douglas, Blair and Wagner2and McEwenll present different methods for solving the one-dimensional case for incompressible fluids with capillarity (the former using finite differences, the latter with an approach based upon characteristics). Fayers and Sheldon4 and Hovanesian and Fayers8 have extended these studies to include the effects of gravity. West, Garvin and Sheldon, l4 in a pioneer paper, treat linear and radial systems with both capillarity and gravity and they also include the effects of compressibility. Douglas, Peaceman and Rachford3 consider two-dimensional, two-phase, in compressible flow with gravity and capillarity and Blair and Peaceman1 have extended this method to allow for compressible fluids. No one, however, has examined the case of three-phase flow, even for the relatively simple case of one-dimensional flow of incompressible fluids in the absence of gravity and capillarity. In obtaining a numerical technique for simulating forward in situ combustion laboratory experiments, Gottfried5 has developed a method for solving the one-dimensional, compressible flow equations with any number of flowing phases. Gravity and capillarity are not included in the formulation. The method has been used successfully, however, for two- and three-phase problems in a variable-temperature field with sources and sinks. This paper examines the algorithm of Gottfried more critically. Two numerical methods are presented for solving the one-dimensional, multi-phase flow equations with variable physical properties. Both methods are analyzed mathematically, and are used to simulate two- and three-phase waterflooding problems. The numerical solutions are then taken as a basis for comparing the utility of the methods.

Jan 1, 1967

By J. C. Martin

Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators in which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the aduantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir. INTRODUCTION In recent years, computers have been used to calculate the performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The situation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array. A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; and, for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations. Most reservoir performance calculations currently are based on one- or two-dimensional flow re1ations.l-3 In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation.l.2. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased. For this analysis, the ratio of the reservoir's thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoir. This requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate. The results presented here provide a firm theoretical foundation to Coats' et aL5 assumption of vertical equilibrium and extend the relations to three-phase flow. Coats' assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations. DISCUSSION SATURATION AND PRESSURE DISTRIBUTIONS Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate: (1) that the fluids along any line perpendicular to such a reservoir's upper surface are in static capillary equilibrium (vertical equilibrium); (2) that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; and (3) that hydrostatic pressure gradients exist along any line perpendicular to the reservoir's upper surface. The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together.

Jan 1, 1969

By H. J. de Haan, L. Schenk

Scope for Thermal Recovery in Shell's Heavy Oil Fields in Venezuela The main heavy oil reservoirs on the East coast of Lake Maracaibo (Fig. I), known as "Bolivar Coast", initially contained some 20 billion bbl of oil in place with gravities in the range of 10 to 15° API. The current total production rate is about 400,000 B/D. Since the recovery to date is on an average only 12.5 percent of the initial oil in place, these fields offer a vast potential for secondary recovery methods. The reservoirs are characterized by moderate depth (generally 1,000 to 3,000 ft; maximum 5,000 ft), good formation properties such as net oil sand thickness (50 to 300 ft), high porosity (30 to 40 percent) permeability (1 to 3 darcies) and oil saturation (initially about 80 percent) and high oil viscosity (100 to 10,000 cp in situ), all of which are favorable for the application of thermal recovery processes. Since 1957, steam drive, steam "soak" and underground combustion have been tested or are being tested in this area.1-3 This paper deals with a major steam drive test in the Tia Juana field. The test was commenced in Sept., 1961, on the basis of encouraging results of laboratory experiments and pilot field tests, which will be discussed briefly. Early Laboratory Investigation of the Steam Drive Process In 1956 a series of model experiments was carried out in the Koninklijke/Shell Laboratorium, Amster- dam, to investigate the displacement of heavy oil by steam.' The prototype studied was a horizontal, un-consolidated heavy oil reservoir, subjected to a linear steam drive. The model, representing a slab from this reservoir along the main flow direction, consisted of a steel tube filled with oil-saturated sand, provided with layers of rock at the top and bottom to simulate conductive heat losses to cap and base rock. Lateral heat losses, perpendicular to the main flow direction, were minimized by insulating the sides of the model. Heat flow and viscosity and gravity forces were scaled by taking the steam injection rate and sand permeability inversely proportional to the geometric reduction factor of the model, and using materials (oil, steam, cap rock, etc.) identical with those of the prototype. The main feature of the process was a frontal displacement mechanism that was observed within wide ranges of injection pressure and rate, oil viscosity, initial oil saturation and sand permeability. The high stability of this front can be attributed to the high volumetric steam flow rates, combined with the reduction in oil viscosity. High recoveries were obtained due to the low residual oil saturation of about 15 percent in the zone swept by steam. The latter value pertains to the specific test conditions (heavy oil, uncon-solidated sand) and may be lower for light oil, due to the effects of steam distillation. The applicability of this experimental approach is limited, as it does not take into account the fact that

Jan 1, 1970

By R. W. Parsons

The over-all apparent single-phase permeability of fracture-rock systems was studied using two different two-dimensional models. In a strict sense the results are applicable only to these models, yet may give some insight into real reseruoir behavior. The regular fracture-matrix model can have any number of regular fracture sets. All fractures within each set have the same orientation, width and spacing. The gross behavior is amlytically equivalent to that of an anisotropic permeable medium. In the heterogeneous fracture systems, individual fracture conductivities vary within a two-dinensioml network pattern. A computer program calculates the overall permeability after the conductivities are randomly placed in the network. The calculated permeability depends upon the distribution of these conductivities and the particular fracture pattern into which they are placed. Although detailed information on actual reseruoir fracture systems is not available, this study shows how some of the possible variables may affect obserued gross reservoir characteristics. INTRODUCTION Characteristics of reservoir rocks are determined by direct (core analysis) or indirect (logging and production tests) examination. Both methods, singly and together, are widely used in reservoir evaluations. For fractured reservoirs a problem exists in relating characteristics of cores to the in situ reservoir properties. Knowing fracture characteristics can be of value in determining production mechanisms. Because of the limitations of observation, the intimate details of a fractured rock system will never be known. One approach in analyzing fractures is to study the behavior of some conceptually simple models. The two models analyzed contain features which may be realistic, yet are mathematically tractable. Flow is considered in two dimensions only, with the plane of interest at right angles with the fractures. This concept represents the horizontal flow in a reservoir system with the fractures oriented vertically. Only single-phase, laminar flow is considered. The regular fracture-rock system permeability is calculable from a simple analytical expression. The heterogeneous system requires a computer program for calculating overall permeability.

Jan 1, 1967

By C. F. Weinaug, R. W. Farley, J. F. Wolfe

A rapid, accurate method for predicting the dew points of gas condensate systems and their subsequent normal and retrograde phase behavior with pressure decline has been developed. The method predicts dewpoint pressure, volume percent liquid, condensate content of produced gas, and the liquid and gas compositions at pressures below the dewpoint pressure. Use of a modified form of the BWR equation of state, which was established by regression calculations on experimental data from several gas condensates, provides depletion behavior predictions that are significantly improved over predictions using K-values based on convergence pressures and Standing's correlation for density. INTRODUCTION Prediction of complex hydrocarbon-mixture thermodynamics is fundamental to reservoir and gas process engineering. In particular, accurate descriptions of phase behavior are needed to plan production operations for gas-condensate reservoirs. Although analysis of reservoir behavior during producing and cycling operations can be made for simple reservoir models, extensions of these studies to more complex reservoir systems are still difficult, both with regard to numerical techniques and the phase behavior of the fluids. This report presents a mathematical method for handling the complex phase relationships encountered in retrograde condensates; it represents an advance in reservoir behavior technology in that it will aid the development of closed-form mathematical representations of total production systems. A typical phase diagram for a complex hydrocarbon mixture is shown in Fig. 1. When the reservoir temperature T, is higher than the critical temperature Tc, and lower than the critical-condensation temperature Tct' the mixture is a gas condensate. Decreasing the pressure on a condensate while maintaining nearly constant temperature will result in two phases forming at the dew point. Further decrease in pressure will cause additional liquids to be formed. This phenomenon is called retrograde condensation. As pressure continues to decrease, normal phase behavior regions are encountered (liquid vaporizes as the pressure decreases) until the low-pressure dew point is encountered. When a gas-condensate reservoir is produced, gas is withdrawn and the volume of fluids in the reservoir remains nearly constant as the pressure declines. Heavier components tend to accumulate in the liquid phase as a result of retrograde condensation, while lighter components tend to concentrate in the vapor phase. Thus, as the gas is produced the composition of the gas and the condensate

Jan 1, 1970

By W. A. Klikoff, I. Fatt

The concept of fractional wet wattability is examined. Fractional water wettability of a reservoir rock is defined as the fraction of the internal surface urea that is in contact with water. Capillary pressure and relalive permeability of unconsolidated sand are shown to be functions of fractional wettability. INTRODUCTION The petroleum industry has long recognized that wettability of reservoir rock has an important effect on multiphase flow of oil, water and gas through reservoirs. As early as 1928 the American Petroleum Institute sponsored a study of wettability as part of API Project 27 at the U. of Michigan.' Despite 30 years of research, there is still little exact knowledge of the wettability of reservoir rocks. There are two parts to the wettability problem. After agreeing to a uniform nomenclature in regard to wettability,' the first question to be answered is, "What is the in situ wettability of a given reservoir rock?" If this can he answered the next question is, "What part does wettability play in determining the characteristics of multiphase fluid flow through the rock?" This paper represents an oblique attack on the problem of wettability. No attempt is made here to answer the basic question of wettability in situ. instead the consequences of the concept of fractional wettability are examined. Multiphase flow in sandpacks is shown to he highly influenced by fractional wettability. Jennings' has given 3 definition of wettability and the other terms used in discussing wettability. These terms must be applied to the physical situation existing In reservoir rock. A survey of the pertinent literature from 1928 to 1956 indicates that the concept of a contact angle was applied to reservoir rock in the same way it would be applied to a flat, homogeneous surface. Attempts were made to state quantitatively the wettability of a reservoir rock in terms of a contact angle which was presumably constant at all points on the very rough and heterogeneous interior surface of a porous rock. Calhoun, et al,3-6 prepared synthetic consolidated and un-consolidated porous media in which they claimed there was a known uniform contact angle. They then showed the effects on the capillary pressure and relative permeability characteristics of varying this angle. The API Project 47 at the U. of Texas' and others' have made extensive studies of an indirect approach to the contact angle through the use of heat of wetting data. Even if successful. however, this approach also states the wettability of porous rock in terms of a contact angle which is uniform over the entirc surface. If the angle varies from one part to another on the internal surface, there is no way of determining from the measurements the area distribution of contact angles. In 1956 Brown and Fatt5 suggested that the concept of a contact angle, as applied to reservoir rock, be abandoned. This suggestion was made because it is known that the internal surface of most reservoir rocks is composed of many different minerals, cach with a different surface chemistry and a different capacity to adsorh surface active materials from reservoir fluids. Furthermore, the operation of a contact angle in determining the form of a fluid-fluid interface is difficult to picture in the very complex geometry of a pore. Brown and Fatt proposed that the wettability of reservoir rock be stated in terms of the fractional internal surface area that is in contact with water or oil. All surfaces on which there is water are called water-wet; surfaces on which there is oil are called oil-wet. The fractional water wettability is then stated as a number which represents the fraction of the internal surface that is in contact with water. A symmetrical statement can be made for the fractional oil wettability. The concept of a fractional wettability as previously stated has in its favor the recognition of the heterogeneous mineral composition of most reservoir rocks. Another point in its favor is that fractional wettability can be measured quantitatively with relative ease. Hol-brook and Bernard10 se a simple dye adsorption test to obtain fractional wettability of reservoir rocks. Amott11 uses a combination of imbibition and displacement to arrive at a wettability index of reservoir rocks which seems related to fractional wettability in the range 0.25 to 0.75 fractional water wettability. Jennings" has shown the changes in relative permeability that take place when a porous material is changed from unity to zero fractional water wettability. He also shows that reservoir rocks in their natural state, but at room temperature and atmospheric pressure, have relative permeability characteristics which would indi-

By H. J. Ramey

Conventional calculation of total system isothermal compressibility for a system containing a free gas phase involves, among other things, evaluation of the change of oil and gas formation volume factors and the gas in solution with pressure. Preferably, this information should be obtained from laboratory measurements made with particular oils and gases. Often, experimental measurements are not available. In this case, it is necessary to obtain pressure-volume-temperature relationships from general correlations such as those of Standing' for California oils. In order to speed estimates of compressibility, generalized plots have been prepared of the change of both oil formation volume factors and gas in solution, with pressure from Standing's correlations. A generalized plot for estimating the change in the two-phase (oil and gas) formation volume factor with pressure is also presented. Usually, the effect of gas dissolved in reservoir water upon the total system compressibility is neglected for gas saturated systems, due to the low solubility of gas in water. Results of this study indicate that the increase in total system compressibility caused by solution of gas in water is often as large as the compressibility of wafer, and can be magnitudes larger for low pressure systems. Generalized results for estimating the change of gas in solution in water with pressure are presented in tabular and graphical form. INTRODUCTION During the past decade, pressure build-up and drawdown techniques have gained an important place in reservoir engineering. Build-up and drawdown analyses are only two special applications of the broad field of transient fluid-flow theory. All solutions of transient fluid-flow problems contain a parameter called the total system isothermal compressibility. This property of fluids and porous rock is a measure of the change in volume of the fluid content of porous rock with a change in pressure, and it may vary considerably with pressure. Evaluation of total system isothermal compressibility is not difficult, but it is tedious and time-consuming. Often compressibilities are estimated roughly, or transient flow methods are neglected completely. The benefits of using accurate system compressibility in properly-executed build-up or drawdown analyses are: 1. Better planning of pressure build-ups may be achieved to avoid unnecessary loss of revenue due to excessively long shut-in periods, or to shut-in periods too short to yield useable data. 2. Better and more reliable estimates of static formation pressures for reserves estimates and rate performance estimates. 3. Reliable information for evaluation of well completion effectiveness, and planning and interpretation of well stimulation efforts. The purpose of this paper is to clarify the nature of the total system isothermal compressibility, and to present useful methods for estimation of compressibility, particularly for systems containing a gas phase. DEVELOPMENT Numerous publications have presented solutions to transient single-phase flow of slightly compressible fluids, stressing pressure build-up applications. In transient flow, a compressibility* term arises to permit volume content of fluids in porous rock to change as pressure changes. The basic nature of the compressibility term is usually taken for granted. Problems arise in practical applications of transient fluid theory because most published works consider only one flowing fluid—in an ideal porous system containing only one fluid. In 1956, Perrine2 presented an intuitive extension of single-phase flow pressure build-up methods to multiphase flow conditions. Later, Martin- established conditions under which Perrine's multiphase build-up method had a theoretical foundation. Perrine has shown that improper use of single-phase build-up analysis in certain multiphase flow situations can lead to gross errors in estimated static formation pressure, permeability and well condition. It is likely that, much pressure build-up data for oil wells should be analyzed on the basis of multiphase flow. For both single-phase and multiphase build-up analysis, the isothermal compressibility term in dimensionless time groups often should be interpreted as the total system compressibility. All real reservoirs contain one or more compressible fluid phases. In addition, rock compressibility can contribute in an important way to the total system compressibility. The proper total system compressibility expression may contain terms for compressibility of oil, gas, water, reservoir rock and terms for the change of solubility of gas in liquid phases.

Jan 1, 1965

By J. D. Rice, S. C. Pitzer, C. E. Thomas

Interpretation of pressure build-up data obtained in the conventional manner has often been difficult because of the deviation from theoretical behavior. Major causes of this deviation have been attributed to damage and afterflow, and to fluid redistribution in the wellbore which, in extreme cases, can result in a pressure hump in the early portion of the build-up curve. Theoretical investigations show that bottom-hole pressure is definitely influenced by phase redistribution in the tubing column during surface shut-in tests, and that the magnitude of the effect is sensitive to the producing gas-oil ratio and stabilized rate of flow in the well. For field experiments a wire-line tubing packer, which can be run in the tubing against a stabilized flow rate, was developed for bottom-hole shut-in tests. By use of this bottom-hole .shut-in method, pressure humps previously observed in surface shut-in tests were completely eliminated and the effects of afterflow minimized. Analog and digital computer studies have been made to obtain theoretical curves for comparison with field curves, and remarkable agreement between the results of bottom-hole shut-in tests and theoretical curves has been obtained. INTRODUCTION Pressure build-up data from shut-in wells have been used by the petroleum industry to determine the permeability of the formation, to estimate wellbore damage, to evaluate the static reservoir pressure and to speculate reservoir volume. Calculation of these factors is based on methods of analysis developed for theoretical systems whose build-up curves have a charac- teristic shape when the wells are shut-in at the sand faCe.1,2,3,5,8,10 However, field curves obtained by conventional surface shut-in methods do not always exhibit this characteristic. Such factors as stratification, rock heterogeneities and irregular reservoir geometry can cause the character of a build-up curve to deviate from that predicted by theory for a simple system. In addition, all field build-up data are affected by the methods utilized in obtaining the measurements. Conventional surface shut-in techniques allow two effects to occur which contribute directly to the manner in which pressure builds up at the sand face. The first, afterflow, has been recognized for some time and methods are available for estimating the magnitude of its effect and for correcting its influence.' The second, phase segregation in the tubing column after shut-in, has been reported only recently and appears to exert a considerable influence on the character of field build-up curves.!' In extreme cases, phase segregation can produce "pressure humps" in the early portion of the build-up curve. Such curves have been considered anomalous and have defied analysis. It is the purpose of this paper to discuss the effect of phase segregation in the tubing string during build-up and to present a bottom-hole shut-in technique for obtaining field build-up data which minimizes the influence of afterflow and eliminates phase segregation effects. It will be shown that reliable reservoir information can be calculated from build-up measurements obtained using this method, even though conventional surface shut-in tests on the same wells yield anomalous data. DISCUSSION OF PHASE REDISTRIBUTION EFFECTS Some of the effects of phase redistribution on pressure build-up curves have been described previously by Matthews and Stegemeier" who have presented evidence that phase segregation is responsible for the pres-

By A. G. Weber, K. H. Coats, M. H. Terhune, R. L. Nielsen

Two computer-oriented techniques for simulating the three-dimensional flow behavior of two fluid phases in petroleum reservoirs were developed. Under the first technique the flow equations are solved to model three-dimensional flow in a reservoir. The second technique was developed for modeling flow in three-dimensional media that have sufficiently high permeability in the vertical direction so that vertical flow is not seriously restricted. Since this latter technique is a modified two-dimensional meal analysis, suitably structured three-dimensional reservoirs can be simulated at considerably lower computational expenses than is required using the three-dimensional analysis. A quantitative criterion is provided for determining when vertical communication is good enough to permit use of the modified two-dimensional areal analysis. The equations solved by both techniques treat both fluids as compressible, and, for gas-oil applications, provide for the evolution of dissolved gas. Accounted for are the effects of relative permeability, capillary pressure and gravity in addition to reservoir geomety and rock heterogeneity. Calculations are compared with laboratory waterflood data to indicate the validity of the analyses. Other results were calculated with both techniques which show the equivalence of the two solutions for reservoirs satisfying the vertical communication criterion. INTRODUCTION Obtaining the maximum profits from oil and gas reservoirs during all stages of depletion is the fundamental charge to the reservoir engineering profession. In recent years much quantitative assistance in evaluating field development programs has been provided by computerized techniques for predicting reservoir flow behavior. Because of the spatially distributed and dynamic nature of producing operations, automatic optimization procedures, such as those now in use for planning refining operations, are not now available for planning reservoir development. However, present mathematical simulation techniques do furnish powerful means for making case studies to help in planning primary recovery operations and in selecting and timing supplemental recovery operations. A number of methods have been reported which simulate the flow of one, two or three fluid phases within porous media of one or two effective spatial dimensions>-4 However, applying computer analyses to actual reservoirs have been limited mostly to two-dimensional areal or cross-sectional flow studies for two immiscible reservoir fluids. To obtain a three-dimensional picture of reservoir performance using such two-dimensional techniques, it has been necessary to interpret the calculations by combining somehow the results from essentially independent areal and cross-sectional studies. To the authors' knowledge, the only other three-dimensional computational procedure, in addition to those presented here, was developed by Peaceman and Rachford8 to simulate the behavior of a laboratory waterflood. Two computational techniques which may be used to simulate three-dimensional flow of two fluid phases are described in this paper. The first method, called the "three-dimensional analysis", employs a fully three-dimensional mathematical model that treats simultaneously both the areal and cross-sectional aspects of reservoir flow. The second method, called the vertical equilibrium (VE) analysis", is applicable

By C. F. Weinaug, P. M. Blair

Many difference equations used to approximate reservoir flow problems treat the phase pressures implicitly but not the mobility-density coefficients. Such difference equations are neither wholly explicit nor implicit, but might be described as mixed. Mixed equations are relatively easy to apply. But the associated time truncation error is relatively large, and when used to solve problems characterized by high flow rates, these equations may be unstable for practical size time steps. This paper outlines the development of a completely implicit difference analogue for reservoir simulation, along with a Newtonian iterative method for solving the resulting nonlinear set of algebraic equations that arise at each time step. While this implicit equation requires two to three times more work than does the mixed equation, it is shown to markedly decrease the time truncation error and to yield a stable solution for much larger time steps than does the mixed equation. INTRODUCTION Calculation of multiphase, multidimensional flow in porous media is generally accomplished by numerical methods that involve approximating systems of partial differential equations by systems of partial difference equations. The early development of difference equation techniques was directed toward the solution of linear differential systems. However, the equations of multiphase fluid flow through porous media are highly nonlinear in that mobility and density often are strong functions of pressure and saturation. Thus, solution of the flow problems by difference equations involves solving sets of algebraic equations whose coefficients change from step to step of the calculations. Current practice is to evaluate most coefficients at the beginning of a time step and then to apply difference methods well suited for solving linear problems. At least two major difficulties arise from the use of this practice. First, evaluation of coefficients at the old time level and evaluation of pressures at the new time level results in larger time truncation errors near displacement fronts than if all quantities in the distance derivative were evaluated at the same time level. Second, evaluation of mobility coefficients at old time levels results in an unstable difference equation in regions of high flow rate. The first of these difficulties often results in optimistic recovery behind the displacement front, and the second requires the use of small time steps or special techniques in solving coning problems1 or problems of gas percolation.2 Evaluation of all quantities in the distance differences at the new time level results in a completely implicit difference system. Such a difference system has a lower time truncation error than equations in which mobilities are evaluated at the old level and pressures evaluated at the new level. The algebraic equations resulting from the fully implicit difference equation are, in this case, nonlinear and require some iterative method for their solution. This paper develops fully implicit difference equations for two-phase flow in porous media, describes their solution by Newtonian iteration, and gives examples of problems more easily solved by the new method than by previous methods. EVOLUTION OF DIFFERENCE EQUATIONS FOR NONLINEAR PROCESSES The completely implicit difference equation used in these calculations results from evolution beginning with explicit difference equations, proceeding through what we will call "mixed" equations, and arriving now at a completely implicit equation. Let us illustrate the evolution by considering different analogs of the nonlinear equation

Jan 1, 1970

By R. C. Earlougher

Jan 1, 1970

By B. S. Gottfried

This paper is concerned with possible transport mechanisms which occur during segregated burning (i.e., burning in an oil reservoir in which the oil-bearing formation is overlain by a "clean" porous zone containing only gas). ,Two analytical models are presented. In the first, burning is assumed to occur at the gas-oil interface, and the dominant mechanism is taken to be the rate at which oxygen diffuses to the interface. The second model assumes that combustion occurs within the clean zone, with the dominant mechanism being the mechanism of fuel deposition from the oil-bearing formation to the clean zone. ,Comparison of the results of the analyses with limited experimental data indicates that the latter mechanism is much more plausible. Both models are idealized representations of the actual process and are somewhat conjectural, however, owing to the lack of more definitive laboratory or field data. INTRODUCTION In recent years there has been considerable theoretical and experimental research directed toward understanding the recovery of crude oil by underground combustion. Although two different recovery schemes have been proposed, known as forward and reverse combustion, it is only forward combustion that shows promise of becoming a practical recovery technique.17 Most of the analyses of the forward combustion process have considered it to be pistonlike; i.e., oil is displaced by a vertical combustion zone which, with its ensuing gases, pushes die oil forward as the flame front is propagated through the reservoir.l-8,ll,l2,14-16 However, information gained in field pilot studies indicates that, at least to a certain extent, forward combustion proceeds in a segregated manner in oil reservoirs.10 An ideal segregated system is one in which the oil-bearing formation is overlain by a "clean" porous zone containing only gas. The location of the combustion zone in such a system is not immediately evident, although combustion is believed to occur either at the gas-oil interface or within the clean zone. In either case, the combustion zone may be horizontal and it moves in parallel rather than in series with the produced oil. The purpose of this paper is to explore the likely mechanisms by which segregated burning can occur, and to determine the sensitivity of the resulting models to changes in various physical parameters. Two governing mechanisms are considered: the case of a horizontal combustion zone at the gas-oil interface with the rate of combustion governed by the rate at which oxygen diffuses to the interface, and the case of combustion within the clean zone with the combustion process controlled by the transport of fuel from the oil-bearing formation to the clean zone. Both models are somewhat hypothetical and involve considerable simplification of the actual process. However, the results allow some preliminary understanding of segregated burning which is useful in the absence of more definitive laboratory or field pilot data.

Jan 1, 1967