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ECMOR III - 3rd European Conference on the Mathematics of Oil Recovery
- Conference date: 17 Jun 1992 - 19 Jun 1992
- Location: Delft , Netherlands
- ISBN: 978-90-6275-785-5
- Published: 17 June 1992
21 - 35 of 35 results
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A New Method for Reducing Numerical Accumulation and Dispersion - Application to Enhanced Oil Recovery
By N. Van QuyThe numerical dispersion of components in EOR problems generally makes forecasting pessimistic or impossible. This problem has been known for a long time, and efforts are still being made to solve it. To reduce this difficulty, some authors have used highorder numerical schemes (two-point upstream weighting, Van Leer, Leonard, etc.). By their overcorrecting, these schemes often lead to a reverse error called numerical accumulation. This paper proposes a New Flux Correcting (NEC) method based on an original correction of the mass flux in each grid-block. The corrected flux will be a weighted average between the flux at the time considered (upper approximate value) and the flux at the preceding time (lower approximate value). Bounded by these limits, the corrected flux generates neither excessive numerical dispersion nor accumulation. A theoretical study was made to determine the suitable value of the weighting factor. The stability conditions of the new process have also been determined. Several examples of applications have been performed. In particular, the injection of a surfactant slug followed by polymer injection has been considered. Each oil displacement (1D or 2D) was simulated several times to test and compare the various methods of dispersion correction. The NEC method has been revealed to be very effective. It is a robust method, easily added to existing industrial simulators.
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Finite-Analytic Cubature Based Numerical Method for Reservoir Simulation
By Farek CivanThis paper presents the application of the method of integro-differential cubature for efficient solution of reservoir models. The application of this method is illustrated by solving the Buckley-Leverett model for water flooding of naturally fractured oil reservoirs. It is shown that the cubature method yields accurate solutions even using 10 times less number of grid blocks and 3000 times larger time steps than the finite difference method. In addition, the numerical dispersion problem that is apparent in the finite difference solution is eliminated.
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Explicit Representation of Flow in a Fractured Porous Medium with Boundary Fitted Coordinates
Authors A. N. Folefac, J. Wang, T. S. Daltaban and J. S. ArcherThe accuracy of representing flow in a fractured rock system depends on the ability to describe precisely the fracture geometry in terms of width, orientation and length. Such a detailed representation requires special numerical gridding techniques. In this paper we describe a new boundary fitted coordinate system in which the complex fracture and matrix shapes are transformed into a simple rectangular system in 2-D. Thus the problem of solving the flow equations in the physical plane is converted to two problems: (i) Calculating the transformation factors. (ii) Solving the flow equations in the transformed plane.
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Hydrocarbons Migration in Basin Modelling: is the Combined Use of Finite Element and Control Volume Possible?
Authors F. Schneider, S. Wolf, I. Faille, T. Gallouët and W. ChoueiriThe problem we consider in this paper concerns the treatment of hydrocarbons migration in basin simulators. Generaly, migration is computed using the classical numerical schemes developped for reservoir simulators. The classical approach has been used in order to perform a reference solution. In this paper, we present a new approach where a pressure equation is established and solved with a finite element method; the saturation equation is solved with a control volume method. We show that mixing these two numerical approaches can only be possible if the numerical scheme is localy conservative for the total flux (total flux = water flux + hydrocarbons flux). In order to respect this constraint, we suggest to distribute the fluid mass balance error on the total flux approximations. Results on several 1D examples are very close to those obtained on the reference solution. Future works are necessary to validate this approach, for 2D and 3D cases, and for compressible fluids.
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A Direct Impes-Type Volume Balance Technique for Adaptive Implicit Steam Models
Authors Eva Farkas and Péter VaIkóIn this paper an IMPES-type thermal volume balance technique is presented applying a direct solution of the set of mass and energy conservation equations. The effect of the convective energy transport can be considered in the isothermal pressure (volume balance) equation by a modification of the coefficients. The solution yields the pressure implicitly, component masses and energy explicitly (IMPEME technique) while keeping the mass, and energy balance. The explicit treatment demands timcstep limitation. Volume balance errors are used to control timesteps. Similarly to the direct isothermal volume balance approach, however, significant volume balance errors occur only at the front and at well regions. This is the portion of the reservoir in an adaptive implicit method (AIM) where the fully implicit handling is utilized. The IMPEME method is proposed to be the explicit approximation of a thermal AIM. The method can also be incorporated by black oil or compositional isothermal volume balance models in order to include thermal effects.
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Multilevel FAC and Asynchronous FAC Techniques for Reservoir Simulation
Authors R. Boyer and K. SaikoukIn order to catch localized phenomena in reservoir simulation, composite grid discretization (with local refinement) seems to be necessary. The F.A.C method (McCORMICK[1986,1990] is certainly among other methods an efficient technique to solve elliptic problems (e.g. pressure equation) on such composite mesh. In this paper we will focus on the asynchronous and multilevel version of this method; the application is developped on a waterflooding simulator. The obtained numerical results show the efficiency of such methods and, in particular, that is more efficient when used as preconditionners. We implemented also asynchronous version of this method on multiprocessor distributed memory machine iPSC.
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Acceleration Techniques for Nonlinear Domain Decomposition Methods
More LessLarge reservoirs usually consist of several distinct drainage regions. Moreover, for simu lation it may be desirable to use different grids in some regions. Such cases motivate a domain decomposition strategy at the reservoir level. The problem is split into several subproblems, corresponding to different drainage regions or grid types (or whatever other physical reasons there may be that motivate such a splitting). The subproblems can be solved separately, and by exchanging boundary information, the error at the interfaces can be reduced iteratively.
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The Influence of Rock Heterogeneity on Water Coning Below Horizontal Wells
More LessWhen producing oil from an underground reservoir, a water cone will form below the production well leading to early breakthrough. In particular for the production of thin oil nms, a good understanding and estimate of the sensitivity for water coning is required. In heterogeneous reservoirs where horizontal wells are applied, not much knowledge is present to relate heterogeneity to production behaviour. In this paper we investigate the effects of dimensionless numbers arid of reservoir heterogeneity on the breakthrough time of a horizontai production well. We develop a model for the simulation of water coning in heterogeneous oil reservoirs. We consider flow of two immiscible and incompressible fluids in a porous medium. The model consists of a coupled system of a stream function and a saturation equation. The saturation equation is solved using a finite volume method with the aid of the Godunov flux. First we validate the numerical simulator by testing convergence under grid refinement. Then the influence of the dimensionless numbers and various geological settings on the breakthrough time is considered. From the results we conclude that knowledge of the influence of reservoir heterogeneity on water coning is needed in determining optimal production strategies and to minimize the occurrences of early water breakthrough.
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A New Method for Modelling Wellbore Crossflow in Reservoir Simulators
Authors J. H. K. Masters and R. E. MottIn highly stratified reservoirs, welibore crossflow can have an important impact on reservoir behaviour, and crossflow effects need to be modelled accurately in numerical simulation. The simple approach of shutting in crossflowing layers can lead to incorrect predictions. This paper describes two methods for calculating welibore crossflow in a reservoir simulator. The first model assumes that the fluids in the weilbore are completely mixed, while the second model is a new method based on an exact solution for the fluid composition at each level in the well, using material balance calculations in the wellbore. These methods have been implemented in a generalised compositional simulator, but are equally suitable for black oil models. They can be used with both IMPES and fully implicit solution schemes. The solution of the wellbore equations is decoupled from the main solution algorithm, and does not affect the numerical stability of the simulator. An example is presented to compare results from the two models.
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Multigrid Methods Applied to Near-Well Modelling in Reservoir Simulation
Authors D. A. Collins and F. M. MouritsA new approach to multigrid computation for three-phase black-oil reservoir simulation is presented. The approach employs patch refinement techniques developed within the context of multigrid and domain decomposition methods. In this study, the approach developed involves using the coarse grid as a predictor for the fine grid in any given timestep. The fine grid solution is calculated using the coarse grid as Dirichlet boundary conditions. The fine grid is used to update the coarse grid equations in order to match fine grid behavior. Multigrid convergence techniques are applied. The method is described first in a general setting including a novel approach to local timestepping. An application of the method, employing multiple grids but not local timestepping, to a specific multigrid patch refinement problem (Collins et al 1991a) based on the Seventh SPE Comparative Solution Project on modelling horizontal wells (Nghiem et al 1991) is described.
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Modelling a Transient Flow in a Well and Coupling it with the Flow in the Reservoir
Authors C. Abchir, A. Bourgeat, V. Pelce and R. NabilIt is now widely recognized that a correct reservoir model must incorporate a precise des cription of well physics. Multiphase flow distribution in a multi-layered reservoir depends on a good description of the bottom hole pressure evolution (time and space). The conventional methods used in most reservoir simulators perform the coupling of transient flow in the reservoir area with stationnary flow in the well. The model described in this paper performs the well calculations (from the bottom hole to the well head) in transient flow conditions (gas-water flow). Thus our simulator provides both a realistic well behavior and a correct description of the well reservoir exchanges. In the case of an imposed well head pressure, starting the well-flowing involves transient flow in both well and reservoir areas. In the diphasic case, the flow in the well is modeled with a one-phase equivalent fluid. METHOD: The Euler equations (in id, assuming energy constant and taking into account the friction term) are the equations governing the well flow. The well and reservoir coupling is perfor med through the massic flow rate exchange between the two areas and the continuity rela tion of the pressures on the welibore. A new implicit scheme derived from the ROE scheme has been developped to discretise the equations. Both efficiency and interest of our method are shown in real case-studies.
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A Thirteen-Point Semi-Implicit Scheme to Reduce the Effect of Grid Orientation
Authors Y. Ding and P. LemonnierThe grid orientation effect encountered in numerical reservoir simulation has received considerable attention in the literature. Several nine-point schemes have been presented, which have not proved entirely satisfactory, especially for high mobility ratio in immiscible displacements where hornlike unrealistic fronts were observed. A thirteen-point semi-implicit fmite difference scheme is presented in this paper which greatly alleviate the gnd orientation effect without producing unrealistic saturation profiles. Theoretical analysis of the scheme is given and the relation between grid orientation and numerical dispersion is discussed. Some numerical simulations were performed. The numerical results show that the thirteen-point semi-implicit scheme is much more accurate than the nine-point one. The CPU time is almost on the same order of magnitude. The thirteen-point scheme can reduce efficiently the effect of grid orientation for high mobility ratio displacements. This new scheme can be used in any reservoir simulator. It would be very efficient for all reservoir studies with high mobility contrasts as in miscible or thermal oil recovery processes.
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The Dispersive-Fingering Transition and Flow Upscaling for Miscible Displacements in Heterogeneous Systems
Authors K. S. Sorbie, F. Feghi, G. E. Pickup, P. S. Ringrose and J. L. JensenThe nature of fluid flow through porous media is determined by the interaction between the fluid mechanics of the displacement process and the rock heterogeneity. The interplay between the various forces may be described by dimensionless scaling groups e.g. mobility ratio, capillary number, viscous/gravity ratio etc. A quantitative measui~ of rock heterogeneity is more difficult to define in real geological systems but it is central to the prediction of the fluid displacement process. In this work, the permeability heterogeneity is described by a correlated random field which is the simplest model into which we can introduce both variability and structure in a systematic way. We consider mainly viscous displacements at adverse mobility ratios in first contact miscible floods. The various flow regimes that are possible in such fluid displacement-heterogeneity combinations are described and, in particular, we focus on the fingering-dispersive transition. The results presented here extend previously reported work on this topic. In particular, the importance of certain “shape” scaling groups in detennining the global flow regime within the system are demonstrated. These scaling groups are related to the vertical equilibrium concept. The significance of the flow regime is discussed in the light of how the flow parameters in miscible displacement may be scaled up. The pseudo-isation technique, which is commonly used in two phase flow, is applied and some observations are made on the resulting averaged or “upscaled” parameters which relate to the overall flow regime in the displacement process.
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Modeling Hydraulic Fracture in Finite Difference Reservoir Simulator
More LessHydraulic fracturing is essential for the development of low-permeability reservoirs (especially gas). A successful fracturation increases several folds the well productivity. Numerical simulations are necessary to predict post-fracturing performances, but fracture representation in a reservoir model leads to some difficulties. If the fracture is modeled by cells of the exact fracture thickness (a few millimetres), an implicit scheme and many cells are required, so the computation cost is high. If the fracture is included in much larger reservoir cells, a methodology has to be defined to get similar results to the first approach. This paper presents a low-cost simulation method tested for single phase, gas-water and oil-water flow in vertical fractures. The fracture is included in cells 1 to 10-meter thick. Pseudo absolute permeabilities are computed from the reservoir and fracture permeabilities and from the geometrical characteristics. Simulation results show that formation relative permeabilities and capillary pressure data may be kept unchanged for these cells. The single phase flow results agree with analytical results. All the 2-phase simulations were performed first with the real scale of the fracture then with the fracture successively included in 1 and 10 m thick blocks. The fluid productions of these comparative simulations are close. The method applies to oil and gas reservoirs over a large pressure and permeability range. The method easily predicts the time dependency of the well productivity index.
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A Miscible Displacement Simulator with Interphase Mass Transfer Resistance
Authors Sonia Embid and Orlando R. RivasThe modeling of miscible displacement processes in petroleum reservoirs, such as Car bon Dioxide Injection or Enriched Gas Drive, has been traditionally done with composi tional simulators in which the distribution of the different components or pseudocomponents among the phases is given by equilibrium constants which are determined with a suitable Equation of State, or given as imput to the simulator in the form of tables or correlations as function of pressure, temperature or composition. In other words, the partial differeutial equations describing the flow of each phase through the reservoir are solved by finite difference techniques assuming thennodynarnic equilibrium among the phases in each grid block, every time step. It is shown here that this assumption leads to large errors since at typical conditions interphase mass transfer resistances are large and thermodynamic equilibrium in not nearly reached during the time steps of interest. In this work, a one dimensional compositional simulator is developed where mass tranfer ki netics is taken into account by including equations which describe the interfacial molar flow of each component in terms of overall mass tranfer coefficients, and the difference between the actual phase concentration and that it would have at equilibrium. Sensitivity studies are carried out to determine mass transfer resistance contributions to process performance. The simulator was validated by modelling CO2 displacement experiments carried out in a laboratory slim tube. In this case, the relative permeability curves were kept constant, and the overall mass transfer coefficient was changed to reproduce the oil production of the ex periments. The experimental results were also simulated with a conventional compositional simulator, changing the relative permeability curves to adjust oil production.
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