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
Reservoir description data largely determine the validity of simulated reservoir performance. This paper presents a method that employs the least paper presents a method that employs the least squares and linear programming techniques to determine a reservoir description from given performance data. The method bandies multiphase performance data. The method bandies multiphase as well as single-phase flow Problems. The description parameters determined by the method may be any physical properties that influence calculated field performance. We believe The technique offers considerably greater efficiency than previously reported techniques. Example applications presented include cases of single-phase gas flow, single-phase oil flow and two-phase gas-water flow. In these particular applications the method gave accurate results with a large range of uncertainty in the reservoir parameters, and with a small number of simulation parameters, and with a small number of simulation runs. Introduction The purpose of reservoir simulation is estimation of future reservoir performance under alternative well configurations or operating conditions. This estimation is increasingly being performed using rather complex, numerical reservoir models. Reservoir description data constitute the bulk of the required input data for these models, and the accuracy of these data largely determine the validity of the calculated results. Thus an obvious problem is the determination of an accurate problem is the determination of an accurate reservoir description. We treat the problem of determining a reservoir description that, when used as input data to a reservoir simulator, results in close agreement between calculated and observed field performance. Field history or performance data are presumed available for some period of time designated the "match period". The available field history may reflect single- or multiphase, multidimensional flow, and the performance data to be matched may be any mix of observed pressures, producing rates, gas-oil and/or water-oil producing ratios. The observed field performance may correspond to a period of depletion and/or injection, or to an period of depletion and/or injection, or to an interference test. Our method for determining a viable reservoir description requires a number of runs using a reservoir simulator, each run using a reservoir description that is random within limits specified by the engineer. We then use a second, small program, that utilizes least squares and linear program, that utilizes least squares and linear programming; techniques, to process the data output programming; techniques, to process the data output from those runs to determine a reservoir description. To illustrate and test this new method, we constructed three example reservoirs experiencing single-phase gas, single-phase oil and two-phase (gas-water) flow, respectively, in two spatial dimensions. Simulator runs were made using a given set of reservoir description parameters. The results of these runs were then treated as "data" and the description parameters considered unknown. The automatic history matching method described in this paper was applied to back out description parameter values from the performance "data". parameter values from the performance "data". The agreement between these values and the true parameter values is given below. parameter values is given below. Reed et al. present an actual field case where the manual approach to matching production history proved prohibitive in both man and machine time. proved prohibitive in both man and machine time. Our least squares, linear programming technique was then used to achieve a satisfactory and economical match of the reservoir performance data. SPEJ P. 66
Dempsey, J.R., SPE-AIME, International Computer Applications Ltd. Patterson, J.K., SPE-AIME, Patterson, J.K., SPE-AIME, International Computer Applications Ltd. Coats, K.H., SPE-AIME, International Computer Applications Ltd. Brill, J.P., SPE-AIME, U. of Tulsa This simulation model permits accurate and efficient evaluation of gas field gathering system design. It provides simultaneous integration of three pressure drops - reservoir, gathering-system - associated with gas production. This complete simulation permits more accurate determinations production. This complete simulation permits more accurate determinations of deliverability than are possible with the standard on studies. Introduction It has long been recognized that gas well deliverability is a function of three pressure drops; these occur in the reservoir, in the production strings, and in the surface piping and compressor network. Actual gas well deliverability and, consequently, total field deliverability can be computed only when all three pressure drops are considered simultaneously. pressure drops are considered simultaneously. Because each of the pressure drops is associated with a different flow system, three different simulation equations are involved. To obtain meaningful results from compression studies, reservoir studies, or gas gathering system design, one must integrate these three simulation segments in such a manner that the flows and pressures balance at each node in a multiwell gathering system. The most common approach to gathering system design does not account for interwell interference and its effect on a well's deliverability. At best, the standard approach consists of imposing one or more backpressure curves on a piping network system. So long as all the wells are being produced at constant rates, this approach does not introduce large errors. However, in general, individual well rates do fluctuate for various reasons. Many systems are produced by floating part of the wells (that is, producing at capacity) and choking others, and in the course of a performance prediction many of the wells are floating performance prediction many of the wells are floating on the system in order to meet total contract obligation. When this occurs, the calculated deliverability of each well must be updated according to the transient reservoir pressures, and the appropriate backpressure of each well must be used at all times during the prediction. One shortcoming of the older approach to prediction. One shortcoming of the older approach to design studies is that a steady-state backpressure curve fixes the drainage radius of a well, and when used over long prediction periods it can introduce large errors in the determination of compression location and timing. Further, the standard approach does not -readily permit the evaluation of infill drilling as an alternative for enhancing gas-field deliverability. A rigorous approach to gathering system design must consider all the reservoir, piping, and compression data together. By subjecting this total-system description to a calculation procedure that integrates the various components, the influence of a modification to any one component is properly taken into account throughout the entire system. Consequently, compression alternatives, variations in line sizes and loops, infill drilling, and combinations of these can be evaluated while the effects of interference with the reservoir are being considered. Calculation Approach Since the flow rates and pressures must balance at each node in the system, one can choose either of these as the iterate and compute the remaining variable directly. An approach that considers flow rate as the iterate gives the best results, and the discussion below is based on the formulation. JPT P. 1067
The recovery of cushion gas upon ultimate depletion of an aquifer storage reservoir is dependent upon reservoir heterogeneity, aquifer strength, production rate, and fluid and rock properties. This study illustrates the use of multidimensional, two-phase, compressible fluid flow calculations to simulate the depletion. Results that illustrate the non-exhaustive examination of the efJects of heterogeneity, aquifer strength, and gas production rate are presented. The study indicates a strong dependence of recovery upon reservoir heterogeneity. The multi-dimensional type of calculation employed appears necessary to reliably estimate recoverable cushion gas for any particular reservoir.
Performance of gas storage reservoirs is affected by a combination of well placement and operational strategy. This paper illustrates the use of a two-dimensional, single phase dry gas reservoir simulator in the study of such a phase dry gas reservoir simulator in the study of such a reservoir. Results are presented that illustrate the effects of well spacing and operational planning on the ability of a reservoir to meet certain requirements. Introduction In the design and operation of gas storage reservoirs, the placement of wells with respect to the most advantageous operational stratagem is of prime importance. The economic consequences of failing to give adequate attention to the effects of interference among wells can be great. In addition, the impact of operational stratagems on the desired performance of the reservoir must be considered. The level of season-end performance is subject to the effects of the method of operation during the season. In this study, the placement of new wells in an existing dry gas storage reservoir is treated concurrently with the search for an operational stratagem that would permit the reservoir to meet certain withdrawal requirements. These investigations were carried out using a two-dimensional, single-phase reservoir simulator based on Eq. 1. .......(1) (The reservoir model is described in detail in the Appendix.) This study also illustrates one approach to the problem of extracting a workable reservoir description from problem of extracting a workable reservoir description from meager data. Definition of Problem The object of the study was a nearly depleted dry gas Oriskany sand reservoir that had been converted to gas storage. In operation as a storage field, the main portion of the reservoir contained 41 wells and, under a given seasonal operational plan, was capable of delivering 140 MMcf/D on the last day of withdrawal. It was required that turnover of gas be increased and that the last-day capacity be upgraded to 300 MMcf/D. This was to be accomplished by drilling additional wells and adding compressor horsepower so that field gathering-line pressure could be lowered. Plans were made for extra compression and 38 new wells, so the problem reduced to one of correct well placement with the possibility that perhaps one or more wells could be eliminated from the plan and yet the required deliverability could be met. Description of Reservoir The computing grid was superimposed on the reservoir map as shown in Fig. 1. The block size was chosen as a compromise between computing speed and definition. Also shown in Fig. 1 are deliverability areas defined from observed performance of the wells. These areas are labeled in order of quality; i.e., Area 1 contains the wells with the highest deliverability and Area 6 contains those with the lowest. These area definitions show that the reservoir decreases in quality outward from a central zone. These data, along with geologic interpretations, led to a contour map of gas-filled porosity that generally followed the same pattern. Values of porosity were entered on the grid at pattern. Values of porosity were entered on the grid at selected control points and a statistical regression technique was used to obtain values over the entire grid. Log picks indicated that a constant value of 7 ft for net pay thickness over the entire grid was reasonable. Obtaining a representation of permeability posed some problems. Core data were available for one well and problems. Core data were available for one well and pressure vs injection/production data were recorded only on a pressure vs injection/production data were recorded only on a cumulative reservoir basis. Thus, in the absence of individual well drawdown or buildup data, the only available means of obtaining a detailed reservoir description was through the use of deliverability curves for each well. Initial values of permeability were selected on the basis of flow capacity, and the regression routine was used to obtain values over the full grid, with the wells serving as control points. Using constant pressure boundary conditions, we then set up the model to compute deliverability curves for each well. We computed three points on these curves during each pass, each time restoring the grid to initial pressure and using a different limiting pressure. To shift the computed curve so as to match the observed curve, we made hand adjustments to grid block values of permeability at wells. To smooth the entire grid, we used an arithmetic moving average procedure that held the values at these control points as constant. JPT P. 1239
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