Layer | Fill | Outline |
---|
Map layers
Theme | Visible | Selectable | Appearance | Zoom Range (now: 0) |
---|
Fill | Stroke |
---|---|
Collaborating Authors
Results
Coats, K.H.; SPE; Intercomp Resource Development and Engineering Inc. Simulation of complex recovery processes often is performed using a single pattern, which may be a five-, seven-, or nine-spot. This approximation of pattern, which may be a five-, seven-, or nine-spot. This approximation of a total-field multiple-pattern development by a representative pattern significantly reduces computing time and cost. This Forum is limited to a single pattern in a network of repeated five- or nine-spot patterns in an areally homogeneous formation. The formation may have arbitrary vertical heterogeneity and any number of layers. The single pattern contains two wells in the five-spot case and four wells in the nine-spot case. The four sides of the square pattern are no-flow or insulated boundaries. The single-pattern simulation should be performed for the smallest symmetrical element of the pattern to minimize computing cost. This well-known element is one-eighth of a pattern. Nevertheless, simulations of one-half and one-fourth pattern elements with areal homogeneity and square (uniform) grid blocks continue in practice and occasionally appear in the literature. Owing to symmetry within the pattern, the pressure (and saturation, composition, etc.) distribution within the one-eighth element defines the distribution throughout the pattern. Transmissibility and block PV alterations necessary for exact reproduction of five- or nine-spot results by a one-eighth pattern calculation are well-known for the five-point difference scheme and are omitted here. This Forum describes a simple alteration necessary for identity of one-half or one-fourth pattern and one-eighth pattern results using the nine-point difference scheme with either parallel or diagonal grids. Figs. 1a and 1b show parallel and diagonal, uniform block-centered grids for the five- or nine-spot. The wells marked by an empty circle are absent in the five-spot. The x-and y-directions and (nine-point) diagonal transmissibilities discussed in detail by Yanosik and McCracken are noted on Fig. 1a by the symbols x, y and d, respectively. Let the normal five-point scheme x- or y-direction transmissibility associated with Grids 1a and 1b be normalized to 1.0. For the nine-point scheme, then, the x-and y-direction transmissibilities are two-thirds and the diagonal transmissibilities are one-sixth. Because of the half-blocks along the horizontal (vertical) edges, the nine-point x(y)-direction transmissibilities along these edges are one-half of two-thirds or one-third. For the one-eighth pattern cases, the nine-point diagonal transmissibilities along the 45 triangle sides must be halved. This one-half factor is necessary simply because the 45 triangular boundary is a streamline. However, all internal nine-point x(y)-direction transmissibilities must remain equal to two-thirds. Obviously, all diagonal or x(y)-direction transmissibilities connected to blocks outside the one-eighth pattern grid are zero. This identity of one-eighth and full-pattern nine-point simulations is not limited to the uniform grids shown. For the diagonal grid, any N x N grid with variable x(I), y(J) can be used provided y(J)= x(I) for each I=J. For the parallel grid, any N x N grid (on a one-half pattern basis) suffices, provided that y(J)= x(I) for each I=J and that the y(J) are symmetrical about A-A. Again, the only requirement for identity of nine-point one-eighth and full-pattern simulations is halving the normal diagonal transmissibilities along the 45 edges in the one-eighth pattern case. For the square one-half and one-fourth pattern elements, use of the one-eighth pattern element reduces computing time by factors of roughly four and two, respectively. Thus, the importance of simulating the triangular elements is obvious. These ratios can be larger if direct solution is used since matrix bandwidth increases from the one-eighth to one-fourth or one-half pattern simulations and direct solution time is roughly proportional to the square of bandwidth. p. 902
Abstract This paper describes a three-dimensional, highly implicit numerical model for simulating steamflooding with distillation or solution gas. The model uses direct solution to solve simultaneously three and four equations for the dead oil and two-component oil cases, respectively. The model is compared in stability and computing time with a model reported earlier. The paper includes comparative discussion of alternate steamflood model formulations, one of which we have adopted as a highly stable, isothermal, black-oil model formulation. Introduction A brief review of published descriptions of steamflood models is given in an earlier paper. That paper described a partially compositional, three-dimensional model that solves first a single-variable pressure equation, then two simultaneous saturation equations. In our experience with dead-oil steamflood problems, that model exhibits adequate stability in most cases and marginal stability in some cases. In some compositional problems, the formulation of that model leads to problems, the formulation of that model leads to deteriorating material balances for light hydrocarbon components. The model described here was developed to gain improved stability for all types of steamflood problems and to eliminate the material balance problems and to eliminate the material balance shortcoming of the earlier model formulation in compositional problems. This highly implicit, three-dimensional model treats oil as a two-component mixture to accommodate problems involving solution or inert gas or distillation. The model simultaneously solves three equations for the dead-oil case and four equations for the compositional case. Transmissibilities, capillary pressures, and production terms are treated pressures, and production terms are treated implicitly in saturations and composition; they also are treated implicitly in temperature in grid blocks where no free gas is present. The term "implicit" refers to evaluation of interblock flow terms and production rates at the new time level, n + 1. We have found insensitivity to explicit or implicit dating of molar densities and viscosities in these terms and therefore simply evaluate them explicitly. We evaluate relative permeabilities at time level n + 1 by the first-order permeabilities at time level n + 1 by the first-order approximation, which ignores second- and higher-order Taylor series terms . Temperature dependence of relative permeability, if present, is treated explicitly. We present the model equations, and describe the highly implicit formulation and method for solution. This model is compared with the earlier steamflood model in stability and efficiency through discussion and example field problems. MODEL DESCRIPTION BASIC EQUATIONS The model consists of five equations expressing conservation of energy, conservation of mass, and phase equilibrium. The mass conservation equations phase equilibrium. The mass conservation equations apply to water and to two hydrocarbon components. In finite-difference form, these equations are Energy Balance (1) Mass Balance on H2O (2) SPEJ P. 369
- Well Completion > Completion Installation and Operations (1.00)
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Improved and Enhanced Recovery > Thermal methods (1.00)
Abstract This paper describes a three-dimensional numerical model for simulating steam-injection processes. The model accounts for solution gas and steam distillation of oil. The relative-permeability treatment presented includes a flexible but simple representation of temperature dependence and a history-dependent hysteresis in gas relative permeability. Since computational stability is a major difficulty in steamflood simulation, an implicit treatment of transmissibilities and capillary pressure is presented in detail. Model applications include comparisons with laboratory data, sensitivity experiments, and a field steam-injection test. Introduction Shutler and Abdalla and Coats described two-dimensional, three-phase flow numerical models for simulating steam-injection processes. Weinstein et al. described a one-dimensional model that accounted for steam distillation of oil. Coats et al. described a three-dimensional steamflood model that neglected steam distillation of oil, release of solution gas at elevated temperatures, and temperature dependence of relative permeability. This paper describes an extended formulation that includes these three phenomena and uses a more implicit treatment of capillary pressures and transmissibilities in the fluid-saturation calculations. The extended formulation represents a step toward a fully compositional thermal model without incurring the computational expense of the latter. The relative-permeability treatment described includes a rather flexible but simple representation of temperature dependence and incorporates a hysteresis in gas-phase relative permeability that varies with the historical maximum grid-block gas saturation. The phase-behavior representation is the weakest element of this work. We have found insufficient data relative to PVT behavior of a heavy-oil/steam system to justify sophisticated schemes of the type used in isothermal hydrocarbon systems. The PVT treatment presented is the simplest we could construct subject to the objectives of "directional correctness," reasonable quantitative accuracy, and ability to obtain required parameters from laboratory data either normally parameters from laboratory data either normally available or readily determinable. Model results presented include a comparison with laboratory data for a steamflood of a distillable oil; sensitivity results indicating effects and relative importance of various types of input data; and a comparison between calculated and observed injection rates for a Cold Lake (Alta.) steam-injection test. The latter is of interest in regard to reservations we have had regarding a model's ability to predict steam-injection rates into virtually immobile oil (100,000 cp). The field-test data showed initial and sustained steam-injection rates of 1,400 STB/D (cold-water equivalent). We discuss several reservoir-fluid parameters that had little effect and one independently measured parameter that had a pronounced effect on the calculated injection rate. pronounced effect on the calculated injection rate. MODEL DESCRIPTION The model consists and sewn equations expressing conservation of energy, conservation of mass, and constraints on sums of liquid and gas phase mol fractions. The mass-conservation equations apply to water and to each of three hydrocarbon components. In finite-difference form these equations are the following. SPEJ P. 235
- North America > United States (0.28)
- North America > Canada (0.28)
Coats, K.H., Member SPE-AIME, Intercomp Resource Development and Engineering, Inc., Houston, Texas George, W.D., Chu, Chieh, Member SPE-AIME, Getty Oil Co., Houston, Tx. Marcum, B.E., Member SPE-AIME, Getty Oil Co., Los Angeles, Calif. Abstract This paper describes a three-dimensional model for numerical simulation of steam injection processes. The model describes three-phase flow processes. The model describes three-phase flow of water, oil, and steam and heat flow in the reservoir and overburden. The method of solution simultaneously solves for the mass and energy balances and eliminates the need for iterating on the mass transfer (condensation) term.Laboratory data are reported for steamfloods of 5,780-cp oil in a 1/4 five-spot sand pack exhibiting three-dimensional flow effects. These experiments provide additional data for checking accuracy and provide additional data for checking accuracy and assumptions in numerical models. Comparisons of model results with several sets of experimental data indicate a need to account for effects of temperature on relative permeability. Calculated areal conformance of a steamflood in a confined five-spot depends strongly upon the alignment of the x-y grid axes relative to the diagonal joining injection and production wells. It has not been determined which, if either, of the two grid types yields the correct areal conformance.Model calculations indicate that steamflood pressure level strongly affects oil recovery. pressure level strongly affects oil recovery. Calculated oil recovery increases with decreasing pressure level. An example application illustrates pressure level. An example application illustrates the ability of the model formulation to efficiently simulate the single-well, cyclic steam stimulation problem. problem Introduction The literature includes many papers treating various aspects of oil recovery by steamflooding, hot waterflooding, and steam stimulation. The papers present laboratory experimental data, field papers present laboratory experimental data, field performance results, models for calculating fluid performance results, models for calculating fluid and heat flow, and experimental data regarding effects of temperature on relative permeability. The ultimate goal of all this work is a reliable engineering analysis to estimate oil recovery for a given mode of operation and to determine alternative operating conditions to maximize oil recovery.Toward that end, our study proposed to develop and validate an efficient, three-dimensional numerical model for simulating steamflooding, hot waterflooding, and steam stimulation. Laboratory steamflood experiments were conducted to provide additional data for validation. Desired model specifications included three-dimensional capability and greater efficiency than reported for previous models. Omitted from the specifications were temperature-dependent relative permeability and steam distillation effects.This paper describes the main features of the three-dimensional, steamflood model developed. Those features include a new method of solution that includes implicit water transmissibilities, that simultaneously solves for mass and energy balances, and that eliminates the need for iteration on the condensation term. Laboratory data are reported for steamfloods in a 1/4 five-spot model exhibiting three-dimensional flow effects. Numerical model applications described include comparisons with experimental data, a representative field-scale steamflood, and a cyclic steam stimulation example. REVIEW OF PREVIOUS WORK Early efforts in mathematical modeling of thermal methods concentrated on simulation of the heat flow and heat loss. Gottfried, in his analysis of in-situ combustion, initiated a series of models that solve fluid mass balances along with the energy balance. Davidson et al. presented an analysis for well performance during cyclic steam injection. Spillette and Nielsen treated hot waterflooding in two dimensions. Shutler described three-phase models for linears and two-dimensional steamflooding, and Abdalla and Coats treated a two-dimensional steamflood model using the IMPES method of solution. SPEJ P. 573
Abstract During the past decade, efforts in reservoir modeling have focused on the three areas of capability efficiency, and reliability. Capability means the ability to handle larger and more complex problems where complexity includes physical problems where complexity includes physical phenomena, such as gas percolation and variable phenomena, such as gas percolation and variable PVT properties, and severe heterogeneity due to PVT properties, and severe heterogeneity due to property variation or geometry, or both. Efficiency property variation or geometry, or both. Efficiency is increased by improving model formulations anti solution techniques to increase tolerable time-step size and reduce computer time per time step. Reliability refers to ease of use and minimum burden in selecting or experimenting with time-step size, solution technique options, iteration parameters, and closure tolerances. parameters, and closure tolerances. The single facet of a reservoir simulator that has the greatest combined influence in all three categories is the technique used to solve the large systems of equations arising from the numerical approximation of the nonlinear fluid flow equations. Available techniques include both direct solution and iterative methods such as ADIP, SOR, and SIP. Iterative methods are currently used almost to the exclusion of direct solution because of the significantly higher computer storage and time requirements of the latter. This paper describes some new ordering schemes for Gaussian elimination that reduce computing time and storage requirements by factors as large as 6 and 3, respectively, relative to more standard orderings. Computational work estimates are given for these methods, for the standard Gaussian ordering, and for several iterative methods. These work estimates are checked by comparisons of actual run times using different solution techniques. Numerical examples are given to illustrate the increased efficiency and reliability that can be achieved in many cases through use of the new direct solution methods. Introduction It is well known that the way we number or order the unknowns of a sparse system of linear algebraic equations can drastically affect the amount of computation and storage for a direct solution. However, until recently the best ordering scheme that appeared in the literature numbered the points of a three-dimensional grid first along the shortest direction - i.e., the dimension with the fewest number of grid points - then in the next shortest direction, and finally in the longest direction. This ordering, which we shall call the standard ordering for Gaussian elimination is still widely used even though it is substantially slower than many other orderings. Ogbuobiri et al. present a survey of the literature related to ordering schemes that exploit matrix sparsity. These schemes are grouped into the two classes of matrix-banding schemes and optimal or pseudo-optimal schemes. The latter schemes pseudo-optimal schemes. The latter schemes purport to yield generally greater efficiency. purport to yield generally greater efficiency. In a recent paper, Georges has shown that for five-point difference approximations on square n x n two-dimensional grids, the total work for certain orderings of the grid points is less than C1n3 and the storage is less than C2n2 log n, compared with n4 and n3, respectively, for the standard ordering. Moreover, George has shown that no ordering scheme can require less work than the order of n3. For the special case of n - 21 he shows that work W is less than 10n3 and the storage S is less than 8ln2 for symmetric matrices. For nonsymmetric matrices these results become W less than 20n3 and S less than 16ln2, respectively. In this paper we describe some specific orderings in the matrix-banding class. Analyses of work and storage requirements are given for these orderings as applied to the diffusivity-type pressure equation that arises in reservoir simulation problems. These work and storage requirements are compared with those of the standard Gaussian ordering and of some iterative methods. These comparisons are performed for problems ranging from simple performed for problems ranging from simple homogeneous squares to practical reservoir problems of typical heterogeneity and irregular problems of typical heterogeneity and irregular geometry. The work requirements of the orderings presented here are also compared experimentally with those of one of the leading pseudo-optimal schemes. SPEJ P. 295
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
- Reservoir Description and Dynamics > Reservoir Simulation (1.00)
- Reservoir Description and Dynamics > Reservoir Fluid Dynamics > Flow in porous media (1.00)
- Reservoir Description and Dynamics > Reservoir Characterization (1.00)
- Reservoir Description and Dynamics > Formation Evaluation & Management (1.00)
Patton, J.T., Member AIME, Patton, J.T., Member AIME, Computer/Bioengineering Institute, Inc., Houston, Tex. Coats, K.H., Member AIME, International Computer Applications Ltd., Houston, Tex. Colegrove, G.T., Kelco Co., Houston, Tex. Abstract This experimental and numerical study was performed to estimate the incremental oil recovery performed to estimate the incremental oil recovery by pattern polymer flooding in a California viscous-oil reservoir. Results indicate that adding 270-ppm Kelzan to the normal flood water will boost oil production by 42 percent (at 1 PV injected) and production by 42 percent (at 1 PV injected) and will reduce water handling costs sharply. This corresponds to $8.35 incremental oil/$1.00 polymer injected, taking into account the 30 percent pore volume bank of polymer solution. The 28.6 percent additional oil recovery predicted at 0.5 PV injected yields a return of $4.60 incremental oil/$1.00 polymer injected. polymer injected. The field predictions are based onlaboratory measurements of polymer solution viscosity, adsorption and dispersion upon displacement by normal water in a sand representative of the reservoir, linear laboratory oil displacement experiments using brine and polymer solution, and a numerical model developed to simulate linear or five-spot polymer floods in single-layer or stratified reservoirs. The paper presents an analytical solution to the linear polymer flood problem, which provides a check on accuracy of the numerical model and a quick estimate of additional oil recovery by line-drive polymer floods. The numerical model developed indicates that additional oil recovery by polymer flooding is sensitive to polymer bank size polymer flooding is sensitive to polymer bank size and adsorption level and is insensitive to the extent of dispersion active at the trailing edge of the polymer slug. polymer slug Introduction The benefits of improving the mobility ratio, lambda o/lambda w, on waterflood performance is well documented and research on how best to effect this improvement has been considerable. Both producers and chemical manufacturers, spurred on producers and chemical manufacturers, spurred on by the vast reserves of oil which will be otherwise abandoned, have sought to resolve the problem. Currently, two types of additives are being marketed and field tested with promising results. Both additives increase oil recovery by lowering the mobility of the flood water, lambda w. However, they effect this lowering by distinctly different mechanisms. Mobility of the flood water is given by: lambdaw = kw/ w . Hence, one may elect to either increase viscosity, mu w, or decrease effective permeability, kw. Viscosity can be increased by adding small amounts of a water-soluble polymer. To be effective at the flood front this additive should exhibit minimum adsorption on the pore surfaces. Polymers showing minimal adsorption are generally a combination nonionic-anionic type. The negative charge repels the clay platelets to reduce adsorption and the nonionic portion provides the brine tolerance required for reservoir applications. A polymer of this type, Kelzan M, was chosen for the study. The alternate method of lowering mobility is equally well known. It consists of adding to the flood water a polymer designed to adsorb on the pore surfaces, thereby physically reducing the available flow area. This study was performed to estimate the additional oil recovery by pattern polymer flooding using Kelzan in a California viscous-oil reservoir. Laboratory experiments were performed to estimate polymer solution viscosity, adsorption and polymer solution viscosity, adsorption and dispersion upon displacement by normal injection water. Waterflood and polymer flood oil recovery curves were obtained for a laboratory core packed with sand representative of the reservoir. A numerical model was developed to simulate polymer floods in linear or five-spot patterns in single-layer or stratified reservoirs. An analytical solution to the linear polymer flood problem was developed to provide a quick estimate of incremental oil provide a quick estimate of incremental oil obtainable by polymer flooding and to provide a check on the accuracy of the numerical model. SPEJ P. 72
- North America > United States > Texas > Harris County > Houston (0.64)
- North America > United States > California (0.44)
Abstract This paper describes the use of a multiphase, multidimensional mathematical model to predict two- and three-phase coning behavior. Severe computational instability in the form of saturation oscillations in grid blocks near the wellbore is commonly encountered in the mathematical simulation of coning. This instability is due to the explicit (dated at the beginning of a time step and held constant for that time step) handling of saturation - dependent transmissibilities and production terms in the finite-difference solution of production terms in the finite-difference solution of the flow equations. An analysis of stability with respect to explicit handling of saturation-dependent transmissibilities is presented in this paper. This analysis shows why explicit transmissibilities can result in a severe time-step restriction for coning simulation. The use of implicit production terms in the difference equations to reduce instabilities is discussed and examples are given. These examples show that the implicit handling of production terms alone can result in a fivefold increase and permissible time step for a coning simulation with virtually no increase in computing time per time step. A laboratory water-coning experiment was simulated and excellent agreement was obtained between computed and observed results. A three-phase coning example for a gravity-segregation reservoir is also presented. Introduction Simulation of coning behavior is normally done by numerically solving the flow equations expressed in cylindrical (r, z, theta) coordinates with symmetry in the theta direction. The finite-difference technique of numerical solution of differential equations requires that the portion of the reservoir being simulated be divided into grid blocks as shown in Fig. 1. Since coning is a well phenomenon and not a gross reservoir phenomenon, the grid blocks must necessarily be relatively small in the vicinity of the wellbore because both pressures and saturations vary rapidly in this region. Severe computational instability is commonly encountered in the simulation of coning due to the relatively small grid-block sizes and high flow velocities in the vicinity of the wellbore. During a time step that would be considered normal for most reservoir simulation problems, a block near the wellbore is required to pass a volume of fluid many times its pore volume. SPEJ P. 257
- Research Report > New Finding (0.68)
- Research Report > Experimental Study (0.68)
Abstract 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
Abstract This study was performed to compare the capability and computing efficiency of successive overrelaxation (SOR) and alternating-direction (ADI) techniques in simulating pressure maintenance by water and gas injection. The calculations simulated two-phase flow and accounted for effects of capillary pressure, relative permeability, gravity and reservoir heterogeneity. The two techniques investigated were applied to the iterative, simultaneous solution of the two flow equations. Several variations of the SOR method were used: point (PSOR), point symmetric (PSSOR), line (LSOR) and line symmetric (LSSOR). The SOR methods were applied in simultaneous solution of the two partial difference equations describing the two-phase flow. Results showed that, for the oil-water simulation problems investigated here, the ADI iterative technique is superior to all variations of the SOR technique employing single relaxation factors. For all three oil-water problems the best single-value relaxation factor in the SOR technique was found to be unity. The total computing time required for simultaneous solution with ADI ranged from approximately 45 to 75 percent of that required using the best SOR technique, namely, LSOR, when the unity relaxation factor was employed in the latter technique. A significant improvement in the SOR computational requirements was obtained in the PSOR and LSOR simulation of one of the three oil-water problems-a 100 grid point two-dimensional simulation. The improved program, using combinations of relaxation factors, resulted in the reduction of LSOR computing requirements to approximately 94 percent of that required using ADI. Due to the relative complexity of the procedures involved in producing the improved SOR simulation programs, it was not considered feasible to apply these methods to the simulation of the other oil-water problems. Comparative results indicate that similar improvements in the LSOR simulation of the 300 and 625 grid point oil-water problems would still leave LSOR inferior to ADI on a computing time basis. In the simulation of a 100 grid point gas-oil cross-section, an optimized LSOR simulation using a number of relaxation factors required approximately 76 percent of the computing time that was used in the ADI simulation. The best LSOR run employing a single relaxation factor (w= 1.65) required approximately 83 percent of the ADI computing time. A satisfactory PSOR simulation of this problem could not be obtained. Introduction A variety of mathematical techniques are available for numerical solution of the partial differential equations governing multidimensional multiphase fluid flow in reservoirs. This work was performed to compare the capability and computing efficiency of two such techniques. The model (set of equations) employed simulates the three-dimensional, unsteady-state flow of two immiscible, incompressible phases and is applicable to pressure maintenance-type problems involving flank or pattern water injection or gas injection. The equations account for effects of gravity, capillarity, relative permeability and arbitrary reservoir geometry and heterogeneity. The model consists of two partial differential equations expressing conservation of mass of each flowing phase. The model equations were expressed in implicit finite difference form and solved simultaneously for the wetting and nonwetting phase flow potentials. SPEJ P. 47ˆ
- Energy > Oil & Gas > Upstream (1.00)
- Water & Waste Management > Water Management > Lifecycle > Disposal/Injection (0.34)