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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
35 results
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Object-Based Modelling of the Spatial Distribution of Fluvial Sandstone Deposits
Authors A. G. Chessa and A. W. MartinusNumerical flow simulations in hydrocarbon reservoirs are based on a detailed knowledge of the geological variability in the subsurface. When well data are sparse compared to geological variability, it is difficult to predict reservoir characteristics in the interwell area. Therefore, it is argued that data of reservoir analogues should be used to support the characterization of sparsely sampled reservoirs. An integrated approach consisting of geological data acquisition and stochastic geological modelling is presented for the problem of predicting the spatial distribution of sandstone deposits (object-based modelling). Outcrop data of the distal part of the TOrtola alluvial fan are used to estimate the parameters of a Boolean model, that is used in a simulation algorithm to generate realizations of a 2D cross-section of the studied outcrop area. The simulations are conditioned on synthetic well data and on the stratigraphic sequence in which sand bodies are deposited. The results have shown that the outcrop area is appropriately reproduced with a mean well spacing of 230 m. The proposed method should also be validated for producing reservoirs that are analogues of the Tdrtola fan. The stochastic method has several shortcomings that arise from statistical considerations.
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The Use of Correlation Statistics for Modelling Immiscible Displacements in Petroleum Reservoirs
Authors P. S. Ringrose, G. E. Pickup, J. L. Jensen and K. S. SorbieMost statistical models which are used when simulating oil reservoir performance employ a correlation function. We have found that fields in which correlation, as a function of direction, is represented only as a positive variable do not give an adequate representation of immiscible displacements in realistic geological formations. A better representation can be achieved by introducing negative correlation in one or more directions. We describe the water/oil displacement efficiency in a selection of deterministic fields, based on typical sediment bedform structures, and then show how this performance may be reproduced in random correlated fields with varying amounts of positive and negative correlation. Negative correlation needs to be considered when the heterogeneity displays significant periodicity; for example, in layered systems. Positive correlation, which represents the tendency for (local) similarity, results in favourable viscous-capillary interactions and better displacement efficiency. Negative correlation results in poorer displacement efficiency. Clastic sedimentary formations, which are characterised by contrasting layers, are better represented, statistically, by anisotropic positively/negatively correlated permeability fields in which displacement efficiency is strongly directiondependent. The implications of negative correlation for numerical flow models are also assessed.
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A Program for 3-D Modelling of Heterogeneities in a Fluvial Reservoir
Authors K. Tyler and A. Henriquezability is distributed in a fme grid with the values generated from parameters obtained from values at the wells. Finally, the permeability values are used to calculate effective permeabilities for grid blocks used in the reservoir simulator. This algorithm uses the conservation of mass and Darcy’s law in order to generate the values for the user defined corner point geometry grid used in the black oil simulator.
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Conditioning Permeability Fields by Simulated Annealing
Authors A. Ouenes, I. Bahralolom, A. Gutjahr and R. LeeA new approach for conditioning stochastic fields is proposed. In this paper, the Simulated An nealing Method (SAM) is applied to laboratory and field permeability data. A generated field of a desired rock property preserves the spatial distribution and honors the actual measured data from field and/or laboratory. In this approach, in contrast to the traditional conditional simula tion, no interpolation technique is used to smooth data at unknown locations. The least-squares objective function is the difference between the spatial variability of the generated field and that of the actual spatial structure of the known data. Hence, theoretical models for fitting experimental variograms are no longer required in this approach. The simulated annealing algorithm is tested on an exhaustive permeability data measured by minipermeameter on a slabbed carbonate core. Comparison of the generated permeability field with the actual data indicates that this approach is efficient and the conditioning is sensitive to the spatial distribution of the known points. A field application is also presented in this study. The anisotropic permeability field generated by SAM on a North Sea Oil Reservoir indicates the capability of SAM to preserve the spatial variability and to represent the uncertainty that exists more realistically than the smoothed representation used in most geostatistical approaches.
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Scaling of Lognormally Distributed Permeability
Authors Lars Holden and Oddvar LiaEffective permeability or large scale limit of lognormaily distributed permeability is calculated in this paper for different kinds of anisotropy. The effect of auisotropy both in the permeability tensor as weli as in spatial correlation structure seems to be important. A method is used to estimate the upscaled permeabilities based on Darcy’s law, and most of the results are calculated in two dimensions. The results show that a rule of thumb may be constructed for calculating the large scale limit or effective permeability without having to simulate the upscaled permeability.
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On the Object-Based Method for Simulating Sandstone Deposits
By A. G. ChessaStochastic simulations of sandstone deposits in hydrocarbon reservoirs should honour the available well data and geological knowledge about their setting. In this paper, the problem of conditioning the simulations on well data is considered. This is done in the case where a Boolean model is used to describe the spatial distribution of sandstone deposits in a reservoir (object-based modelling). It is argued that the conventional simulation algorithms generate sandstone deposits that do not possess the correct statistics for their size at the conditioning wells. The reason for this is closely related to the so-called waiting-time paradox. A statistically correct simulation approach is proposed for the problem of simulating random arcs on a circle, given that certain points on the circle are covered by arcs while other points remain uncovered. Intensity functions for the arc centres and probability distributions of an arc radius are derived for the conditional distribution of arcs on the circle. The results obtained for this problem might be partly used to solve the 2D and 3D conditional simulation problem.
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Calculating the Relative Permeability and the Capillary Pressure Functions from In Situ and Effluent Measurements: An Error Analysis
Authors Jan-Erik Nordtvedt, Magnar Aga and Kristofer KolltveitIn this paper we analyze the impact of different types of data on the accuracy of the estimates of the relative permeability and capillary pressure functions from displacement experiments. We calculate the confidence intervals around true relative permeability and capillary pressure curves, and use the area between the upper 95% confidence limit and lower 95% limit to decide which type of data has the largest impact on the accuracy of the estimate. We show that the most sensitive information is provided by the in situ pressure measurements as a function of time. Other in situ type data, such as the saturation as a function of time or several saturation profiles, proves not to be crucial for good estimate, hence the confidence intervals around the true curves for this type of data is quite wide.
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Multiphase History Matching by Finite Element Approximation in Porous and Naturally Fractured Reservoirs
Authors B. Palatnik and I. ZakirovThis paper describes an application of optimal control meth ods to a wide range of history matching problems arisen dur ing oil and gas field development. A conjugate gradient met hod is used to minimize an objective function value. To det ermine a step value along the corresponding conjugate direc tion, a new high effective procedure is set up. The procedu re significantly increases the convergence rate. A new, more accurate form of adjoint problem is obtained that allows to improve the gradient accuracy calculation for the problems with high unlinearities. The parameterization of k(x) and m(x) distributions is considered and the results of numeric al testing of the algorithms are presented.
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Geostatistical Simulation and Flow Modeling of a Fluvial Reservoir: A Case Study
Authors Antonella Godi and Luca CosentinoA geostatistical simulation study aimed at dynamic modeling was carried out on a fluvial sandstone reservoir. A number of realizations of the spatial distribution of the petrophysical properties of the reservoir was generated and tested in the numerical flow model on the basis of the production history of the field. The realization that gave the best agreement in the history match was selected for the forecast phase.
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An Application of Geostatistics in Subsurface Characterization using Hard Data and Soft Information
Authors M. M. Nobre and J. F. SykesThe reliability of flow and transport models depends on the understanding of the characteristics and impacts of spatial structures of subsurface formations. Preferential pathways are controlled by the geometries of these units. In subsurface characterization, it is the current practice to use both interpolation and extrapolation using a series of observations from well logs. Classical techniques, such as weighted interpolation and least-squares polynomial, are not recommended because they assume independence of sample data. Geostatistics methods bring a new horizon for reservoir engineering studies. Spatial dependence or autocorrelation of data exist, given the fact that neighbouring points tend to present similarities. In addition to this fact, reservoir heterogeneities are caused by usually known geologic processes such as deposition, sediment diagenesis and fracturing. Thus geologic data and models should also be helpful in characterizing spatial variability of flow properties. This work applies a Bayesian Kriging technique which includes an expertise guess with a given uncertainty in the estimation procedure. This technique assures that observation data (hard data) prevail in areas close to field measurements, whereas in areas without observations the expert’s experience (soft data) have greater influence. Maps of the estimates and the associated uncertainties are shown to be key tools in reflecting the quantity and quality of the available. Simple Kriging and Universal Kriging become subsets of this procedure.
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Stochastic Discrete Fracture Modelling of Heterogeneous and Fractured Reservoirs
Authors William Dershowitz, Neil Hurley and Ken BeenDiscrete fracture analysis provides the crucial link between the complex geometries and structural control of reservoir geology and the finite difference continua of reservoir engineering. This approach is applicable to reservoirs where heterogeneity is the result of the combination of a primary permeability defined by features such as faults, fractures, thin beds, and karsts and a secondary permeability, with primary or secondary porosity, in the rock blocks. This paper describes the implementation, validation, and application of this approach in the FracMan model, utilizing an interactive stochastic geologic simulator and a conjugate gradient fmite element solver.
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Riemann Problem for Two-Phase Four- and Morecomponent Displacement (Ideal Mixtures)
Authors P. Bedrikovetsk and M. ChumakThis paper considers the two-phase multicomponent displacement in porous media. We focus on the ideal systems (mixture volume and entropy are additive). This assumption is extensively used for oil-gas and gas-condensate systems at low pressure. For the actual n-component system the problem of the approximation by ideal mixture is solved - density and entropy of every component is found as a phenomenological parameter from the equation of state. Explicit formulae for the Riemann invariants, shock and rarefaction waves are found. Invariant solutions for one-dimensional displacement problem is obtained. Description of the displacement of oil by gases is given.
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Analytical Solution for Four Component Gas Displacements with Volume Change on Mixing
Authors B. Dindoruk, R. T. Johns and F. M. OrrAnalytical solutions obtained by the method of charac teristics are reported for the material balance equations that describe one-dimensional flow of four-component mixtnres in which components change voinme as they transfer between phases. Volume change induces variations in local flow ve locity, but we show that the variations can be decoupled from composition variations. We report example solutione for dis placements of a methane(CH4)/butane(C4)/decane(Cie) mix ture by carbon dioxide (C02) or nitrogen (N2). They show that flow behavior is controlled by three key tie lines: the in jection tie line, the initial tie line, and a crossover tie line. We calculate the eigenvalues and eigenvectors for the eigen value problem in terms of ruled surfaces of tie lines and the tie-line envelopes. That representation shows when continu ous composition variations connect pairs of key tie lines and when self-sharpening shocks are required. We also prove for ne-component systems with volume change that if there is a self-sharpening wave between two tie lines, the tie line exten sions must intersect. Finally, we show that development of miscibility can be controlied by any of the three key tie lines.
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Use of Fractional Derivatives for Fluid Flow in Heterogeneous Media
By R. LenormandAn original approach for modeling the dispersion of a chemical tracer through heterogeneous porous media is described in this paper. It is shown that, instead of introducing a time dependent dispersion coefficient, the presence of heterogeneities strongly modifies the structure of the transport equation. The presentation and the solution of this complicated equation are simplified by using the concept of fractional derivatives. Fractional derivatives are generalizations of the standard integration for a noninteger order q. In this study, the one-dimensional equation describing the transport of a tracer in a strongly heterogeneous medium (correlation at all the length’s scales) is demon strated. In such a medium, characterized by a parameter a (which can be related to a fractal dimension), the spreading width of a tracer varies as tα/², instead of t¹/² for standard dispersion, or t for pure convection in a layered medium. A gen eral transport equation for the concentration is derived using Laplace and Fourier transforms, which yield derivatives of a fractional order a in time and a second order in space. This form of transport equation can be reduced to the two limiting cases: (1) dispersive flow, which gives the standard convection-diffusion equation (a = 1), and (2), convective flow in layered media, which gives a wave equation (a = 2). The general equation represents a continuous crossover between dispersive and convective flow with only one tuning parameter (1
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Network Simulations of Flow Processes in Strongly Wetting and Mixed-Wet Porous Media
Authors S. R. McDougall and K. S. SorbieThe main objective of this work is to derive two phase flow parameters, such as relative permeability and capillary pressure, from microscopic considerations using a network modelling approach. A steady-state network method is used in order to elucidate the most important factors affecting these flow parameters under various imbibition and drainage cycles and for porous media which are either strongly wetting or mixed wet in character. The network model is used to examine how phenomena such as phase-trapping, saturation history and film-flow affect relative permeability and capillary pressure curves. Both primary and secondary displacements have been simulated and results demonstrate that the accepted way in which relative permeability curves are constructed and presented tends to disguise the underlying dynamics of the process under investigation. There is some disagreement in the literature regarding the effects of wettability on waterflood recovery efficiency. Recent studies have indicated that this may be optimal at close to neutral conditions in mixed wet porous media. By explicitly incorporating pore wettability effects into the network model, it has been possible to explain these experimental observations from a microscopic standpoint. Results are presented which show how a (the fraction of pores which are assigned oil wet characteristics) affects the calculated relative permeability curves. These relative penneabilitites have been used to calculate waterflood displacement efficiencies for a range of wettability conditions, and have been shown to be in very good qualitative agreement with related experimental results. This appears to be the first time that a network model has been used in the study of both fractionally- and mixed-wet media in order to explain waterflood perfonnancc. The implication is, therefore, that the essential physics underlying the relative permeability concept is included in our rule based steady-state simulator.
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Analytical Studies for Processes at Vertical Equilibrium
More LessSome new results are presented for systems in Vertical Equilibrium (VE) which extend our recent work on this subject (Yortsos, 1991). The linearized stability of a prototypical miscible displacement at VE is considered and shown to correspond to the short wavelength regime of previous studies in unbounded geometries. Concepts from VE are applied to model general displacement processes in various geometries. Under conditions of gravity-segregated flow we investigate and suggest simple methods for the construction of the solution. In particular, we extend our previous work to flows involving dip and show that the latter dominates the solution. Prepared for The 3rd European Conference on the Mathematics of Oil Recovery, 17-19 June 1992, Delft University of Technology, The Netherlands.
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Effect of Layer Ordering on Viscous Crossflow in Stratified Systems
Authors Dagrun Kjønsvik and Jostein AlvestadAn analytical treatment of viscous crossflow in unordered layered systems is performed under the assumption of viscous dominated vertical equilibrium (VVE) and piston-like displacement. The method developed is essentially a generalization of earlier approaches (Hearn, 1971) to unordered systems. It is shown that the fluid interface is influenced by a non-monotonic fluid front profile, leading to formation of “extra zones” of oil or water between the layer fronts. The internal ordering of layers in a system may have a significant effect on fluid crossflow and interfaces, and thereby on the reservoir performance. The procedure of ordering layers before calculating reservoir performance or VVE pseudo fliinctions (Ream 1971, El-Khatib 1985) may therefore lead to significant errors.
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Permeability Prediction from Geological Models
Authors Steven Bryant and Christopher CadeWe present a new approach to permeability modelling and prediction in sandstones. By simulating a range of geological processes which operate to form a sandstone, we obtain physically representative model porous media. The geometry of the models is completely specified, so that permeability can be calculated directly using a flow path network approach. In contrast to previous efforts to predict permeability, our approach is based on first principles. There are no adjustable parameters in the calculations, and no requirement for additional measurements or correlations (e.g. capillary pressure data or pore system data from thin sections). Model predictions match measurements on Fontainebleau sandstone samples whose permeabilities span nearly five orders of magnitude. We have also been able to correctly predict pore throat size distributions to match mercury injection measurements. Pore-scale geometrical features of the model are found to be spatially correlated, and this departure from randomness significantly affects macroscopic properties. The agreement between predictions and measurements suggests that spatial correlation is inherent in granular porous media, and that uncorrelated (or arbitrarily correlated) models of transport in such media are unlikely to be physically representative.
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A Method for Calculating Permeability Tensors using Perturbed Boundary Conditions
Authors G. E. Pickup, J. L. Jensen, P. S. Ringrose and K. S. SorbieFor reservoir simulation, it is usually necessary to represent fine-scale permeability heterogeneities by larger scale effective permeabilities. The effective permeability of a heterogeneous medium is a tensor and depends on the boundary conditions which dictate the direction of flow through the medium. We have reviewed current methods for determining effective permeability tensors, and find that existing methods either apply one type of boundary conditions, or give approximate results for a range of boundary conditions. This paper presents a new method for calculating the effective permeability tensors for single phase flow. The method is based on a pressure perturbation scheme which uses two flow cases. The first case uses boundary conditions which reflect the actual flow conditions for the medium. The second case uses a perturbation of the pressures calculated from the first case. This perturbation is applied first in the horizontal and then in the vertical directions. By using perturbed pressures, the flow is not distorted by unrepresentative boundary conditions. Each term of the effective permeability tensor is proportional to the ratio of the increment in flow to the increment in pressure gradient. This method has been tested using a variety of 2D permeability fields, both stochastic and deterministic, and gives good agreement with analytical results. We have applied the method to study the effects of no-flow boundaries in deterministic fields, representing certain types of elementary bedform. We have also investigated the effect of coarse-block size on the effective permeability in correlated random fields.
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A Dynamically Adaptive Higher Order Godunov Scheme in Two Dimensions for Reservoir Simulation
More LessCurrent reservoir simulators are generally inefficient due to the use of low order accurate schemes on regular grids. The numerical diffusion inherent in the singlepoint upstream weighting scheme can cause large errors in the predicted solutions, which can only be removed by the use of fine grids with many grid blocks. The aim of this work is to develop a method which can produce results of high order accuracy efficiently. A high order Godunov scheme is combined with grid adaptivity, where grid blocks are inserted in highly active regions of the flow field thereby concentrating the computational effort where it is most needed. The results presented demonstrate the benefits of the method in reservoir simu lation. The superiority of the adaptive high order scheme over that of the first order scheme is demonstrated. The quality of the results obtained by the adap tive scheme is comparable with those of the standard scheme, while great savings in computer time are obtained. The adaptive high order scheme also uses less computer time than the corresponding adaptive low order scheme.
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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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