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ECMOR XI  11th European Conference on the Mathematics of Oil Recovery
 Conference date: 08 Sep 2008  11 Sep 2008
 Location: Bergen, Norway
 ISBN: 9789073781559
 Published: 08 September 2008
1  20 of 105 results


Continuous Darcyflux Approximations with Full Pressure Support for Structured and Unstructured Grids in 3D
Authors H. Zheng and M.G. EdwardsThis paper presents the development of families of controlvolume distributed fluxcontinuous schemes with full controlvolume surface pressure continuity for the general three dimensional pressure equation. Most effort to date has focused on schemes with pointwise continuity where the local continuity conditions are relatively compact. While full surface pressure continuity requires an increase in the number of local continuity conditions the new schemes prove to be relatively robust. Families of full pressure continuity schemes are developed here for structured and unstructured grids in three dimensions. The resulting Darcy flux approximations are applied to a range of three dimensional test cases that verify consistency of the schemes. Convergence tests of the threedimensional families of schemes are presented, for a range of quadrature points. Mmatrix conditions are presented and the schemes are tested for monotonicity. The full pressure continuity schemes are shown to be beneficial compared with earlier pointwise continuity schemes [1] both in terms improved monotonicity and convergence. The schemes are applied to challenging three dimensional test cases including heterogeneity for both structured and unstructured grids. TECHNICAL CONTRIBUTIONS Extension of full controlvolume surface pressure continuity flux continuous schemes to structured and unstructured grids in three dimensions. Mmatrix and monotonicity analysis of three dimensional methods. Benefits are demonstrated for challenging three dimensional test cases including heterogeneity on structured and unstructured grids.



A General Framework for Non Conforming Approximations of the Single Phase Darcy Equation
Authors D. Di Pietro, L. Agelas, R. Eymard and R. MassonIn this paper we present an abstract analysis framework for nonconforming approximations of heterogeneous and anisotropic elliptic operators on general 2D and 3D meshes. The analysis applies to discontinuous Galerkin methods, to the popular MPFA Omethod, as well as to hybrid finite volume methods. A number of examples is provided in the paper. The guidelines of the analysis can be summarized as follows: (i) each method is rewritten in weak formulation by introducing two discrete gradient operators. Penalty terms may be required for non $H^1$conformal spaces to ensure coercivity; (ii) a discrete $H^1$norm is introduced such that a discrete version of the Rellich theorem holds; (iii) an a priori estimate on the discrete solution in the discrete $H^1$norm is derived using the coercivity of the bilinear form; (iv) the convergence of the discrete problem to the continuous one is eventually deduced, thus proving the convergence of the approximation. In order for the above analysis to hold, the gradient operator applied to the discrete solution should be consistent, whereas the one applied to the test function should be weakly convergent in $L^2$. According to the method, further assumptions on the mesh may be required. This analysis framework easily extends to nonlinear problems like the ones encountered in the study of multiphase Darcy flows, and it allows to weaken regularity assumptions on the coefficients of the problem as well as on the exact solution. From a practical viewpoint, its main interest lies in the possibility to analyze and to design new methods as well as to improve existing ones.



On Efficient Implicit Upwind Schemes
Authors J.R. Natvig and K.A. LieAlthough many advanced methods have been devised for the hyperbolic transport problems that arise in reservoir simulation, the most widely used method in commercial simulators is still the implicit upwind scheme. The robustness, simplicity and stability of this scheme makes it preferable to more sophisticated schemes for real reservoir models with large variations in flow speed and porosity. However, the efficiency of the implicit upwind scheme depends on the ability to solve large systems of nonlinear equations effectively. To advance the solution one time step, a large nonlinear system needs to be solved. This is a highly nontrivial matter and convergence is not guaranteed for any starting guess. This effectively imposes limitations on the practical magnitude of time steps as well as on the number of grid blocks that can be handled. In this paper, we present an idea that allows the implicit upwind scheme to become a highly efficient. Under mild assumptions, it is possible to compute a reordering of the equations that renders the system of nonlinear equations (block) lower triangular. Thus, the nonlinear system may be solved one (or a few) equations at a time, increasing the efficiency of the implicit upwind scheme by orders of magnitude. Similar ideas can also be used for highorder discontinuous Galerkin discretizations. To demonstrate the power of these ideas we show results and timings for incompressible and weakly compressible transport in real reservoir models.



Spatiallyvarying Compact Multipoint Flux Approximations for 3D Adapted Grids with Guaranteed Monotonicity
Authors J.V. Lambers and M.G. GerritsenWe propose a new singlephase local transmissibility upscaling method for adapted grids in 3D domains that uses spatially varying and compact multipoint flux approximations (MPFA), based on the VCMP method previously introduced for 2D Cartesian grids. For each cell face in the coarse upscaled grid, we create a local fine grid region surrounding the face on which we solve three generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve up to ten neighboring pressure values. They are required to honor the three generic flow problems as closely as possible while maximizing compactness and ensuring that the flux approximation is as close as possible to being twopoint. The resulting MPFA stencil is spatially varying and reduces to twopoint expressions in cases without fulltensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a localglobal upscaling method. We also present a corrector method that adapts the stencils locally to guarantee that the resulting pressure matrix is an Mmatrix. This corrector method is needed primarily for cases where strong permeability contrasts are misaligned with the grid locally. The corrector method has negligible impact on the flow accuracy. Finally, we show how the computed MPFA can be used to guide adaptivity of the grid, thus allowing rapid, automatic generation of grids that can account for difficult geologic features such as faults and fractures and efficiently resolve finescale features such as narrow channels.



Analytical Estimation of Advanced IOR Performance
Authors R.A. Berenblyum, A.A. Shchipanov, E.M. Reich, L. Surguchev and K. PotschIn the recent years of high oil prices many companies turned their attention to the advanced IOR/EOR methods including cyclic water flooding, miscible gas and steam injection. This paper summarizes experience accumulated in International Research Institute of Stavanger over several years of using inhouse analytical prescreening tool for evaluation of possible IOR/EOR strategies. An analytical models of cyclic water, miscible gas and steam injection is presented. Strong and week points of these methods are analyzed. The results of field case studies both on the Norwegian Continental Shelf and outside Norway are presented and discussed. We conclude that the analytical models may be successfully used for prescreening of the IOR methods. The analytical methods can, in most cases, correctly handle the behavior of the reservoir system as a function of IOR process critical parameters. However final decision to apply a certain IOR procedure and its optimization should be done utilizing fullscale reservoir models. The analytical methods can significantly decrease the decision making time by either: a) decreasing a number of computationally expensive fullscale simulations for mature developed reservoirs; or b) quickly screening new prospect where reservoir knowledge is limited.



Shortest Path Algorithm for Porescale Simulation of Wateralternatinggas Injection
Authors M.I.J. van Dijke, J.E. Juri and K.S. SorbieFor simulation of threephase flow processes, such as wateralternatinggas injection (WAG), accurate descriptions of the threephase capillary pressures and relative permeabilities as functions of the phase saturations are essential. Instead of deriving these functions from experiments, which is very difficult, they may be obtained from porescale network simulations. During multiple injection processes, especially under mixedwet conditions, isolated clusters of the three phases occur. These clusters may still be mobilized, even under the assumption of capillarydominated flow, as part of socalled multiple displacement chains of adjacent clusters stretching from inlet to outlet of the model. To determine the most favourable displacement chain requires implementation of an efficient shortest path algorithm. For the present problem the distances or costs are the capillary entry pressures between the various phase clusters. Because these entry pressures can also be negative, socalled negative cost cycles arise, which invalidate traditional algorithms. Instead, we have implemented an efficient shortest path algorithm with negative cycle detection If negative cycles arise, the corresponding cyclic displacement chains are carried out before any linear displacement from inlet to outlet. Negative cost cycles correspond to spontaneous displacements, for which the prevailing invading phase pressure is higher than required. Additionally, and probably even more significant, the shortest path algorithm determines in principle the pressures of all phase clusters in the network. These pressures are in turn used in the accurate calculation of the threephase entry pressures, as well as the film and layer volumes and conductances in pore corners. Some WAG network simulations in mixedwet media have been carried out using the new algorithm to demonstrate the occurrence of long displacement chains and the much improved efficiency, also for increasing network sizes. A significant number of negative cycles occurred, but the corresponding displacement chains were short. Additionally, we have analysed the obtained values of the cluster pressures.



Fractionalflow Theory Applied to NonNewtonian IOR Processes
Authors W.R. Rossen, R.T. Johns, K.R. Kibodeaux, H. Lai and N. Moradi TehraniThe method of characteristics, or fractionalflow theory, is extremely useful in understanding complex IOR processes and in calibrating simulators. One limitation has been its restriction to Newtonian rheology except in rectilinear flow. Its inability to deal with nonNewtonian rheology in polymer and foam IOR has been a serious limitation. We extend fractionalflow methods for twophase flow to nonNewtonian fluids in onedimensional cylindrical flow, where rheology changes with radial position r. The fractionalflow curve is then a function of r. At each position r the "injection" condition is the result of the displacement upstream; one can plot the movement of characteristics and shocks downstream as r increases. We show examples for IOR with nonNewtonian polymer and foam. For continuous injection of foam, one can map the 1D displacement in time and space exactly for the nonNewtonian foam bank, and approximately, but with great accuracy, for the gas bank ahead of the foam. The fractionalflow solutions are more accurate that finitedifference simulations on a comparable grid and can be used to calibrate simulators. Fractionalflow methods also allow one to calculate changing mobility near the injection well to a greater accuracy than simulation. For polymer and SAG (alternatingslug) foam injection, characteristics and shocks collide, making the fractionalflow solution complex. Nonetheless, one can solve exactly for changing mobility near the well, again to greater accuracy than with conventional simulation. For polymer solutions that are Newtonian at high and low shear rates but nonNewtonian in between, the nonNewtonian nature of the displacement depends on injection rate. The fractionalflow method extended to nonNewtonian flow can be useful both for its insights for scaleup of laboratory experiments and to calibrate computer simulators involving nonNewtonian IOR. It can also be an input to streamline simulations.



Fractionalflow Theory of Foam Displacements with Oil
Authors M. NamdarZanganeh, T. La Force, S.I. Kam, T.L.M. van der Heijden and W.R. RossenFractionalflow theory has proven useful for understanding the factors that control foam displacements and as a benchmark against which to test foam simulators. Most applications of fractionalmethods to foam have excluded oil. Recently, Mayberry and Kam (SPE 100964) presented out fractionalflow solutions for foam injection with a constant effect of oil on the foam. We extend fractionalflow methods to foam displacements with oil, using the effects of oil and water saturations on foam as represented in the STARS simulator. There can be abrupt shifts in the composition paths at the limiting water and oil saturations for foam stability. In the immiscible threephase SAG displacements examined, if foam collapses at the initial oil saturation in the reservoir, there is a verysmallvelocity shock from the injected condition to complete foam collapse. The displacement is nearly as inefficient as if no foam were present at all. It does not matter in these cases whether foam is weakened by low water saturation. The displacement is efficient, however, if foam is unaffected by oil but weakened at low water saturation. These results may reflect our foam model, where foam is only partially destroyed at low water saturations but is completely destroyed by high oil saturation. Two idealized models for threephase displacements can be represented at twophase displacements with chemical or miscible shocks: A model for a firstcontact miscible gas flood with foam suggests an optimal water fraction in foam that puts the gas front just slightly ahead of the foam (surfactant) front. An idealized model of a surfactant flood with foam for mobility control suggests it is important to inject a sufficiently high water fraction in the foam that the gas front is behind the surfactant front as the flood proceeds. We present simulations to verify the solutions obtained with fractionalflow methods.



Doublefamilies of QuasiPositive Fluxcontinuous Finitevolume Schemes on Structured and Unstructured Grids
Authors M.G. Edwards and H. ZhengThis paper focuses on fluxcontinuous pressure equation approximation for strongly anisotropic media. Previous work on families of flux continuous schemes for solving the general geometrypermeability tensor pressure equation has focused on singleparameter families e.g. [1]. These schemes have been shown to remove the O(1) errors introduced by standard twopoint flux reservoir simulation schemes when applied to full tensor flow approximation. Improved convergence of the schemes has also been established for specific quadrature points [2]. However these schemes have conditional Mmatrices depending on the strength of the crossterms [1]. When applied to cases involving full tensors arising from strongly anisotropic media, these schemes can fail to satisfy the maximum principle. Loss of solution monotonicity then occurs at high anisotropy ratios, causing spurious oscillations in the numerical pressure solution. New doublefamily fluxcontinuous locally conservative schemes are presented for the general geometrypermeability tensor pressure equation. The new doublefamily formulation is shown to expand on the current singleparameter range of existing schemes that have an Mmatrix. While it is shown that a double family formulation does not lead to an unconditional Mmatrix scheme, an analysis is presented that classifies the sensitivity of the new schemes with respect to monotonicity and doublefamily quadrature. Convergence performance with respect to a range of doublefamily quadrature points is presented for some well known test cases.



AdaptivelylocalizedcontinuationNewton; Reservoir Simulation Nonlinear Solvers that Converge All the Time
Authors R.M. Younis, H. Tchelepi and K. AzizGrowing interest in understanding, predicting, and controlling advanced oil recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The Fully Implicit Method offers unconditional stability in the sense of discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations which need to be solved at each timestep. Current reservoir simulators apply safeguarded variants of Newton’s method, and often can neither guarantee convergence, nor provide estimates of the relation between convergence rate and timestep size. In practice, timestep chops become necessary, and are guided heuristically. With growing complexity, such as in thermally reactive compositional models, this can lead to substantial losses in computational effort, and prohibitively small timesteps. We establish an alternate class of nonlinear iteration that both converges, and associates a timestep to each iteration. Moreover, the linear solution process within each iteration is performed locally. By casting the nonlinear residual for a given timestep as an initialvalueproblem, we formulate a solution process that associates a timestep size with each iteration. Subsequently, no iterations are wasted, and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as a standard Newton counterpart whenever it does converge. Finally, by exploiting the local nature of nonlinear waves that is typical to all multiphase problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify the minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging largescale simulation examples are presented, and the results demonstrate the robustness of the proposed method as well as its performance.



Study of a New Refinement Criterion for Adaptive Mesh Refinement in Sagd Simulation
Authors M. Mamaghani, C. Chainais and G. EncheryThe SAGD (Steam Assisted Gravity Drainage) is an enhanced oil recovery process for heavy oils and bitumens. To simulate this thermal process and obtain precise forecasts of oil production, very smallsized cells have to be used because of the fine flow interface. But the use of fine cells throughout the reservoir is expensive in terms of CPU time. To reduce the computation time one can use an adaptive mesh refinement technique which will use a refined grid at the flow interface and coarser cells elsewhere. The first numerical tests of SAGD modelling, using a criterion based on two threshold temperatures, showed that this adaptivemeshrefinement technique could reduce significantly the number of cells [LLR03]. Nevertheless, the refined zone remains too wide. In this work, we introduce a new criterion for the use of adaptive mesh refinement in SAGD simulation. This criterion is based on the work achieved in [KO00] on a posteriori error estimators for finite volume schemes for hyperbolic equations. It enables to follow more precisely the flow interface. Through numerical experiments we show that the new criterion enables to further decrease the number of cells (and thus CPU times) while maintaining a good accuracy in the results.



Mimetic MPFA
Authors A.F. Stephansen and R.A. KlausenThe analysis of the multi point flux approximation (MPFA) method has so far relied on the possibility of seeing it as a mixed finite element method for which the convergence is then established. This type of analysis has been successfully applied to triangles and quadrilaterals, lately also in the case of rough meshes. Another well known conservative method, the mimetic finite difference method, has also traditionally relied on the analogy with a mixed finite element method to establish convergence. Recently however a new type of analysis proposed by Brezzi, Lipnikov and Shashkov, cf. [3], permits to show convergence of the mimetic method on a general polyhedral mesh. We propose to formulate the MPFA Omethod in a mimetic finite difference framework, in order to extend the proof of convergence to polyhedral meshes. The general nonsymmetry of the MPFA method and its local explicit flux, found trough a splitting of the flux, are challenges that require special attention. We discuss various qualities needed for a structured analysis of MPFA, and the effect of the lost symmetry.



Streamline Tracing on General Tetrahedral and Hexahedral Grids in the Presence of Tensor Permeability Coefficients
Authors S.F. Matringe, R. Juanes and H.A. TchelepiModern reservoir simulation models often involve geometrically complex (distorted or unstructured) threedimensional grids populated with fulltensor permeability coefficients. The solution of the flow problem on such grids demands the use of accurate spatial discretizations such as the multipoint flux approximation (MPFA) finite volume schemes or mixed finite element methods (MFEM). The extension of the streamline method to modern grids therefore requires a streamline tracing algorithm adapted to these advanced discretizations. In this paper, we present a new algorithm to trace streamlines from MPFA or MFEM on general tetrahedral or hexahedral grids and in the presence of fulltensor permeabilities. Our approach was already used successfully in 2D and this paper presents the extension to 3D of the previous work by the authors. The method is based on the mathematical framework of MFEM. Since MPFA schemes have recently been interpreted as MFEM, our streamline tracing algorithm is also applicable for the MPFA finite volume methods. Using the mixed finite element velocity shape functions, the velocity field is reconstructed in each grid cell by direct interpolation of the MFEM velocity unknowns or MPFA subfluxes. An integration of the velocity field to arbitrary accuracy yields the streamlines. The method is the natural extension of Pollock’s (1988) tracing method to general tetrahedral or hexahedral grids. The new algorithm is more accurate than the methods developed by Prévost (2003) and Haegland et al. (2007) to trace streamlines from MPFA solutions and avoids the expensive flux postprocessing techniques used by both methods. After a description of the theory and implementation of our streamline tracing method, we test its performance on a fullfield reservoir model discretized by an unstructured grid and populated with heterogeneous tensor coefficients.



On Front Tracking for Compressible Flow
Authors H.M. Nilsen and K.A. LieStreamline simulation is particularly efficient for incompressible twophase flow, for which one can use front tracking to solve the 1D transport problems along streamlines. Here we investigate the extension of this method to compressible (and immiscible) flow and discuss some of the difficulties involved, and in particular the choices one has in writing the 1D transport equation(s). Our study is motivated by the simulation of CO2 injection, and we therefore also develop methods that are particularly suited for solving compressible flow where one phase is incompressible. Altogether, we present four fronttracking methods that are based on a combination of solving ordinary Riemann problems and Riemann problems with discontinuous flux.



Adaptive Control for Solver Performance Optimization in Reservoir Simulation
Authors I.D. Mishev, N. Fedorova, S. Terekhov, B.L. Beckner, A.K. Usadi, M.B. Ray and O. DiyankovRuntime performance of a reservoir simulator is significantly impacted by the selection of the linear solver preconditioner, iterative method and their adjustable parameters. The choice of the best solver algorithm and its optimal parameters is a difficult problem that even the experienced simulator users cannot adequately solve by themselves. The typical user action is to use the default solver settings or a small perturbation of them that are frequently far from optimal and consequently the performance may deteriorate. There has been extensive research to develop automatic performance tuning and selfadaptive solver selection systems. For example SelfAdapting Largescale Solver Architecture (SALSA) developed at the University of Tennessee requires running of large number of problems to initialize the system before using it. In contrast we propose an adaptive control online system to optimize the simulator performance by dynamically adjusting the solver parameters during the simulation. We start with a large set of parameters and quickly choose the best combinations that are continuously adapted during the simulation using the solver runtime performance measurements (e.g. solver CPU time) to guide the search. This software system, called the Intelligent Performance Assistant (IPA), has been successfully integrated into ExxonMobil’s proprietary reservoir simulator and deployed with it worldwide. The system can handle a large number of combinations of solver parameters, currently in the order of 108, and consistently improves run time performance of real simulation models, frequently by 30% or more, compared to the performance with the default solver settings. Moreover, IPA includes a persistent memory of solver performance statistics. The runtime statistics from these individual runs can be gathered, processed using data mining techniques and integrated in the IPA system, thus allowing its continuous improvement.



A Novel Percolative Aggregation Approach for Solving Highly IllConditioned Systems
Authors H. Klie and M.F. WheelerA key aspect in any algebraic multilevel procedure is to be able to reliably capture the physical behavior behind system coefficients. However, the modeling of complex reservoir scenarios generally involves the computation of extremely discontinuous and nonlinear coefficients that, in turn, compromise the robustness and efficiency of smoothing and coarsening strategies. In addition, the need of dealing with large discretized domains leads to highly illconditioned systems that represent a true challenge to any linear solver technology known today. In this work, we exploit the fact that flow trend information can be used as a basis for developing a robust percolative aggregation (PA) twostage preconditioning method. To this end, we identify coefficient aggregates by a means of an efficient version of the HoshenKopelman percolation algorithm suitable for any flow network structure. By partitioning and reordering unknowns according to these aggregates, we obtain a set of highconductive blocks followed by a lowconductive block. Diagonal scaling allows for weakening the interaction between these high and low conductive blocks plus improving the overall conditioning of the system. The solution of the highconductive blocks approximates well the solution of the original problem. Remaining sources of errors from this approximation are mainly due to small eigenvalues that are properly eliminated with a deflation step. The combination of the algebraic solution of the highconductive blocks with deflation is realized as a twostage preconditioner strategy. The deflation stage is carried out by further isolating the aggregate blocks with a matrix compensation procedure. We validate the performance of the PA approach against ILU preconditioning. Preliminary numerical results indicate that the PA twostage preconditioning can be used as a promising alternative to employ existing algebraic multilevel methods in a more effective way.



Mortar Coupling of Discontinuous Galerkin and Mixed Finite Element Methods
Authors G.V. Pencheva, S.G. Thomas and M.F. WheelerGeological media exhibit permeability fields and porosities that differ by several orders of magnitude across highly varying length scales. Computational methods used to model flow through such media should be capable of treating rough coefficients and grids. Further, the adherence of these methods to basic physical properties such as local mass balance and continuity of fluxes is of great importance. Both discontinuous Galerkin (DG) and mixed finite element (MFE) methods satisfy local mass balance and can accurately treat rough coefficients and grids. The appropriate choice of physical models and numerical methods can substantially reduce computational cost with no loss of accuracy. MFE is popular due to its accurate approximation of both pressure and flux but is limited to relatively structured grids. On the other hand, DG supports higher order local approximations, is robust and handles unstructured grids, but is very expensive because of the number of unknowns. To this end, we present DGDG and DGMFE domain decomposition couplings for slightly compressible single phase flow in porous media. Mortar finite elements are used to impose weak continuity of fluxes and pressures on the interfaces. The subdomain grids can be nonmatching and the mortar grid can be much coarser making this a multiscale method. The resulting nonlinear algebraic system is solved via a nonoverlapping domain decomposition algorithm, which reduces the global problem to an interface problem for the pressures. Solutions of numerical experiments performed on simple test cases are first presented to validate the method. Then, additional results of some challenging problems in reservoir simulation are shown to motivate the future application of the theory.



Multipoint Flux Approximations via Upscaling
Authors C.L. Farmer, A.J. Fitzpatrick and R. PotsepaevControlvolume discretizations for fluid flow in anisotropic porous media are investigated. The method uses numerical solutions of the simplest model equation, –div[K(x) grad p(x)] = f(x). The permeability tensor, K(x), is allowed to have discontinuities. Multipoint Flux Approximations (MPFA) are used, and transmissibility coefficients are obtained, in the usual way, from local numerical flow experiments (transmissibility upscaling) for each cell face. For regular Korthogonal grids, with a uniform permeability tensor, the scheme reduces to the standard twopoint flux approximation. Monotonicity of the solution matrix is discussed and a version of the method that provides an Mmatrix is described. This discretization scheme is applied to reservoir simulation on 3D structured grids with distorted geometry, highly anisotropic media, and discontinuities in the permeability tensor. Simulation results are presented and compared with results from other twopoint and multipoint flux approximations.



The Competing Effects of Discretization and Upscaling – A Study Using the qFamily of CVDMPFA
Authors M. Pal and M.G. EdwardsFamilies of fluxcontinuous, locally conservative, finitevolume schemes have been developed for solving the general tensor pressure equation on structured and unstructured grids [1,2]. A family of fluxcontinuous schemes is quantified by a quadrature parameterization [3]. Improved convergence has been observed for certain quadrature points [4]. In this paper a qfamily of fluxcontinuous (CVDMPFA) schemes are used as a part of numerical upscaling procedure for upscaling fine scale permeability fields on to coarse grid scales. A series of data sets [5] are tested where the upscaled permeability tensor is computed on a sequence of grid levels. The upscaling sequence is repeated for three distinct quadrature points (q=0.1, q=0.5, q=1) belonging to the family of schemes. Three types of refinement study are presented: 1. Refinement study with invariant permeability distribution with respect to grid level, involving a classical mathematical convergence study. The same coarse scale underlying permeability map (and therefore the problem) is preserved on all grid levels including the fine scale reference solution. 2. Refinement study with renormalized permeability. In this study, the local permeability is upscaled to the next grid level hierarchically, so that permeability values are renormalized to each coarser level. Hence, showing the effect of locally upscaled permeability, compared to that obtained directly from the fine scale solution. 3. Reservoir field refinement study, in this study the permeability distribution for each grid level is obtained by upscaling directly from the fine scale permeability field as in standard simulation practice. The study is carried out for the discretization of the scheme in physical space. The benefit of using specific quadrature points is demonstrated for upscaling in the study and superconvergence is observed. REFERENCES 1. M.G. Edwards, C.F.Rogers. Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Comput. Geo. (1998). 2. M.G.Edwards, Unstructured, controlvolume distributed, fulltensor finite volume schemes with flow based grids. Comput.Geo. (2002). 3. M. Pal, M.G. Edwards and A.R. Lamb., Convergence study of a family of fluxcontinuous, finitevolume schemes for the general tensor pressure equation. IJNME, 2005. 4. M. Pal, Families of ControlVolume Distributed CVD (MPFA) Finite Volume Schemes for the Porous Medium Pressure Equation on Structured and Unstructured Grids. PhD Thesis, Swansea University, 2007. 5. M.A.Christie, SPE, HerriotWatt University, and M.J Blunt, Imperial College, Tenth SPE comparative Solution Project: A Comparison of Upscaling Techniques, 2001.



Ensemble Level Upscaling of 3D Welldriven Flow for Efficient Uncertainty Quantification
Authors Y. Chen, K. Park and L.J. DurlofskyUpscaling is commonly applied to account for the effects of finescale permeability heterogeneity in coarsescale simulation models. Ensemble level upscaling (EnLU) is a recently developed approach that aims to efficiently generate coarsescale flow models capable of reproducing the ensemble statistics (e.g., P50, P10 and P90) of finescale flow predictions for multiple reservoir models. Often the most expensive part of standard coarsening procedures is the generation of upscaled twophase flow functions (e.g., relative permeabilities). EnLU provides a means for efficiently generating these upscaled functions using stochastic simulation. This involves the use of coarseblock attributes that are both fast to compute and representative of the effects of finescale permeability on upscaled functions. In this paper, we establish improved attributes for use in EnLU, namely the coefficient of variation of the finescale singlephase velocity field (computed during computation of upscaled absolute permeability) and the integral range of the finescale permeability variogram. Geostatistical simulation methods, which account for spatial correlations of the statistically generated upscaled functions, are also applied. The overall methodology thus enables the efficient generation of coarsescale flow models. The procedure is tested on 3D welldriven flow problems with different (Gaussian) permeability distributions and high fluid mobility ratios. EnLU is shown to capture the ensemble statistics of finescale flow results (flow rate and oil cut as a function of time) with similar accuracy to full flowbased methods but at a small fraction of the computational cost.
