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ECMOR X  10th European Conference on the Mathematics of Oil Recovery
 Conference date: 04 Sep 2006  07 Sep 2006
 Location: Amsterdam, Netherlands
 ISBN: 9789073781474
 Published: 04 September 2006
61  78 of 78 results


A Predictor for Accelerated Coupled Rock Mechanics and Reservoir Simulation
By O. PettersenThe impact of the stress field on reservoir fluid flow and production can be significant for many kinds of reservoirs, and hence coupled Rock Mechanics and Reservoir Simulation has been seeing a growing popularity. A much used scheme is iterative coupling, where compaction is computed at each stress step by iteratively updating cell pore volumes in the reservoir simulator by values calculated from strain in the stress simulator.
Although the procedure works satisfactory, it may be slow, as often many iterations are needed. Further, the pore volume corrections will only be performed at selected stress time steps, such that pressure and compaction in the flow simulator are not continuous in time. Many reported schemes assume specific poroelastoplastic models, as e.g. linear elastic, and also require modification of code.
It is well known that compaction is a function of strain, while reservoir simulators use fluid pressure, the only compaction energy available. On this background few if any coupled procedures utilize the compaction vs. fluid relationship at all.
In this paper we show that the relationship can nevertheless be used as basis for constructing a predictor for the actual stress / strain computations, which leads to significant speedup. Many of the features of the predictor can be determined from the first stress time step only, and for later stress steps it can be improved with small effort. The scheme is valid irrespective of the poroelastoplastic model, and is based on information exchange, so no simulator code modification is necessary.
The compaction state is primarily dependent on the materials, boundary conditions, and the production process, with the geometry dependency as the governing. The predictor is constructed by modifying compaction vs. fluid pressure to take account of geometry variation. A good predictor will result in an improved pressure field as computed by the reservoir simulator, hence providing the stress simulator with a better pseudoinitialiser, such that it converges quicker, and in the pore volume iteration scheme fewer if any iterations are required.
In total we have experienced a reduction in total computer time of more than 90% in some cases, and as a bonus the fluid pressure field is continuous in time.



Modelling Induced Fracture Mechanics in Reservoir Simulations – A New Coupled Simulator
Authors D. Zwarts, B. Hustedt and P. J. van den HoekWaterinjection induced fractures are key factors influencing successful waterflooding projects. Controlling the dynamic fracture growth can lead to largely improved water management strategies and potentially to increased oil recovery and reduced operational costs (reduction in well count and water treatment facilities etc.).
The primary tool that a reservoir engineer requires to design an optimal waterflood is an appropriate reservoir simulator that is capable of handling the dynamic fracturing process in complex reservoir simulation grids.
We have developed a new modelling strategy that adds fracturegrowth to our standard fluidflow reservoir simulator. A first prototype simulator was successfully tested and applied to field cases.
The dynamic fractures are free to propagate asymmetrically in length, height, and widthdirection controlled by the pressures and poro and thermoelastic stresses acting on the fracture face. The stresses are calculated in the reservoir simulation grid. The simulator determines new fracture sizes by evaluating fracture propagation criteria, based on a Barenblatt condition on the four fracture tips: left, right, up, and down, and on a volumebalance on the width. This nonlinear fivedimensional coupled set of equations is solved every timestep using a Broyden approach moderated with LevenbergMarquardt techniques to handle nonlinearities. Since the changing fracture sizes affect the pressure and stress profiles in the reservoir, the equations are solved in an implicit scheme.



A New Model for Solution Gas Drive in Heavy Oils
Authors S. Zaleski, M. Chraibi and F. FrancoWe introduce a new Darcy scale model in which the permeability of the gas phase is replaced by a new expression for the gas phase velocity. When the bubbles are smaller that a critical radius, capillary forces anchor the bubbles and the gas phase does not move. Above the critical radius the bubbles move at a velocity of the same order as the liquid phase velocity.
The number of bubbles per unit volume may vary through coalescence between neighboring bubbles. The coalescence rate is an important parameter in the model. When coalescence is fast, the bubbles reach rapidly a large size at which they become mobile. The dependence of the coalescence rate on the fluid parameters and the bubble size is discussed.
The predictions of the model are then compared to several laboratory scale experiments.



Methodology of Fast Transmissibility Determination for MediumScale Fractures Systems
By S. VitelProductivity of fractured reservoirs is mainly affected by the fracture network configuration, which impact on the flow paths cannot be neglected. Today, two main approaches, opposite and complementary, are used to perform flow simulation on fractured reservoirs: (1) the continuum approximations, simple and efficient but approximate in the actual fracture geometry integration, and (2) the discrete model formulations, quite complex to put into practice and much slower, but respectful of the discrete fracture network and thus more accurate. Yet, fractured reservoir simulation still needs to balance the speed up of the calculation processes while keeping the fracture system paths accurate.
This paper presents a methodology to evaluate equivalent transmissibilities in naturally fractured reservoirs, based on a discrete fracture network. It is divided in two parts: (1) first the codiscretization of the fractures and the simulation grid, (2) then the determination of the equivalent transmissibility between two gridblocks of the simulation grid. The first phase consists in representing the reservoir with a connectivity list. Nodes represent control volumes, holding porosity, volume, pressure and saturation properties; pipes stand for connections between those control volumes, holding transmissibility property. First this list is extracted from the simulation grid and from the fractures that have been clipped by the gridblocks. Then both lists are connected together between each fracture piece and the surrounding gridblock. The second phase consists in computing the interblock transmissibilities from this connectivity list. These transmissibilities are evaluated for each couple of gridblocks in each direction by applying electrical simplification theorems.
The discretization tool and the simplification technique make the method really fast. Its performance is demonstrated on a 3D highly fractured reservoir. The main limitation is due to the electricity analogy limitations that does not take into account complex flow mechanisms related to fractures; even though, results show good agreement with those of a fine grid.



TwoPhase Flow in Fractured Media – Homogenized Model with Mixing and Upscaling by a StreamConfiguration Method
Authors S. Skachkov and M. PanfilovThe paper deals with the displacement process of two fluids, immiscible at the molecular scale, through a porous medium. The macroscale mixing between the phases is caused by the medium heterogeneity determined by a fracture network whose scale is much greater than the pore size, but much smaller than the reservoir size. At the heterogeneity scale the flow is described by a twophase flow model with a generalized Darcy law, classical phase permeabilities and capillary pressure. The macroscale flow model is obtained by the twoscale asymptotic homogenization method. To capture the effect of dynamic mixing, the first order model is derived.
The mixing is described by the dispersion effect and the convective velocity renormalization. The dispersion tensor, the effective phase permeabilities and the velocity renormalization are defined through the cell problems as the functions of saturation, viscosity ratio and flow velocity.
To solve the cell problems we have developed the twophase version of the streamconfiguration method proposed earlier for singlephase flow. The method is based on the following features of a system of fine fractures: a) the limit flow is locally onedimensional; b) the saturation and heterogeneity can be factorized, so the cell problem becomes independent of saturation. At the same time, the limit solution is shown to be nonunique due to a loss of information about the stream configuration geometry in the nodes of fracture intersection. The regularization procedure is developed which proposes additional conditions describing the type of stream configuration in each node. The types of configuration are determined from the principle of local energy minimum or entropy maximum. For a periodic regular fracture network, the method allows obtaining analytical solution for the dispersion tensor. For disordered networks, the method reduces the cell problem to an algebraic system of a rank equal to the number of nodes in a unite cell. A version of method is developed, based on a combination between the streamconfiguration and the bordering techniques.
A singular dispersion regime is revealed in which the dispersion tensor becomes unbounded due to arising of stagnant zones in several fracture segments and fluid trapping. The data on comparison of our results with those obtained using other methods are illustrated.
The industrial application of this research consists of a new numerical algorithm of upscaling twophase flow in fractured media, which is much faster than other methods.
The research is supported by the Schlumberger Technology Center in Abingdon.



Mathematical Model for ThreePhase Compressible Dual Porosity Streamline Simulation
Authors A. Myasnikov, A. Kozlova, F. Bratvedt and K. BratvedtFlow simulation of fractured reservoirs usually is performed using a dual porosity model. The dual porosity system is modeled by using two coupled grids: one for matrix and one for fracture. These two continua communicate by transfer functions. Until now, there were no mathematical models of dual porosity, threephase, compressible flow for streamline simulators. To realize this model, it was necessary to reformulate the matrix and fracture pressure equations The conventional transfer function has been incorporated as a source/sink term, not only in the streamline saturation equations (as it was in incompressible case), but also in the pressure equation.
The dual porosity model has been implemented into a streamline simulator. This tool has its main application in the geological modeling domain for analyzing uncertainty, model ranking and screening of geologically detailed models, including fractures.
This paper describes the mathematical model for a threephase compressible dual porosity streamline simulator and compares the results and run times of the streamlinebased approach with a conventional dual porosity gridbased commercial simulator. The results from the streamline simulator for dual porosity show good agreement with those produced by a commercial finite difference simulator with order of magnitude improvement in simulation time.
Streamline methods as a reservoir simulation tool have generated much interest in petroleum engineering because of the capability to calculate fluid flow in multimillion cell geological models with reasonable CPU times. However, important physical properties of geoscale fluid flow models are still not properly modeled by streamline methods. Enhancing the range of physical properties that can be simulated accurately in a timely manner will enable improved workflows in the geological modeling domain.



Rigorous Derivation of the KazemiGilmanElsharkawy Generalized Dual Porosity Shape Factor
Authors Z. E. Heinemann and G. M. MittermeirKazemi et al.[7] suggested to use an empirical matrix/fracture transfer function, verified based on experimental data of Mattax and Kyte[9]. Kazemi et al. showed that for rectangles and cylinders the formula reduces to the well known forms of the shape factor. In the mean time many authors indicated the validity of the formula, but no theoretical proof was offered so far.
This paper derives the KazemiGilmanElsharkawy (KSE) shape factor using Control Volume Finite Difference discretization on the fracturematrix dual continuum.
It will be shown that the KSE shape factor is valid for multiphase fow of compressible fluids irrespective of matrix block shape. Additionally an extension to tensorial matrix permeability is given.



A Learning Computational Engine for History Matching
Authors R. E. Banchs, H. Klie, A. Rodríguez, S. G. Thomas and M. F. WheelerThe main objective of the present work is to propose and evaluate a learning computational engine for history matching, which is based on a hybrid multilevel search methodology. According to this methodology, the parameter space is globally explored and sampled by the simultaneous perturbation stochastic approximation (SPSA) algorithm at a given resolution level. This estimation is followed by further analysis by using a neural learning engine for evaluating the sensitiveness of the objective function with respect to variations of each individual model parameter in the vicinity of the promising optimal solution explored by the SPSA algorithm.
The proposed methodology is used to numerically determine how additional sources of information may aid in reducing the illposedness associated with permeability estimation via conventional history matching procedures. The additional sources of information considered in this work are related to pressures, concentrations and fluid velocities at given locations in a reliable fashion, which in practical scenarios might be estimated from high resolution seismic surveys, or directly obtained as in situ measurements provided by sensors. This additional information is incorporated, along with production data, into a multiobjective function that is mismatched between the observed and the predicted data. The preliminary results presented in this work shed light on future research avenues for optimizing the use of additional sources of information such as seismic or sensor data in history matching procedures.



Hybrid Mesh Generation for Local Grid Refinement in Reservoir Flow Simulation
Authors C. Bennis, H. Borouchaki and N. FlandrinMesh generation becomes a crucial step in reservoir flow simulation of new generation. The mesh must faithfully represent the architecture of the reservoir and its heterogeneity. For more accuracy, the mesh must also follow the flow directions in the well vicinity. The current industrial standard meshes based on Corner Point Geometry (CPG) grids have already shown their limits. They are very practical and easy to use, but they fail to represent complex objects due to their structured aspect. More recently, other approaches have been proposed, in particular using the PErpendicular BIssector (PEBI) grids, which are completely unstructured. These grids are very flexible and can model most complex shapes. But, they are often difficult to manage in 3D due to their lack of structure.
A three dimensional hybrid mesh model was proposed, in ECMOR 9, to capture the radial characteristics of the flows around the wells. In this hybrid mesh, the reservoir is described by a non uniform Cartesian structured mesh and the drainage areas around the wells are represented by structured radial circular meshes. Unstructured polyhedral meshes are used to connect these two kinds of structured grids. The construction of these transition meshes is based on 3D Power Diagrams to ensure finite volume properties : mesh conformity, dual orthogonality and cell convexity. In this paper, we propose an extension of this hybrid model to the case of real CPG reservoir grids. At first, the CPG grid is mapped, in a reference space, into a non uniform Cartesian grid by minimising the mapping deformation. Then, a hybrid mesh is generated in this reference space using the previous method. Finally, this mesh is mapped back into the real space. This mesh can also be optimised with respect to the deformation metric between the real space and the reference one.



Homogenization for Fe²+ deposition near drink water tube wells during arsenic remediation
Authors J. Bruining and M. I. M. DarwishHomogenization is a well established methodology that can be used for upscaling. The advantage of homogenization over other upscaling methods is that it only requires the choice of a periodic unit cell to overcome the closure problem. The choice of the unit cell is completely flexible and only limited by computer capacity. Our interest is in the upscaling of reactive diffusion convection flows in connection with arsenic contaminated water remediation. We will rederive the homogenized equations with emphasis on the physical aspects. By considering a simple twodimensional example we cover all computational aspects linked to the actual application of the method. However, the method can be easily extended to more complicated configurations in 3D. The computed dispersion coefficient shows a similar (qualitative) dependency on the Peclet number as literature data, albeit that the values for this 2D example are considerably smaller. Moreover in the case of nonlinear adsorption the dispersion coefficient depends on the concentration.



Optimization of the Oil Field Putting into Operation Stage
Authors S. A. Ermolaev and A. I. ErmolaevThe strategy of field putting into operation is to solve the following problem: “Which wells are to be put into operation at the given moment?” There are different strategies which differ greatly by the volumes of oil recovery for the period of the field putting into operation. Consequently there is a problem how to form and choose the rational strategy. By rational strategy is meant the best strategy for given criteria. In the present paper we propose some algorithms to solve this problem.
There are two aspects while solving the problem of creating and choosing the rational strategy. The first aspect is to fulfill detailed hydrodynamic simulation of the field development. The second aspect is to enumerate a great number of possible strategies. At present either nobody realizes detailed simulation or very few possible strategies are analyzed. Thus an irrational strategy may be selected.
The suggested method of creating and choosing rational strategy of putting into operation is a synthesis of simulation and optimization algorithms. This method allows one to take into account main reasons of strategies different efficiency (reservoir heterogeneity, nonuniform watering of field sections). Also we realize purposeful and shortcut enumeration of all possible strategies of field putting into operation.
The main stages of this method are:
1. Wells grouping into blocks (number of wells in blocks is to be approximately equal, not only neighboring wells can be included in one block).
2. Computation of potential volumes of oil recovery produced by wells blocks while the moments of their putting into operation are different (these parameters are input data for optimization; these computations use hydrodynamic simulation software; also the suggested technology allows one to estimate the injection wells deposit into total oil recovery).
3. The optimum moments of putting blocks into operation or their optimum order are found (these problems are formulated as transport models; standard linear programming algorithms may be used to solve).
In this paper we consider examples using suggested method for a real deposit.
The method can be expediently used to choose rational strategies of field putting into operation when it is impossible to follow specialists experience and intuition.



Integrating Dynamic Data into Reservoir Models – A Multiple Point Perspective
Authors S. Srinivasan and K. EskandariConstraining reservoir models such as permeability distribution to dynamic well data has been traditionally accomplished using inverse theory. The resulting illconditioned system of equations needs to be solved for the required model parameters. Gradientbased optimization schemes require efficient computation of sensitivity coefficients (Chu, Reynolds and Oliver, 1992). Markov chain Monte Carlo (Omre and Tjelmeland, 1996), frequently used in earth sciences, is another iterative approach in which a prior distribution of permeability is perturbed into a posterior distribution using a probabilistic perturbation acceptance criteria. However, that technique like the pervious methods is expensive in term of CPU usage.
This paper investigates an approach to alleviate the computational expense associated with the iterative conditioning of reservoir models to dynamic information. A multiple point proxy is proposed that accounts for the configuration and orientation of geological patterns in the reservoir and their relationship to flow. The nonlinear relationship between the multiple point connectivity and the flow response is established by calibration. Once calibrated, the proxy expression acts as a surrogate to the full physics based flow simulators and can be used within an iterative framework such as Markov chain Monte Carlo algorithm to build reservoir models conditioned to well test responses. The mathematical formulation and calibration of such a proxy function for matching well pressure characteristics is presented in this paper.
Building reservoir models based on just static data using multiple point statistics instead of just two point statistics has received a lot of attention recently. The SNESIM algorithm (Strebelle, 2002) is such a multiple point (mp) simulation algorithm. This paper presents a multipoint simulation algorithm that is distinctly different from all other mpbased simulation algorithms available in the literature. The presented approach is extended to integrate dynamic data using an efficient probability perturbation based approach. The proposed method is computationally fast for retrieving mp statistics (scanning) and simulation using a unique pattern growthbased methodology. Dynamic data integration is achieved by gradually perturbing the multiple point probability distribution using a deformation parameter. The perturbed multiple point probability distribution is then merged with the prior multiple point distribution inferred from a training image. The permanence ratio of hypothesis is used to perform the merge. The resultant reservoir model is thus consistent with both the available production data as well as the prior geology.



A Numerical Solution to Groundwater Flow in Full Tensorial Permeability Fields
Authors M. Le RavalecDupin and L. RicardThe description of flow through heterogeneous porous media is of primary interest to many fields such as reservoir engineering or hydrology. Because of porosity and permeability heterogeneity, flow equations are usually solved on the basis of numerical techniques. The discretization technique most commonly used is the finitedifference method. Thus, a conventional discretization scheme involves a 5 point stencil for twodimensional media and a 7 point stencil for threedimensional media. Continuity of flux and pressure is readily incorporated into the discretization by deriving the coefficients at the interface between adjacent gridblocks from the harmonic average of the permeabilities of the two contiguous gridblocks. This approximation holds as far as permeability tensor is diagonal.
Today, very detailed geological models are built on grids containing millions of gridblocks. Although computers are growing ever more powerful, fluid flow simulations on such grids are tremendously CPUtime consuming. To make them tractable, the number of gridblocks must be reduced to about 105. This motivated the development of techniques for upscaling the geological model to a manageable level of detail. Upscaling consists in determining the equivalent permeabilities of the coarse gridblocks. These equivalent permeabilities are usually full tensors. Discretization techniques in which interface fluxes are determined from more than two points have been developed to solve full tensorial flow equations. In this case, the discretization stencil requires 9 points for two dimensional media and even more for threedimensional media. These techniques, known as multipoint flux approximations (MPFA) have recently enjoyed increased popularity. To date, to our knowledge, there is no straightforward generalization of the harmonic average to estimate fluxes at interfaces with MPFAs.
We develop an alternative to MPFAs based upon numerical spectral methods so that the full tensorial pressure equation is solved on regular grids. We also consider source or sink terms as well as gravity. Contrary to MPFAs, the proposed spectral technique does not call for any kind of approximation to estimate fluxes at the interface between adjacent gridblocks. Specific applications are presented and discussed.



Nonlinear Analysis of Fluid Flow Behaviour and Flow Control Factors in Fractured Systems Using Support Vector Machine
Authors J. Ma and G. CouplesPredicting the fluid flow behaviour of systems containing fractures or other flow conduits (which are typically subseismic in scale) is an important element of flow modelling in petroleum exploration and production. Since the fluid flow can be strongly influenced by multiple flow control factors, including the connectivity of the fracture network, the spatial distribution of fractures (e.g. fracture intensity), and the contrast of flow properties between fractures and the matrix, the fluid flow behaviour of a fractured system, with respect to flow control factors, may show multiple distinctive modes. Clearly the existence of different modes indicates a need for a multimodal approach to incorporating effective flow properties in the coarserscale flow simulation. However, the intrinsic nonlinearity of such relationships and the high dimensionality of the factors make it difficult to identify such relationships, let alone to distinguish the distinctive modes among them.
In this work, a nonlinear analysis, based on a machine learning method  Support Vector Machine (SVM), was considered for identifying the relationships. It was applied to two simple fractured systems in 2D, each of which was characterised by a set of distributional parameters and a stochastic fracture modelling procedure. For each system, equallyplausible fracture models were generated, and the singlephase steadystate fluid flow was simulated for each model. The relationships between the simulated fluid flow, in a form of upscaled permeability, and a number of flow control factors, were then analysed. The results showed the existence of complex and multimodal relationships between the upscaled permeability and the control factors in each system, with distinctive features between the systems. The implications of not employing a multimodal approach in a coarserscale simulation are obvious: the upscaled flow properties can be significantly misestimated.



Determining the Filtration and Permeability Reduction Functions for Flow of Water with Particles in Porous Media
Authors D. Marchesin, A. C. Alvarez, G. Hime and P. G. BedrikovetskyDeep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, bacteria, viruses or contaminant transport in groundwater, industrial filtering, etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction.
Models for deep bed filtration contain two coefficients that represent rock and fluid properties: the filtration function, which is the fraction of captured particles per unit of particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities.
The coefficients cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients.
In this work we determine these coefficients from pressure drop and effluent concentration histories, measured in onedimensional laboratory experiments.
The filtration function is recovered by optimizing a nonlinear functional with box constraints. The permeability reduction is recovered likewise, taking into account the filtration function already found.
The recovery method consists of optimizing Tikhonov's functionals in appropriate subdomains.
In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.



Numerical Approach to Optimising Oil and Gas Production in Thin Oil Rim Reservoirs
Authors G. G. Cassarà, C. Monico, D. Castano and S. DresdaThe development of a thin oil rim can involve complications in the oil production due to severe gas and/or water coning events.
This paper deals with a Nigerian oil and gas multilevel field where some of the levels bear a thin oil rim associated with a large gas cap. After a few years of production from three oil levels, it was decided to develop all gas and oil levels, to increase the oil production rate and satisfy the increasing demand for gas. Therefore, the thin oil rim reservoirs were considered as potential sources of oil and gas reserves.
A 3D black oil simulation model was built to study the development of the whole field. Gas PVT data were characterised through the vaporised oil gas ratio to simulate the effects of depletion on the condensate stream during production from gas levels.
Sensitivity runs were performed to investigate the development of the oil rim reservoirs. The model indicated that the wells located in the oil rim were affected by high water cut, whilst the wells in the gas cap area and far away from the oil rim, could produce large gas rates without draining the oil in the rim.
An intermediate location was considered in the gas cap flank close to the oil rim. The wells with this location showed early high gas production with condensate decline, followed by liquid hydrocarbon production increase with gas rate reduction. During the early time, the condensate production fell due to depletion. Afterwards, the liquid hydrocarbon rate started to increase since wells drained the oil rim through coning.
The results of the simulation encouraged the development of the thin oil rim reservoirs mainly through these intermediate locations to optimise both oil and gas production.



New Method to Calculate Water Saturation from LogDerived JFunction in Carbonte Reservoir
Authors T. A. Obeida and Y. S. Al MehaoriCalculation of initial fluid saturations is a critical step in any 3D reservoir modeling studies. The initial water saturation (Swi) distribution will dictate the original oil in place (STOIIP) estimation and will influence the subsequent steps in dynamic modeling (history match and predictions). Complex carbonate reservoirs always represent a quit a challenge to geologist and reservoir engineers to calculate the initial water saturation with limited or no SCAL data available. The proposed method in this study combines core data (permeability) from 32 cored wells with identifiable reservoir rock types (RRTs) and log data (porosity and Swi) to develop drainage logderived capillary pressure (Pc) based on rock quality index (RQI) and then calculate Jfunction for each RRT which was used in the simulator to calculate the initial water saturation in the reservoir.
The initialization results of the dynamic model indicate good Swi profile match between the calculated Swi and the logSwi for 70 wells across the field. The calculation of STOIP indicates a good agreement (within 3% difference) between the geological 3D model (31 million cells fine scale) and the upscaled dynamic model (1 million cells). The proposed method can be used in any heterogeneous media to calculate initial water saturation as a function of height as well as rock property (RQI) and thus estimate proper original oil in place. This method is also applicable to initialize huge fine scale static models.



Multiscale Averaging Algorithms for Flow Modeling in Heterogeneous Reservoir
Authors A. K. Pergament, V. A. Semiletov and M. Y. ZaslavskyThe averaging algorithms for Lebedev’s grids and for nonorthogonal grids by the Samarskii support operator method are considered in the article. The averaging process allows integrating the grid cells in case the approximate solutions in the integrated cell may be determined. The functions describing the features of solutions have been obtained for integrated cells of the Lebedev and arbitrary nonorthogonal grids. It results from approximating fluxes in each cell for linear span elements of the functions mentioned above that the symmetric permeability tensor may be constructed and the flux approximation of the first order may be obtained. The flux approximation leads to the strong convergence of the algorithm. It is essential the system grid cells may be multiscale, i.e. sizes of cells and the values of permeabilities may be very different.
The flux approximation distinguishes this article results from those of other authors of averaging algorithms. Then the methods of flow modeling in anisotropic media have been developed.
