Summary. Three related stream injection problems are presented along with simulation results for them obtained from six organizations. The problem selected for six comparison were intended to exercise many of the problem selected for six comparison were intended to exercise many of the features of thermal models that are of practical and theoretical interest. The first problem deals with three cycles of cyclic steam injection and the other two problems deal with steam displacement in an inverted nine-spot pattern. The first two problems are of "black-oil" type and the third of compositional type. Complete data are presented for these problems. The comparison of solutions indicates good agreement for problems. The comparison of solutions indicates good agreement for most of the results of importance in field operations. Introduction Validation of reservoir simulators for complex recovery, processes is a particularly difficult problem because analytical solutions are available under only a few limiting conditions. While good agreement between the results from different simulators for the same problem does not ensure validity of any of the results. a lack of problem does not ensure validity of any of the results. a lack of agreement does give cause for some concern. Such comparisons can also be useful in the development of new models and in optimizing the performance of existing reservoir simulators. This is the fourth in a series of simulation problems for which results from a number of commercial simulators have been obtained and reported in the SPE literature. The first such study was organized and conducted by Odeh on the simulation of a three-dimensional (3D). two-phase, black-oil case. Seven companies participated in that project. This was followed by a comparative participated in that project. This was followed by a comparative solution project developed b,. the program committee for the 1982 SPE Reservoir Simulation Symposium in New Orleans. The problem selected was a three-phase, single-well radial cross-section coning problem. Eleven companies participated in this study. Chappelear problem. Eleven companies participated in this study. Chappelear and Nolen were responsible for organizing this study and for reporting the results at the symposium. A similar approach was adopted by the program committee for the 1983 SPE Reservoir Simulation Symposium in San Francisco. In this case, the problem selected was to study gas cycling in a rich-gas retrograde condensate reservoir. In the first part of the study, the participants matched their phase-behavior packages to the data supplied, and in the second part they considered two options for the depletion of the reservoir. This study required a 3D, three-phase. multicomponent compositional model. Nine companies participated in this study. Kenyon and Behie organized the project and reported the results at the symposium. The enthusiastic response of industry and the academic community to the three problems encouraged the program committee for the 1985 SPE Reservoir Simulation Symposium to continue the tradition and to develop a set of problems suitable for the comparison of steam injection models. We were given the task of organizing this project. The objective of this paper is to present selected results submitted by the participants in this project with a minimum of commentary. It is worth emphasizing that the type of comparison presented here and in previous reports is different from other kinds of-comparisons discussed in the literature (e.g., Refs. 4 and 5). In the SPE comparisons, the problems are designed by one or more knowledgeable people, and model results are provided directly by those who have built or acquired suitable models. This is different from a study where the person doing the comparison develops new software using published descriptions of several models. It is possible-although we have tried to minimize this-that some of possible-although we have tried to minimize this-that some of the differences in the results compared here could be a result of different interpretations of the problem, while differences in a com-parison of the type discussed in Ref. 4 may be caused by differences in the interpretation of the published procedures. Furthermore, the models used in our comparison are all commercial in nature. Some of these models have been in existence for several years and others are new. Problem Statement Problem Statement We have selected three related but independent problems for the comparison of steam injection models:cyclic steam injection in a nondistillable oil reservoir with a two-dimensional (2D) radial cross-sectional grid,nondistillable oil displacement by steam in an inverted nine-spot pattern by considering one-eighth of the full pattern (see Fig. 1), anddisplacement of an oil consisting of two volatile components and one nonvolatile component in the same pattern as Problem 2. The oil properties were the same in the first two problems. The participants had the option to submit results for one, two, or all three problems. In addition we invited optional runs. A complete statement of the problems as offered to the participants is contained in Appendix A. (We have deleted the section on reporting requirements to save space.)The problems were selected to exercise features of the models that are important in practical applications they do not necessarily represent real field situations. In particular. we wanted to see the influence of grid orientation on the results of the steam displacement problems. The inverted nine-spot appeared to us to be ideal for this purpose. Six companies (see Appendix B) participated in the project with only three submitting results for the compositional case (Problem 3). Four of the other companies contacted indicated an interest in the project during the early stages but were unable to provide results for the comparison for a variety of reasons. In addition to providing numerical results, the participants were also asked to providing numerical results, the participants were also asked to describe their model briefly and to answer a number of questions about the model. These descriptions with only minor editorial changes are given in the next section. Description of Models Used p. 1576
An automatic history-matching algorithm based onan optimal control approach has been formulated forjoint estimation of spatially varying permeability andporosity and coefficients of relative permeabilityfunctions in two-phase reservoirs. The algorithm usespressure and production rate data simultaneously. The performance of the algorithm for thewaterflooding of one- and two-dimensional hypotheticalreservoirs is examined, and properties associatedwith the parameter estimation problem are discussed. Introduction There has been considerable interest in thedevelopment of automatic history-matchingalgorithms. Most of the published work to date onautomatic history matching has been devoted tosingle-phase reservoirs in which the unknownparameters to be estimated are often the reservoirporosity (or storage) and absolute permeability (ortransmissibility). In the single-phase problem, theobjective function usually consists of the deviationsbetween the predicted and measured reservoirpressures at the wells. Parameter estimation, orhistory matching, in multiphase reservoirs isfundamentally more difficult than in single-phasereservoirs. The multiphase equations are nonlinear, and in addition to the porosity and absolutepermeability, the relative permeabilities of each phasemay be unknown and subject to estimation. Measurements of the relative rates of flow of oil, water, and gas at the wells also may be available forthe objective function. The aspect of the reservoir history-matchingproblem that distinguishes it from other parameterestimation problems in science and engineering is thelarge dimensionality of both the system state and theunknown parameters. As a result of this largedimensionality, computational efficiency becomes aprime consideration in the implementation of anautomatic history-matching method. In all parameterestimation methods, a trade-off exists between theamount of computation performed per iteration andthe speed of convergence of the method. Animportant saving in computing time was realized insingle-phase automatic history matching through theintroduction of optimal control theory as a methodfor calculating the gradient of the objective functionwith respect to the unknown parameters. Thistechnique currently is limited to first-order gradientmethods. First-order gradient methods generallyconverge more slowly than those of higher order.Nevertheless, the amount of computation requiredper iteration is significantly less than that requiredfor higher-order optimization methods; thus, first-order methods are attractive for automatic historymatching. The optimal control algorithm forautomatic history matching has been shown toproduce excellent results when applied to field problems. Therefore, the first approach to thedevelopment of a general automatic history-matchingalgorithm for multiphase reservoirs wouldseem to proceed through the development of anoptimal control approach for calculating the gradientof the objective function with respect to theparameters for use in a first-order method. SPEJ P. 521^
This paper describes the design and development of a steamflood pilot consisting of six inverted five-spot patterns in Section 26C of the Midway-Sunsetfield. Steam injection will be in the 330-ft Monarch sand. A steamflood simulation study indicated a potential of 60- to 70-percent oil recovery and defined the relative importance of various steamflood parameters. Introduction The reservoir characteristics and production history of Section 26C of Midway-Sunset field make it a favorable candidate for thermal recovery. Current well productivity under cyclic steaming is only about 0.05 BOPD/ft. The estimated ultimate recovery by this method is only 15 percent because of its continuously decreasing percent because of its continuously decreasing effectiveness. Underground combustion is thought to be undesirable for this section because of the reservoir's large thickness and low dip. Based on its predicted and actual performance in other heavy oil reservoirs, performance in other heavy oil reservoirs, steamflooding is the most promising thermal recovery technique for this property. The high capital and operating costs of steamflooding made a pilot desirable to evaluate the process in this field. Other reasons were uncertainties about the geological structure, sand continuity, and the role of gravity override in steamflooding a very thick reservoir. This paper describes the reservoir and its geology, the production history under cyclic steaming, pilot project production history under cyclic steaming, pilot project details, and the results of a simulation study to optimize steam injection and predict performance. Reservoir Characteristics Geology The structural feature associated with the various productive zones in Section 26C of Midway-Sunset field is a productive zones in Section 26C of Midway-Sunset field is a large southeasterly plunging nose. The main productive sands lie on the southwest flank and plunge, where the dip is about 10 deg. . Little production comes from the northeast flank, where the dip is up to 50 deg. . The two areas are separated by a thrust fault that offsets the oil-water contact. A northeasterly trending sand channel in the northern half of the section is developed and productive mainly over the northwest quarter. No production comes from the southwest comer of the section because of sand truncation. The Monarch sand, considered the main productive zone, is of Miocene age and is at an average depth of 1,300 ft. Its thickness varies from 0 to 600 ft and averages 350 ft in the main productive area. Well logs show that the sand is vertically continuous, without any significant shale breaks. A typical induction-electric log is shown in Fig. 1. However, recent cores contain a number of diatomite beds from 0.25 to 6 in. thick. Their areal extent and ability to restrict vertical fluid movement are not known. Rock and Fluid Properties The reservoir rock is an unconsolidated, poorly sorted sand with very fine to very coarse grains. The productive interval consists of turbidite beds generally 2 to 5 ft thick that exhibit normal upward fining of grain size, and that usually are separated by thin, low-permeability diatomite beds. Average porosity is 27 percent and air permeability is 520 md under reservoir conditions. The over-all net-to-gross ratio for the zone is 0.74. The rock is saturated with a 14 deg. API oil whose viscosity is a strong function of temperature (Fig. 2). Current oil saturation in the main part of the reservoir is 59 percent but only 36 percent in the depleted portions near the top. JPT P. 1559
Failure of Stream Tube Methods To Predict Waterflood Performance of an Isolated Inverted Five-Spot at Performance of an Isolated Inverted Five-Spot at Favorable Mobility Ratios Introduction Since their introduction by Higgins and Leighton in 1962, stream-tube methods (STM) have been used by a number of authors to make approximate waterflood and gas flood calculations. The principal justification for the methods is given by Higgins and Leighton, who present excellent agreement between their STM results and the experimental and numerical result of Douglas et al. for a wide range of M values in a repeated five-spot-pattern waterflood. Doyle presents fair to good agreement between STM results and laboratory model results for five-spot and direct line drive well configurations. The basic assumptions of the Higgins-Leighton STM arethat the streamlines are independent of the mobility ratio, andthat Buckley-Leverett theory can be used to calculate the fluid displacement between adjacent streamlines. The first assumption allows the streamlines to be determined from singlephase flow and to be held fixed as the flood progresses. In their original work, Higgins and Leighton used potentiometric model results to determine the potentiometric model results to determine the streamlines. Other authors, including us, have used analytical and numerical methods involving pressure and stream-function solutions. Discussion Fig. 1 presents calculated oil recovery vs displaceable pore volumes injected for an isolated inverted fivespot pore volumes injected for an isolated inverted fivespot pattern for various values of M. pattern for various values of M. JPT P. 151
·Conjugate-gradient-like (CGL) methods were used to solve the matrix equations for several large-field and fine-grid models. The methods used incomplete LU (lLU) or modified incomplete LU (MILU) preconditioning coupled with orthomin acceleration. The CGL methods were applied to the original matrix or to a reduced system from red/black ordering. Test results from a black-oil model with fault throws, a fractured reservoir model, and three steamflood models are given. It was found that the CGL methods are effective for solving the matrix equations. A more powerful preconditioner improves reliability but may require more work. Guidelines for selecting the most effective method are provided.
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