Pseudo Relative Permeability Research Articles

An Approach to Gas-Coning Correlations for a Large Grid Cell Reservoir Simulator

Summary Generalized gas-coning correlations were developed specifically for a three-dimensional five-layer large grid cell model of the Prudhoe Bay field. These correlations can be used to predict the critical coning rate or to predict the gas/liquid ratio (GLR) of a well after coning has been achieved. Although these correlations have been developed with specific data for the Prudhoe Bay field, the approach and treatment of the parameters should be of use in the evaluation of gas coning in other fields. Introduction Gas-coning behavior can be described accurately in three-dimensional reservoir simulations with sufficient vertical layering. However, for large-scale field simulations, grid blocks are so large that the effects of gas coning are essentially absent. For these cases, it is necessary to use gas-coning correlations. These correlations may be of many forms. Critical coning rate is the maximum oil production rate at which the well produces oil without concurrent production of gas by coning. Critical coning rates may be calculated by standard methods; however, they cannot predict gas production rates if the well intentionally is produced above this rate. Predictions of coning behavior by pseudorelative permeabilities in three-dimensional reservoir simulators have been reported, but grouping of wells into various well types may be necessary. Several other methods for describing well coning behavior have been reported. In all of these methods, the basic purpose of the gas-coning correlations is to predict the gas/oil ratio (GOR) for each well at a given production rate.This study describes a set of gas-coning correlations that have been developed specifically for a three-dimensional five-layer model of the Prudhoe Bay field. The correlations assign a GLR in each well on the basis of the physical parameters of each individual well and the production rate. Although these correlations have been developed specifically for data from the Prudhoe Bay field, the procedure, treatment, and techniques for their development should be valid for more generalized situations.In most previous gas-coning correlations, wells having common characteristics were grouped into a number of well types since it is impractical to simulate individually all the wells in any large field. Gas-coning correlations were developed for each well type, and then either coning behaviors were identical or an interpolation procedure was used for each well of this type. The approach used for these gas-coning correlations was to develop generalized gas-coning correlations that would use individual well parameters rather than to group the wells into various types. By this method it was hoped that each individual well could be modeled more closely. Approach to the Problem Individual coning wells were modeled on a two-dimensional fully implicit radial simulator. Wells were modeled with properties encompassing the range expected from the field and with boundaries that correspond to similar boundaries in the three-dimensional field model. The gas-coning behavior was correlated to one critical parameter - the average oil column height above the perforated interval of the well.We determined the average oil column height above the perforations by first calculating an average oil column height within the drainage area of the well. This oil column height was determined by averaging the oil saturation around the well and calculating an average oil column height as if the gas/oil contact were level. JPT P. 2267^

History Match Analysis of the Little Creek CO2 Pilot Test

Production and pressure data from the Little Creek CO2 miscible pilot test are history matched using a miscible flood simulator. Simulated recovery is dominated by gravity segregation of CO2 and crossflow of oil. A new pseudorelative permeability technique, which decouples gravitational and viscous fingering effects, is developed and used in the history match. Introduction A limited number of carbon dioxide miscible pilot tests have been completed or have enough history to permit a meaningful evaluation. Noteworthy tertiary examples include the SACROC, Twofreds, and Willard tests in west Texas and the Little Creek test in Mississippi. SACROC, Twofreds, and Little Creek were producing pilots; of these, the Little Creek pilot is best suited for a thorough analysis to identify those factors that influenced tertiary oil recovery. The test was well confined and small, with one injector and three producers.The primary objective of this work was to determine the controlling factors in the Little Creek CO2 flood by history matching pressure and production data with a four-component, miscible flood simulator. Additional objectives were to (1) develop, demonstrate, and employ a pseudorelative permeability technique for simulating three dimensional miscible performance with two-dimensional calculations and (2) project performance for different-size CO2 slugs than that tested at Little Creek. Field Description and History The characteristics and history of the Little Creek field are documented elsewhere. A limited discussion of the field is presented here for completeness.The Little Creek field is in the Lower Tuscaloosa trend of southwestern Mississippi. It was discovered in 1958. A peripheral line-drive waterflood was conducted from 1962 to 1970, and CO2 was injected into the pilot area from Feb. 1974 to Feb. 1977. Of the 102 million bbl (16.2 x 10(6) m3) of oil initially in place, 25 million bbl (4.0 x 10(6) m3) were produced by primary depletion and 22 million bbl (3.5 x 10(6) m3) were produced as a result of waterflood operations. More than 120,000 bbl (19 000 m3) were produced in the pilot test.The reservoir's pay zone is the 10,750-ft (3280-m) Lower Tuscaloosa Denkman sand, which is Cretaceous in age. The reservoir is controlled stratigraphically, being limited by rapid sand pinchout on the flanks. The reservoir is said to have "remarkably uniform rock properties ... (with) no correlative high permeability zones." The reservoir responded very well to water injection, reportedly "flooding out uniformly ... going to 100% water within 3 months of flood-front breakthrough and conformance was estimated to be 90%" Table 1 lists average rock and fluid properties.The CO2 miscible pilot test was conducted against a pinchout along the eastern edge of the field. As shown in Fig. 1, the pattern is essentially one-quarter of an inverted nine-spot on 40-acre (16-ha) spacing. The pilot area was confined by five backup water-injection wells whose purpose was to prevent migration of fluids out of the test pattern. JPT P. 2042＾

Reservoir Simulation of the Empire Abo Field: The Use of Pseudos in a Multilayered System

Abstract A three-layered model with vertical equlibrium pseudo-relative permeabilities and pseudo-capillary pseudo-relative permeabilities and pseudo-capillary pressures was used to match the results from a pressures was used to match the results from a 22-layered model of a portion of the Empire Abo field. Results of the two models were almost identical in both history match and prediction phases. Directionally dependent pseudo-relative permeabilities were introduced to match the drainage mechanism of the reservoir. Both horizontal displacement and drainage mechanisms can be modeled simultaneously in a multilayered system using these pseudo functions.The excellent comparison between the finely gridded model and the pseudo model allowed the simulation of alternative recovery schemes using the pseudo model with the same confidence as the pseudo model with the same confidence as the original three-dimensional model. In particular, a comparison between pressure maintenance with carbon dioxide and nitrogen was simulated, with a substantial savings in both manpower and computer time. Development of this pseudo model makes the simulation of the entire Empire Abo field feasible. Introduction For several years vertical equilibrium pseudo-relative permeabilities and pseudo-capillary pressures have permeabilities and pseudo-capillary pressures have been used throughout the industry to collapse a three-dimensional model with many layers to a two-dimensional areal model with a single layer. The obvious advantages of pseudo models are reduced computer and manpower costs. The justification for the use of pseudo models has been the comparison of results from one-dimensional models using pseudos with those from two-dimensional, cross-sectional models. Few comparisons have been made between an areal model using pseudos and a three-dimensional simulation.The Empire Abo field offered an excellent example for such a comparison. The field had been simulated in three dimensions, using 22 layers. An analysis of the simulation results showed that conditions were such that vertical equilibrium existed in most columns of the model. By applying pseudos to the three-dimensional model, it appeared possible to reduce the number of layers and associated computer time substantially.In this paper, the techniques for obtaining a match of the three-dimensional models with a pseudo model are described. The original three-dimensional model is described along with a history of the Empire Abo field. Coning correlations and directionally dependent pseudo-relative permeabilities also are described. Results between the pseudo model and the three-dimensional model also are compared. Finally, the use of the pseudo model for predicting alternative recovery schemes is discussed. Field History The Empire Abo field is located in southeastern New Mexico, about 8 miles (13 km) southeast of Artesia, NM (Fig. 1). The Abo reservoir was discovered by Amoco Production Co. in Nov. 1957. Subsequent development and unitization of the Abo reservoir was documented by Christianson. The producing horizon is a carbonate reef, basal Permian (lower Leonard) in age. The productive reef is about 12 1/2 miles (20.12 km) long and 1 1/2 miles (2.47 km) wide (Fig. 2), covering about 11,339 surface acres (45.9 x 10(6) m2). The reef dips gently from southwest to northeast at about 1 degree. JPT p. 279

The Effect of Small, Discontinuous Shales on Oil Recovery

A mathematical model is presented for analyzing the effect of small, discontinuous shales on oil recovery. The model was validated by calculations with a fine-grid computer model. The study illustrates the importance of teamwork between geologists and engineers when describing a reservoir and when predicting its performance. Introduction A complex problem when predicting reservoir performance is assessment of permeability of a formation performance is assessment of permeability of a formation perpendicular to bedding planes. Vertical permeabilities, perpendicular to bedding planes. Vertical permeabilities, which may be less than horizontal permeabilities because of orientation of rock particles and cementing material, can be measured on core samples. It also is necessary to determine distribution of nonpay intervals because small amounts of impermeable rock can profoundly affect vertical permeabilities, even if the rock is discontinuous and randomly distributed. Lateral dimensions of shale laminae, shale bodies, or other impervious materials and their distribution are difficult to describe solely from data on cores and logs. Proper description requires knowledge of depositional environment and detailed information from geologic studies of outcrops and recent sediments. Correct assessment of vertical permeability is particularly important in gas drive operations because high oil particularly important in gas drive operations because high oil recovery depends on effective oil drainage by gravity. For example, shale bodies may be large enough laterally in some reservoirs to prevent coning and yet be few enough to permit good vertical oil drainage. A laminate barrier in a pinnacle reef (carbonate reservoir) was extensive enough to cause loss of miscibility by collecting oil and part of an LPG bank above it. Recent studies indicate the presence of frequent, small, randomly distributed shale bodies in the upper formation of a large sandstone reservoir, which reduce vertical permeability in that interval. Fortunately, conceptual models based on geologic studies of recent sediments, outcrops, cores, and logs are available for describing the distribution and dimensions of pay and nonpay in both sandstone and carbonate reservoirs. Also, reservoir simulators are available for solving increasingly complex problems in practical times at reasonable costs. Construction of simulators is relatively straightforward when a reservoir is divided into separate zones by continuous barriers. Usually, at least three layers are provided in a computer model for each zone. Thin layers provided in a computer model for each zone. Thin layers are used above impervious boundaries to represent flow from gas-invaded regions. An alternative procedure is to generate pseudorelative permeabilities for layers above boundaries. Layers beneath impervious boundaries in water-invaded regions require similar treatment for accurate simulation of reservoir performance. However, no standard treatment is available for modeling reservoirs containing many small, discontinuous barriers. Adequate simulation of performance in large regions of a reservoir could requite many more small computer blocks than the user could afford. This paper illustrates how geologists and engineers can cooperate in describing and predicting oil recovery from complex distributions of permeable and impermeable intervals. A brief overview is given on geologic modeling of different types of deposits. A simple mathematical model is presented for assessing the impact of a given sized impervious body on oil recovery. JPT P. 1531

Pseudofunctions for Water Coning in a Three-Dimensional Reservoir Simulator

Abstract Three-dimensional numerical models of bottom-water-drive reservoirs show delayed water breakthrough into individual wells when compared with observed well performance and individual-well coning models. This reservoir-model behavior results from masking of the well coning effect by volume-averaging pressure and saturation profiles around a well over a grid block with a large volume. The reservoir-simulator prediction of well performance can be improved by mathematically performance can be improved by mathematically transforming the production performance of a detailed well-coning model into a set of time-independent pseudorelative-permeability and capillary-pressure curves that then can be used in the reservoir model. A procedure for obtaining the required pseudofunctions is described and the results of their application in simple models and in a large reservoir-simulator model are shown. Introduction The prohibitive cost of numerical reservoir simulation with fine-grid definition models of large reservoirs has led to development of techniques whereby vertical saturation distribution and/or localized flow conditions in the vicinity of individual wells can be approximately accounted for in relatively coarse-grid models at an acceptable incremental cost. In particular, vertical cross-section models under capillary and gravity equilibrium have been used to derive pseudorelative permeabilities and capillary pressures for use in two-dimensional, areal models to simulate the average vertical distribution of flow without having to pay the computing price of a full three-dimensional model. Coats et al. described the use of the vertical equilibrium concept for developing pseudorelative-permeability and capillary-pressure pseudorelative-permeability and capillary-pressure functions for simulating the vertical dimension in a two-dimensional, areal simulator model This method assumes gravity-capillary equilibrium in the vertical direction. Also, Coats et al. developed a dimensionless parameter for estimating when these conditions are valid. Martin formed a mathematical basis for pseudofunctions by reducing the equations for pseudofunctions by reducing the equations for three-phase, three-dimensional, compressible flow to two-dimensional relations by partial integration of the equations of flow. Hearn extended the pseudorelative-permeability concept by adapting it pseudorelative-permeability concept by adapting it to stratified reservoirs where viscous rather than gravity and capillary forces dominate the vertical sweep efficiency. Hawthorne studied the effects of capillary pressure on pseudorelative permeability derived from the Hearn stratified model. Jacks et al. further enlarged thepseudorelative-perrneability concept by developing dynamic pseudorelative permeabilities. (Dynamic pseudos, denoting pseudos permeabilities. (Dynamic pseudos, denoting pseudos determined under flowing rather than static conditions, allow one to account for the interaction between viscous and gravity forces resulting from rate variation in the vertical plane.) Kyte and Berry generalized the work of Jacks et al. by introducing the concept of pseudocapillary pressures and improving dynamic pseudofunction calculations to include varying flow potential gradients. Emanual and Cook expanded the concept of vertical cross-section, pseudorelative permeabilities to include the vertical performance of individual wells. Their procedure combines the effect of coning and well pseudorelative permeabilities for use in a two-dimensional, areal model. Chappelear and Hirasaki used a different approach to including of coning effects in a two-dimensional, areal simulator by developing a functional relationship among water cut, average oil-column thickness, and total rate based on an analytical incompressible, steady-state model. The most sophisticated approach to representing well-coning effects in a reservoir simulator has been taken by Mrosovsky and Ridings and Akbar et al. They incorporated detailed numerical well models into the reservoir-model grid. SPEJ P. 251

Reservoir Stimulation Studies Reveal Key to Maximizing Waterflood Oil Recovery of Virginia Hills Beaverhill Lake "A" Pool

Abstract Two comprehensive simulation studies were performed on this reservoir to evaluate waterflood performance to determine the long-term production and the injection policy for the field. The results of these studies revealed that the key to maximizing economic oil recovery from the pool is to improve the vertical conformance of the waterflood. The major biohermal reservoir is found at an average depth of 5600 ft subsea and contains under-saturated crude oil in two partially overlying, non-communicating zones. Fifteen of the field's 117 wells were converted to water injectors in an irregular peripheral water/load pressure maintenance scheme. The simulation models history matched 5 years of primary performance and 10 years of waterflood performance. As it was not practical to simulate the entire field in three dimensions, a four-layer, two-phase, three-dimensional (3-D) reservoir simulation study was done on all area approximately representing the area of influence of the largest infection well. Results indicated that there was a degree of sub-layering within the layers described in this model, Flood-front saturation profiles from the history- matched three-dimensional model were employed to derive pseudo relative permeability relationships for the two-dimensional (2-D) areal model and an excellent history match of the performance of the entire pool was achieved. The study recommended a program of production logging to establish actual production/injection profiles. Results of subsequent attempts to workover wells to improve the flood profiles are discussed. Introduction It has been common practice in the oil industry for a number of yearsto conduct comprehensive reservoir engineering studies to design enhanced recovery schemes for oil reservoirs. A more recent very valuable use for in-depth reservoir studies is to optimize the performance of existing enhanced recovery schemes. Reservoir model studies on reservoirs which have considerable production history can be used to determine ways to improve performance through changes in production and injection policy, to justify remedial action to correct adverse performance und to investigate the ramifications of alternate ourses of action. Reservoir models can also be used as effective performance surveillance tools. The opportunity to improve performance of an enhanced recovery scheme is greatest when the pool is unrestricted by market demand or other arbitrary rate restrictions, because the full benefits of the changes can be realized as soon as he field responds to the change. Whenever the demand for crude oil approaches or exceeds the productive capacity of a reservoir, opportunity to optimize performance by onducting a comprehensive engineering study presents itself. To be most effective, these studies should be interdisciplinary efforts including input from reservoir petrophysical, geological and production engineering disciplines. The studies described in this paper were performed as a cooperative effort involving all of the above disciplines and the results illustrate some of the advantages gained by this cooperative approach.

The Effect of Capillary Pressure in a Multilayer Model of Porous Media

Abstract This paper shows how the Hiatt-Hearn method for computing pseudo-relative permeability functions is modified by capillary pressure. The model is one-dimensional and without gravity forces. It provides another limiting case to computed pseudo-relative permeability functions. pseudo-relative permeability functions.The model was used for theoretical analysis and numerical experiments on water drives. The study shows that capillary forces promote a sharper front and a higher recovery at breakthrough than would be predicted for a noncapillary case. Introduction Reservoir performance prediction by analytical methods is based on some definition of fluid permeability as a function of saturation. In the permeability as a function of saturation. In the past, we have tried to use laboratory tests on past, we have tried to use laboratory tests on cores (relative permeabilities) to predict reservoir performance. Not only do laboratory data on performance. Not only do laboratory data on different cores from the same field vary widely, but data obtained from the same core by different methods are not always the same. Extrapolation of small-scale core measurements to the reservoir scale is questionable. Various averaging and combining schemes are tried, but only sometimes do laboratory flow tests relate to actual field performance. There is extensive literature on attempts to define relative permeability from basic physical principles. A review of this literature is beyond principles. A review of this literature is beyond the scope of this paper; for further information, see Refs. 1 and 2. Alternatively, pseudo-relative permeability curves calculated for layered models have been tried. In practice, the layer methods sometimes work, but practice, the layer methods sometimes work, but often the laboratory core analysis is just as good. The technique presently used to avoid these difficulties is history matching. This is basically extrapolating a curve and, unfortunately, the method depends on knowing a good part of the performance we wish to predict. Two basic layered models are used in petroleum engineering. The Dykstra-Parsons method assumes the layers are isolated except for common pressure at the input and output ends. The Hiatt method assumes crossflow between layers such that the layers are always in pressure equilibrium at any given distance. Both methods assume a sharp displacement front between displacing and displaced fluids. Recently, Hearn showed how to compute pseudo-relative permeability curves based on the pseudo-relative permeability curves based on the Hiatt model. Carrying the idea even further, jacks et al. showed how to compute pseudo-relative permeability functions under dynamic conditions permeability functions under dynamic conditions using a reservoir simulator. It seems possible that the failure of layered models was partly caused by neglecting capillary forces in computing fluid displacement. This study was made to determining the part that capillary forces could play in the Hiatt-Hearn model. The results show how capillary forces change pseudo-relative permeability curves. While thick layers in the permeability curves. While thick layers in the reservoir probably could not reach the pressure equilibrium postulated by this model, the method does impose a limiting condition. With this and other models, it should be possible to bracket extremes of relative permeability (as caused by permeability variation, capillary imbibition, permeability variation, capillary imbibition, crossflows, viscous fingering, etc.) and to limit the range of performance predictions. In the following development of theory, I emphasize the water-drive casethat is, the case for a wetting fluid displacing a nonwetting fluid from the reservoir. Flow is treated as one-dimensional and horizontal so that there are no gravity effects. First, the formulas are given for two parallel capillaries, and then the results are generalized for any number of capillary channels. Sample calculations were made to illustrate the effects of flow parameters on the shape of the displacement front. In the dynamic system, these sample calculations become numerical experiments in fluid flow. Moreover, these numerical experiments did not give entirely the expected results for water drive at an unfavorable mobility ratio. The gas-dove case (nonwetting fluid displacing a wetting fluid) has been studied in an analogous manner. SPEJ P. 467

New Pseudo Functions To Control Numerical Dispersion

Abstract This paper presents an improved procedure for calculating dynamic pseudo junctions that may be used in two-dimensional, areal reservoir simulations to approximate three-dimensional reservoir behavior. Comparison of one-dimensional areal and two-dimensional vertical cross-sectional results for two example problems shows that the new pseudos accurately transfer problems shows that the new pseudos accurately transfer the effects of vertical variations in reservoir properties, fluid pressures, and saturations from the properties, fluid pressures, and saturations from the cross-sectional model to the areal model. The procedure for calculating dynamic pseudo-relative permeability accounts for differences in computing block lengths between the areal and cross-sectional models. Dynamic pseudo-capillary pressure transfers the effects of pseudo-capillary pressure transfers the effects of different pressure gradients in different layers of the cross-sectional model to the areal model. Introduction Jacks et al. have published procedures for calculating dynamic pseudo-relative permeabilities fro m vertical cross-section model runs. Their procedures for calculating pseudo functions are procedures for calculating pseudo functions are more widely applicable than other published approaches. They demonstrated that, in some cases, the derived pseudo functions could be used to simulate three-dimensional reservoir behavior using two-dimensional areal simulators. For our purposes, an areal simulator is characterized by purposes, an areal simulator is characterized by having only one computing block in the vertical dimension. The objectives of this paper are to present an improved procedure for calculating dynamic pseudo functions, including a dynamic pseudo-capillary pressure, and to demonstrate that the new procedure pressure, and to demonstrate that the new procedure generally is more applicable than any of the previously published approaches. The new pseudos previously published approaches. The new pseudos are similar to those derived by jacks et al. in that they are calculated from two-dimensional, vertical cross-section runs. They differ because (1) they account for differences in computing block lengths between the cross-sectional and areal models, and (2) they transfer the effects of different flow potentials in different layers of the cross-sectional potentials in different layers of the cross-sectional model to the areal model. Differences between cross-sectional and areal model block lengths are sometimes desirable to reduce data handling and computing costs for two-dimensional, areal model runs. For very large reservoirs, even when vertical calculations are eliminated by using pseudo functions, as many as 50,000 computing blocks might be required in the two-dimensional areal model to minimize important errors caused by numerical dispersion. The new pseudos, of course, cannot control numerical pseudos, of course, cannot control numerical dispersion in the cross-sectional runs. This is done by using a sufficiently large number of computing blocks along die length of the cross-section. The new pseudos then insure that no additional dispersion will occur in the areal model, regardless of the areal computing block lengths. Using this approach, the number of computing blocks in the two-dimensional areal model is reduced by a factor equal to the square of the ratio of the block lengths for the cross-sectional and areal models. The new pseudos do not prevent some loss in areal flow-pattern definition when the number of computing blocks in the two-dimensional areal model is reduced. A study of this problem and associated errors is beyond the scope of this paper. Our experience suggests that, for very large reservoirs with flank water injection, 1,000 or 2,000 blocks provide satisfactory definition. Many more blocks provide satisfactory definition. Many more blocks might be required for large reservoirs with much more intricate areal flow patterns. The next section presents comparative results for cross-sectional and one-dimensional areal models. These results demonstrate the reliability of the new pseudo functions and illustrate their advantages pseudo functions and illustrate their advantages over previously derived pseudos for certain situations. The relationship between two-dimensional, vertical cross-sectional and one-dimensional areal reservoir simulators has been published previously and will not be repeated here in any detail. Ideally, the pseudo functions should reproduce two-dimensional, vertical cross-sectional results when they are used in the corresponding one-dimensional areal model. SPEJ P. 269

Pseudo-Relative Permeability for Well Modeling

Several authors have described the use of pseudo-relative-permeability curves for modeling pseudo-relative-permeability curves for modeling the vertical performance of formation flow in areal reservoir simulators. The pseudo curves are usually determined from small cross-section or three-dimension models and then incorporated into large-scale areal models. The concept of pseudo-relative permeability can also be extended to the vertical performance of individual wells by a technique analogous to that for formation flow. In a cross-section model, the production of phase p from a well with nc completions can be expressed as(1) Eq. 1 follows the form of the well potential equation of Ref. 4. For an areal model in which the well has only one completion, the production is(2) The cross-section model can be constructed with the vertical detail necessary to account for phase segregation, partial penetration, etc. To obtain the same well performance in the areal model, we desire(3) or combining Eqs. 1 through 3,(4) If the cross-section model has been divided into nl layers, then an areal model block represents a composite of the nl layers and a well a composite of nc completions. Therefore, the following approximations are made.(5) and(6) Combining Eq. 4 through 6, the pseudo-relative permeability for the areal model well is obtained: permeability for the areal model well is obtained:(7) The corresponding saturation is(8) The wellbore pressure in the areal model, PwA, can be assumed to be an appropriate average of the Pw calculated in the cross-section model. Eqs. 6 Pw calculated in the cross-section model. Eqs. 6 and 8 am appropriate for an areal block equivalent to one column of the vertical model. If the areal block is larger, then the pressure and saturation must be averaged over the volume encompassed by the areal block. P. 7

Grand Forks - Modelling a Three-Dimensional Reservoir With Two-Dimensional Reservoir Simulators

Abstract The purpose of this paper is to discuss techniques that can be used to simulatea three-dimensional reservoir using two-dimensional reservoir simulators. Thetechniques were applied in a two-dimensional areal simulation of waterfloodingin the Grand Forks Lower Mannville D Pool to account for gravity segregationand water coning phenomena. A coning correlation was developed for the range of rates expected in the arealmodel using a two-dimensional radial coning model. The coning correlation andpseudo relative permeability curves for modelling vertical segregation wereincorporated into the areal model to match reservoir history and predictwaterflood performance. The predicted water-oil ratios using the coning'correlation appear to be reasonable when compared with the water-oil ratioscalculated using the laboratory relative permeability curves – the usualassumption in areal modelling. It is concluded from this study that it ispossible to generate pseudo relative permeability curves to approximate waterconing in areal simulation of a three-dimensional reservoir. In developingthese curves velocity sensitivity should be evaluated. Introduction NUMERICAL SIMULATION MODELS are currently available to most reservoir engineersfor simulation of oil and gas reservoir performance. In studying a reservoir.where numerical simulation is justified, the engineer must first select theappropriate model. The model selected could vary from a one-dimensionalsingle-phase model to a three-dimensional, fully compositional model. Generally, study costs are directly proportional to the number of dimensionsand phases or components modelled; therefore, in the case of athree-dimensional problem it is desirable to use a two-dimensional simulatorproviding, of course, that three-dimensional performance can be adequatelyapproximated. In simulating three-dimensional performance with two-dimensional areal models, flaw and fluid distribution in the vertical direction and individual wellperformance must be approximated. Several authors(1–3) havesuggested using pseudo relative permeability and pseudo capillary pressurecurves in two-dimensional areal models to approximate flow and fluiddistribution in the vertical direction. However, except for van Poollen etal.(4) who discussed correspondence been model and measuredpressures, none of these investigations were concerned with approximatingindividual well performance in areal models. Modelling individual well performance is perhaps the most difficult problemencountered in two-dimensional areal simulation of a three-dimensional problem.Initial bottom water, slumping or under-running of injected water can result inwater caning, which is difficult to simulate on two-dimensional areal orthree-dimensional models because block geometry does not adequately definesaturation or pressure distribution near the wellbore. Coning behaviour is bestsimulated with two-dimensional radial coning models, which, unfortunately, cannot predict areal conformance. Mrosovsky and Ridings(7) suggested coupling a three-dimensionalmodel and a radial coning model to simulate coning in a field-wide study. Thisapproach would probably be prohibitively expensive for simulation studies withlarge numbers of wells. One could use analytical approximations such as theradial flow equation or correlations proposed by Sobosinsky(8) or Telkov(9) to approximate coning behaviour. However, to model coningthese approximations must be calibrated to actual field performance forreliable results.

The Modeling of a Three-Dimensional Reservoir with a Two-Dimensional Reservoir Simulator-The Use of Dynamic Pseudo Functions

Abstract Dynamic pseudo-relative permeabilities derived from cross-section models can be used to simulate three-dimensional flow accurately in a two-dimensional areal model of a reservoir Techniques are presented for deriving and using dynamic pseudos that are applicable over a wide range of rates and initial fluid saturations. Their validity is demonstrated by showing calculated results from comparable runs in a vertical cross-section model and in a one-dimensional areal model using the dynamic pseudo-relative permeabilities and vertical equilibrium (VE) pseudo-capillary pressures. Further substantiation is provided by showing the close agreement in calculated performance for a three-dimensional model and corresponding two-dimensional areal model representing a typical pattern on the flanks of a large reservoir. The areal pattern on the flanks of a large reservoir. The areal model gave comparable accuracy with a substantial savings in computing and manpower costs. Introduction Meaningful studies can be made for almost all reservoirs now that relatively efficient three-dimensional reservoir simulators are available. In many instances, however, less expensive two-dimensional areal (x-y) models can be used to solve the engineering problem adequately, provided the nonuniform distribution and flow of fluids in the implied third, or vertical, dimension of the areal model is properly described. This is accomplished through the use of special saturation-dependent functions that have been labeled pseudo-relative permeability (k ) and pseudo-capillary pressure permeability (k ) and pseudo-capillary pressure (P ) or, for simplicity "pseudo functions", to distinguish them from the conventional laboratory measured values that are used in their derivation. Two types of reservoir models have been suggested in the past to derive pseudo functions: the vertical equilibrium (VE) model of Coats et al., which is based on gravity-capillary equilibrium in the vertical direction; and the stratified model of Hearn, which assumes that viscous forces dominate vertical fluid distribution. Neither of these models accounts for the effects of large changes in flow rate that take place as a field is developed, approaches and place as a field is developed, approaches and maintains its peak rate, and then falls into decline. This paper presents an alternative method for developing pseudo functions that are applicable over a wide range of flow rates and over the complete range of initial fluid saturations. The functions may be both space and time dependent and, again for clarity and convenience in nomenclature, we have labeled them "dynamic pseudo functions". Description Of Pseudo-Relative Permeability Functions Permeability Functions Methods for developing pseudo functions have been presented in the literature. The distinction between our method and those used by others lies in the technique for deriving the vertical saturation distribution upon which the pseudo-relative permeabilities are based. In our approach, the permeabilities are based. In our approach, the vertical saturation distribution is developed through detailed simulation of the fluid displacement in a vertical cross-section (x-z) model of the reservoir. The simulation is run under conditions that are representative of those to be expected during the period to be covered in the areal model simulations. period to be covered in the areal model simulations. Results of the cross-section simulation are then processed to give depth-averaged fluid saturations processed to give depth-averaged fluid saturations (S ) and "dynamic" pseudo-relative permeability values (k ) for each column of blocks in the cross-section model at each output time. The above approach can result in a different set of dynamic pseudo functions for each column of blocks due to differences in initial saturation, rate of displacement, reservoir stratification, and location. However, differences between columns are frequently minor or they can be accounted for by correlation of the data. In this and several other reservoir studies, it was possible to reduce the complexity of the pseudo function sets through correlations with initial fluid saturations and fluid velocities.

A Numerical Simulation of Kaybob South Gas Cycling Projects

In this study of the relative performance of downdip, line-drive, and crestal injection schemes, a three-dimensional, three-phase model was used to predict project performance and two-dimensional areal and cross-sectional models were used for supplementary calculations. The results of the study made it clear that a different scheme would have to be used with each of the two units in question. Introduction The Kaybob South field is located approximately 160 miles northwest of Edmonton, Alta., Canada. The Beaverhill Lake A pool was discovered in 1961, and subsequent development resulted in the formation of three Unit areas (Fig. 1). In the first of these Units (Unit 1), a gas cycling scheme was started in 1968. This was necessary because the reservoir fluid exhibits retrograde condensation and is extremely rich in condensate and sulfur. (The simulation of a gas cycling project for Unit 1 has already been discussed in a paper that presents a concise description of the development history, geology, data preparation and predicted performance of the area.) predicted performance of the area.) Two subsequent Units (Nos. 2 and 3) were formed to the south of Unit I as a result of additional drilling. Governmental authorities approved cycling schemes with throughputs of 170 MMcf/D and 445 MMcf/D for Units 2 and 3, respectively, with an initial voidage replacement of about 52 percent of withdrawals. To predict better the performance of these two Units predict better the performance of these two Units under a broader range of cycling pattern alternatives, more sophisticated mathematical models were required than those used for Unit 1. The inclusion of gravity override effects in this new work caused a greater areal dispersal of the injected dry gas fronts. To ensure that the results of this study were true to reality, a group of associated studies was performed before the proposed cycling alternatives were investigated. The results of these auxiliary studies provided:interference and water influx information,pressure drawdown restrictions due to water coning,pseudo relative permeability relationships,reasonably sized grid spacings, anddata on vertical communication and gravity effects. These results were incorported into the main study of cycling pattern alternatives. We shall describe here the work performed and present the results obtained. Reservoir Description Following the formation of Unit 1, additional drilling defined the eastern and southern limits of the reservoir. The Beaverhill Lake gas pool extends approximately 32 miles and currently contains 57 gas wells in the area defined as Units 2 and 3. In this area the reservoir can be geologically divided into two stages of reef growth. The two stages have distinctive lithologies and are separated by a correlatable green shale marker that averages about 5 ft in thickness and is referred to as the "B" shale marker (Fig. 2). The porous layer beneath this shale consists of a medium to coarsely crystalline dolomite. The upper porous unit is distinguished by a fine to medium porous unit is distinguished by a fine to medium crystalline matrix. Initial formation pressure data indicate that a gradient exists from north to south across the pool. This pressure gradient, along with a depressed gas-water contact in most of the Unit 3 area, led to the postulation of interreef interference. There appears to be communication with the Pine Creek and the Pine Northwest fields, as illustrated in Fig. 2. JPT P. 1253

Simulation of Stratified Waterflooding by Pseudo Relative Permeability Curves

In two-dimensional areal numerical simulation of injection projects, the use of relative permeability curves that do not properly describe vertical effects in the reservoir call lead to incorrect results. Here is a way to calculate carves that reflect reservoir stratification and that can be used in two-dimensional reservoir simulators to approximate the three-dimensional solution. Introduction Numerical reservoir simulation has seen increasing use as a reservoir engineering tool. An important consideration in using numerical models, particularly in studies involving pressure maintenance, is the vertical definition of the reservoir. An accurate description of vertical sweep efficiency is important if the model is to simulate field performance. The only rigorous way of modeling vertical effects is to use a three-dimensional reservoir simulator, but, this technique is not always practical. To avoid excessive data preparation time and computing expense it is preferable to use a two-dimensional areal model. preferable to use a two-dimensional areal model. Unfortunately, a two-dimensional areal calculation implies uniform reservoir properties and fluid saturations throughout the reservoir thickness - an assumption that clearly is incorrect for most reservoirs. This means the input data to the two-dimensional model must be adjusted so that it will approximate vertical effects. Previous authors have shown that, where vertical effects are dominated by gravity segregation, two-dimensional areal reservoir simulators can be used to approximate the three-dimensional solution through the use of appropriate pseudo relative permeability and capillary pressure functions. permeability and capillary pressure functions. However, vertical sweep may be dominated, not by gravity forces, but by viscous flow forces due to vertical permeability variation. This paper presents a method for developing pseudo relative permeability curves for two-dimensional simulation of fluid displacement projects where vertical sweep is primarily affected by permeability variation. primarily affected by permeability variation. A common method for approximating the effect of vertical permeability variation in displacement projects such as waterflooding is to assume that the projects such as waterflooding is to assume that the reservoir is stratified. To obtain a general model, the stratified model, which describes the vertical efficiency, is usually combined with a method for approximating areal sweep. Experience has shown that such a model can successfully simulate and predict the performance of pattern waterfloods. However, such a model may not be applicable when the well patterns are not uniform or when there is significant areal variation in reservoir properties. In such cases, a two-dimensional numerical model is convenient for simulating reservoir performance, using the pseudo relative permeability performance, using the pseudo relative permeability concept presented here. The pseudo relative permeability functions are based on a mathematical model for calculating vertical efficiency using a stratified reservoir concept. The stratified model is basically similar to those described by Hiatt and by Warren and Cosgrove. Viscous rather than gravity forces are assumed to dominate the vertical sweep. The model is thus more applicable to water-drive than to gas-injection projects where density differences may be large. The theoretical basis for using the pseudo relative permeability curves in numerical simulators can be permeability curves in numerical simulators can be seen by analysis of the stratified model. JPT P. 805

The Use of Vertical Equilibrium in Two-Dimensional Simulation of Three-Dimensional Reservoir Performance

Abstract This paper discusses the use of the Vertical Equilibrium (VE) concept in simulating heterogeneous reservoirs. Where VE criteria are met, this technique allows two-dimensional (2-D) simulation of three-dimensional (3-D) problems with equivalent accuracy, and with attendant substantial savings in data preparation and machine time. The paper presents the VE concept itself and a new dimensionless group as a possible criterion for the validity of VE as applied to thick reservoirs or to reservoirs where the capillary transition zone is a small fraction of thickness. A description of the generation of the appropriate pseudo relative permeability and capillary pressure curves is permeability and capillary pressure curves is presented. presented. In addition to the dimensionless group criterion, an actual comparison of the results of an x-z cross-section and a one-dimensional (1-D) areal run with VE illustrates the validity of the VE concept. Numerical results of such a comparison along with the attendant machine-time requirements are presented. More than an order of magnitude difference in machine-time requirements was experienced. Finally, an actual field case example shows the utility of VE as applied to a reservoir containing one or multiple gas pools residing on a common aquifer. Introduction Numerical simulation of reservoir performance currently encompasses a wide variety of recovery processes, reservoir types and purposes or questions processes, reservoir types and purposes or questions to which answers are sought. A feature common to virtually all reservoir simulation studies, however, is the choice of simulation in one, two or three dimensions. Most frequently this choice is one between an areal (x-y) study and a 3-D study. While the areal study is considerably cheaper than a 3-D simulation, the validity or accuracy of the former is often questioned in light of its apparent inability to simulate flow and fluid saturation distributions in the vertical direction. Areal studies are frequently performed with little attention to or understanding of the extent to which the x-y calculations do or can be made to account for this vertical flow and fluid distribution. Previous papers describe a VE assumption or concept which leads to the definition of pseudo relative permeability and capillary pressure curves to be used in areal studies to simulate 3-D flow. A dimensionless group proposed as a criterion for the assumptions validity primarily treats the case of a reservoir where the capillary transition zone is an appreciable fraction of reservoir thickness. This paper neats the case of a reservoir where the capillary capillary transition zone is a small fraction of reservoir thickness (e.g., less than 10 percent). We propose to describe the VE concept as percent). We propose to describe the VE concept as applied to thick reservoirs or to reservoirs where capillary transition zone is a small fraction of thickness; to describe the generation of appropriate relative permeability and capillary pressure curves for such reservoirs to represent 3-D performance by 2-D areal calculations; to propose a new dimensionless group as a criterion for the VE assumptions' validity, obtained from an analysis of countercurrent gravity segregation; and finally, to present a cross-sectional vs 1-D (VE) comparison and a 2-D areal field case study. THE VERTICAL EQUILIBRIUM CONCEPT Most oil and gas reservoirs extend distances areally which are at least two orders of magnitude greater than reservoir thickness. Viewed in perspective, these reservoirs appear as "blankets" perspective, these reservoirs appear as "blankets" For a variety of reasons, some valid and some invalid, numerical simulations of such reservoirs are performed occasionally in three dimensions as opposed to only two areal (x-y) dimensions. SPEJ P. 63