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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
21  40 of 105 results


Indirecterrorbased, Dynamic Upscaling of Multiphase Flow in Porous Media
Authors S.H. Lee, H. Zhou and H. TchelepiWe propose an upscaling method that is based on dynamic simulation of a given model in which the accuracy of the upscaled model is continuously monitored via indirect measures of the error. If the indirect error measures are larger than a specified tolerance, the upscaled model is dynamically updated with approximate finescale information that is reconstructed by a multiscale finite volume method (Jenny et al., JCP 217: 627641, 2006). Upscaling of multiphase flow entails detailed flow information from the underlying fine scale. We apply adaptive prolongation and restriction operators for the flow and transport equations in constructing an approximate finescale solution. This new method reduces the inaccuracy associated with traditional upscaling methods, which rely on prescribed boundary conditions in computing the upscaled variables. This dynamic upscaling algorithm is validated for incompressible twophase flow in two dimensional heterogeneous domains. We demonstrate that the dynamically upscaled model achieves high numerical efficiency compared with finescale computations and also provides excellent agreement with reference finescale solutions.



Vertical Aggregation of Reservoir Simulation Models for Numerical Well Testing
Authors R.A. Archer and J. KohUpscaling reservoir simulation models in an appropriate manner is important for any well test interpretation workflow which involves numerical reservoir simulation. Approaches based on steadystate flow behaviour are not necessarily optimal when trying to capture the details of pressure transient phenomena. This study proposes a new layer aggregation strategy which forms a weighted sum of the permeability differences between cells in neighbouring layers. The weighting uses a kernel function (G) presented by Oliver (1990). This approach was successfully applied in four upscaling experiments using a permeability distribution based on the Tarbert formation. These cases included a standard well test, an interference test and a test from a partially penetrating well. The results showed that coarsened models with as few as five layers can reproduce the pressure transient behaviour of a twenty layer fine grid model.



Upscaling of Large Reservoir Systems by Using a Controlrelevant Approach
Authors S.A. VakiliGhahani, R. Markovinovic and J.D. JansenIn reservoir simulations, we use an ensemble of finescale geological models that are upscaled to coarser representations for flow simulations. The primary objective of the upscaling process is to meet the computational limits of the simulator. However, we argue that from a systemtheoretic point of view, a more fundamental underlying reason for upscaling is that the complexity level of a model should be adjusted to the amount of available information from measurements and the extent of control (input) exercised by adjusting the well parameters. That is because for a given configuration of wells, a large number of combinations of state variables (pressure and saturation values in the gridblocks) are not actually controllable and observable and accordingly they are not affecting the inputoutput behavior of the model. Therefore, we propose a “controlrelevantupscaling” (CRU) approach that determines equivalent coarsescale permeabilities based on the actual system’s inputoutput behavior. The coarsescale parameters are obtained as the solution of an optimization problem that minimizes the distance between the inputoutput behaviors of the fine and coarsescale models. This distance is measured by using Hankel or energy norms, in which we use Hankel singular values and Markov parameters as a measure of the combined controllability and observability, and response of the system, respectively. This work focuses on singlephase flow upscaling, where we develop a CRU algorithm for reservoir systems. Moreover, we address the potential benefits of using proper orthogonal decomposition (POD) in combination with our CRU method to obtain a reducedorder CRU algorithm that accelerate the upscaling procedure. In the cases considered, the CRU algorithm shows superior inputoutput behavior as compared to upscaling algorithms commonly used in reservoir simulators.



The MPFA G Scheme for Heterogeneous Anisotropic Diffusion Problems on General Meshes
Authors L. Agelas, D. Di Pietro, I. Kapyrin and R. MassonFinite volume cell centered discretizations of multiphase porous media flow are very popular in the oil industry for their low computational cost and their ability to incorporate complex nonlinear closure laws and physics. However, the trend towards a more accurate description of the geometry and of the porous medium requires to dispose of flux approximations handling general polyhedral meshes and full diffusion tensors. The convergence on general meshes as well as the robustness with respect to anisotropy and heterogeneity of the diffusion tensor should come at a reasonable computational cost. An important property on which the analysis relies is coercivity, which ensures the stability of the scheme and allows to prove convergence of consistent discretizations. Meeting all these requirements is still a challenge and this is an active field of research. The Lmethod, which has been recently introduced by Ivar Aavatsmark and coworkers, displays enhanced monotonicity properties on distorted meshes and for anisotropic diffusion tensors. In this work, we present a generalization of the Lmethod based on a discrete variational framework and on constant gradient reconstructions. The coercivity of the discretization requires a local condition depending both on the mesh and on the diffusion tensor to be satisfied. The monotonicity and convergence properties of the scheme are assessed on challenging singlephase problems with distorted meshes and anisotropic and heterogeneous diffusion tensors. The proposed method is compared with the standard O and Lmethods as well as with an unconditionally symmetric, coercive cell centered finite volume scheme using a larger stencil.



Monotonicity for Control Volume Methods on Unstructured Grids
Authors E. Keilegavlen and I. AavatsmarkIn reservoir simulation the pressure is the solution of an elliptic equation. It follows from the maximum principle that this equation satisfies a monotonicity property. This property should be preserved when discretising the pressure equation. If this is not the case, the pressure solution may have false internal oscillations and extrema on noflow boundaries. Thus, there is a need for sufficient conditions for the discretization methods to be monotone. These conditions will depend on the permeability tensor and the grid. Previously monotonicity for control volume methods has been studied on quadrilateral grids and on hexagonal grids. The former was done for general methods which reproduces linear potential fields, while the latter was done in a Control Volume Finite Element setting. These analyses have given sharp sufficient conditions for the discretisation methods to be monotone. However, for methods whose cell stencils include cells which do not have any edges common with the central cell in the discretisation scheme, no work has been done on unstructured grids. In this work, we study monotonicity on triangular grids for control volume methods which are exact for linear potential fields. We derive sufficient conditions for monotonicity of the MPFAO and L methods. The found monotonicity regions for the MPFA methods are also tested numerically. The tests are done both on uniform grids in homogeneous media, and on perturbed grids, which corresponds to heterogeneous media. The investigations are done for single phase flow only. However, the results are relevant for multiphase simulations. The results obtained in this work may be utilised in grid generation. In this way we can construct grids where the discretisation of the pressure equation is guaranteed to be monotone.



Higher Dimensional Upwind Schemes for Flow in Porous Media on Unstructured Grids
Authors M.S. Lamine and M.G. EdwardsStandard reservoir simulation schemes employ singlepoint upstream weighting for approximation of the convective fluxes when multiple phases or components are present. These schemes rely upon upwind information that is determined according to the grid geometry. As a consequence directional diffusion is introduced into the solution that is grid dependent. The effect can be particularly important for cases where the flow is across grid coordinate lines and is known as crosswind diffusion. Novel higher dimensional upwind schemes that minimize crosswind diffusion are presented for convective flow approximation on structured and unstructured grids. The schemes are free of spurious oscillations and remain locally conservative. The higher dimensional schemes are coupled with fulltensor Darcy flux approximations. Benefits of the resulting schemes are demonstrated for classical convective test cases in reservoir simulation including cases with full tensor permeability fields, where the methods prove to be particularly effective. The test cases involve a range of unstructured grids with variations in orientation and permeability that lead to flow fields that are poorly resolved by standard simulation methods. The higher dimensional formulations are shown to effectively reduce numerical crosswind diffusion effect, leading to improved resolution of concentration and saturation fronts. TECHNICAL CONTRIBUTIONS Locally conservative unstructured multidimensional schemes coupled with fulltensor Darcy flux approximations are presented for reservoir simulation. The multidimensional schemes are developed for general unstructured grids. The schemes are proven to be positive subject to conditions on the tracing direction and permit higher CFL conditions than standard schemes. Comparisons with single point upstream weighting scheme are made on a range of unstructured grids for different grid orientation and aspect ratios in cases involving full tensor coefficient velocity fields. The numerical results demonstrate the benefits of the higher dimensional schemes both in terms of improved front resolution and significant reduction in crosswind diffusion.



Grid Optimization to Improve Orthogonality of Twopoint Flux Approximation for Unstructured 3D Fractured Reservoirs
More LessToday's geological models are increasingly complex. The use of unstructured grids is an efficient way to account for this complexity especially when dealing with large network of fractures and faults. Flux based approximations are commonly used in reservoir simulation community to discretize the flow equations. The twopoint flux approximation (TPFA) is the simplest and the most robust discretization technique. Application of TPFA requires an orthogonal grid. When applied to an unstructured model, Perpendicular Bisector (PEBI) grids are usually used as they are orthogonal by construction. But the construction of PEBI grids can become challenging for fully 3D models containing a large number of fractures and faults. In this work we propose to use an unstructured grid which matches exactly all discrete features (fractures and faults) but may not be orthogonal. The objective is to relax the orthogonality criteria to simplify the grid generation step and to deal with the orthogonality at the flux approximation step. Typical discretization techniques for nonorthogonal grids are based on multiplepoint flux approximation (MPFA) where more than two pressure points are used to evaluate the flux. Although substantial progress has been made in improving MPFA, they remain more complex and less robust than TPFA. In this work we present a simple methodology to apply a TPFA to unstructured nonorthogonal grids. The idea is to determine the location of the pressure node inside each controlvolume to make the connections with the surrounding cells as orthogonal as possible. Although it is not always possible to make all connections perfectly orthogonal, this optimization procedure improves systematically the grid quality. In addition, this purely geometrical optimization is done at the preprocessing level and does not require any flow information and can be used with any connectivity based flow solver.



High Performance Dual Mesh Method to Simulated Twophase Gravity Dominated Flows in Porous Media
Authors M.A. Ashjari and B. FiroozabadiThis paper presents a new combined method for accurate upscaling of twophase displacements in highly heterogeneous reservoirs. The method has the capability to retain its high performance for various flow regimes, from viscous to gravity dominant displacements, without the need for further modifications and computational steps. Two different grids are incorporated for simulation. The grid on fine scale is used to recognize the complicated physics of flow which depends on dominated driving forces and their interaction with heterogeneity. However, to achieve a fast simulation, the global flow calculation is performed on the coarse scale grid using upscaled equivalent properties. The communication between two different scale grids is achieved by the dual mesh method (DMM) procedure. Since DMM performance is still dependent on the accuracy of the coarse scale simulation, vorticitybased coarse grid generation technique is also incorporated to limit the upscaling errors. The technique optimizes the coarse grid distribution based on vorticity preservation concept where singlephase vorticity is attempted to be preserved among fine and coarse grid models. To demonstrate accuracy and efficiency, the combined DDMvorticity method is applied to highly heterogeneous systems in two dimensions with and without gravity. The results reveal that the flow regime has only minor impact on the performance of the combined method.



Discretisation Schemes for Anisotropic Heterogeneous Problems on Nearwell Grids
Authors S.S. Mundal, E. Keilegavlen and I. AavatsmarkIn reservoir simulation, a proper treatment of wells is crucial for the performance of fluid flow models. Important well parameters such as the well flow rate and the wellbore pressure are highly sensitive to the computational accuracy of nearwell flow models. Nearwell regions are characterised as high flow density regions, and the dominating flow pattern exhibits a radiallike nature with large pressure gradients and, for multiphase flow, large saturation gradients. Skew or horizontal wells will also in general imply a strong effect of anisotropy and heterogeneities due to geological layering of the reservoirs. Due to difference in the reservoir scale and the wellbore radius, reservoir flow models do not capture the true flow behavior in the well vicinity. Thus, more flexible models with local grid refinement in nearwell regions are needed. The models should also be adapted to handle complex geological nearwell structures and multiphase flow simulations. Existing nearwell models are in general based on homogeneous media in the well vicinity. Anisotropies and strong heterogeneities are less accounted for in nearwell flow simulations. In this work, we construct analytical solutions for nearwell flow which is not aligned with a radial inflow pattern. These solutions resemble strongly heterogeneous, possible anisotropic media. We compare different control volume discretisation schemes and radialtype grids for such cases and give their convergence behavior. The objective is to obtain a clearer view on the accuracy of the nearwell grids and discretisation schemes for large contrast in permeability, and hence, to determine which is the preferable grid and a suitable numerical scheme given certain nearwell conditions. The simulations are performed for singlephase flow, however, the results will also have relevance for multiphase flow simulations.



A stochastic trapping model for nonwetting phase ganglias in two phase flow through porous media
Authors M. Tyagi, H.A. Tchelepi and P. JennyResidual trapping is one of the CO2 retention mechanism during CO2 storage in saline aquifers. In a region where the CO2 saturation falls below the residual saturation, isolated ganglias of CO2 are formed which can become immobile (trapped). In this paper, we propose a stochastic model for trapping and release of nonwetting phase ganglias. Opposed to traditional trapping models, which only take into account of the mean quantities, the proposed stochastic model contains additional statistical informations, e.g., correlation time. By selecting appropriate model parameters, both drainage and imbibition can be described without a priori knowledge of the displacement direction.



A Twoscale Operator Splitting for Flow Problems in Porous Media
Authors Y. Cao, A. Weiss and B. WohlmuthOperator splitting time discretization techniques are getting more and more popular for the numerical simulation of nonlinear problems in porous media. The main idea is to split complex operators in evolution equations into simpler ones which are successively solved in each time step. For porous media flow a natural splitting is given by the diffusive and the advective part of the flux. Then for each part, optimal solvers for the particular evolution equation can be used, i.e., a parabolic solver for the diffusion equation and a hyperbolic solver (e.g., characteristics methods) for the advection equation. In this talk, we first present an operator splitting discretization using an implicit finite volume scheme for the diffusive part and a semiLagrangian method for the advective part. Hence, the computational cost of the semiLagrangian method can be neglected compared to the one of the implicit finite volume scheme. In the second part, we introduce a twoscale operator splitting method where the diffusion part is solved on a coarse grid while the advection part is considered on a fine grid. This is motivated by the fact that the diffusion and the advection act on different scales. Then, the resulting operator splitting discretization is equivalent in performance to a fast hyberbolic solver. In numerical examples, our proposed methods are compared with standard implicit finite volume methods. We will demonstrate that by using the operator splitting time discretization, the number of iterations in the implicit finite volume scheme may be significantly reduced.Moreover, for the twoscale method, it turns out that despite of the poor approximation in the diffusive part, the solution is still of equal quality compared to that of standard methods on fine grids. To summarize, operator splitting techniques provide a very flexible and powerful framework which makes it quite attractive for an efficient solution strategy for flow problems in porous media.



A General Multiscale FiniteVolume Method for Compressible Multiphase Flow in Porous Media
Authors H. Hajibeygi and P. JennyThe Multiscale FiniteVolume (MSFV) method was originally proposed to solve elliptic problems arising from incompressible multiphase flow in highly heterogeneous porous media at reduced computational cost. However, when phases are compressible, especially when the reservoir contains gas, mathematical formulations lead to a parabolic equation to be solved for pressure. In this paper we introduce a general MSFV method to deal with such parabolic problems. In this scheme, the basis and correction functions are numerical solutions of the full parabolic problems in localized domains. Hence compressibility effects are represented by a stencil in the coarsesystem matrix, i.e. not only by a diagonal entry. Furthermore, to enhance the computational efficiency of the scheme, the basis functions are kept independent of the pressure field. As a result, only the correction functions (1 per dualcoarse cell) have to be updated during the iterative procedure. It is an important property of this approach that it requires no additional simplifications. Finally, its good efficiency is demonstrated for a number of challenging test cases.



Multiscale Asynchronous Algorithms Based on the Superelements Method for Multiphase Flow
Authors A.K. Pergament, V.A. Semiletov and P.Y. TominThe typical scales of pore pressure and fluid saturations variations are different both for space and temporal variables. The multiscale method is based on using the coarse grid for pore pressure equation and fine grid for saturation equations. The time step for these equations may be different too. The essential feature of the method is the basic function construction by solving the one phase stationary equations. These basis functions have the same peculiarities determined by the fine grid structures. The solution of pore pressure equations may be constructed as linear span of these basic functions for BuckleyLeverett equations. In general case it is possible to construct the tensor total permeability coefficients and to upscale the relative permeability. The method of calculating these variables is the dissipative energy integral approximation. As a result we have the nonlinear parabolic equation for pore pressure with tensor coefficients. The implicit schemes on the fine grid are considered for saturation equation. The general method is high resolution method. It is essential this method is a high resolution one. The results of modeling some problems are represented.



Multiscale/Mimetic Pressure Solvers with Nearwell Grid Adaptation
Authors B. Skaflestad and S. KrogstadMultiscale methods have proved to be an efficient means to achieving fast simulations on large reservoir models. The main characteristic of these methods is high resolution flow fields obtained at relatively low computational cost. We are thus able to resolve large scale flow patterns as well as finescale details that would be impossible to obtain for models for which direct simulation using traditional methods would be prohibitive. However, there are still a number of open problems in applying these methods to reservoir simulation. In particular, we observe some discrepancy in the performance of wells when compared to direct simulation on the finescale grid. To improve the multiscale method's predictive power for individual wells, we consider two direct opposite strategies: First, by resolving the nearwell flow in the coarse grid by adaptive grid refinement in regions near wells. Second, by ensuring that nearwell flow is sufficiently captured in the corresponding multiscale basis functions. For the latter strategy, we consider both adaptive alignment of coarse pressure grid to well trajectories, and an oversampling method for the computation of the multiscale basis functions corresponding to wells. In this paper we will study the effects of such nearwell grid adaptations, and state pros and cons for the approaches considered.



Spontaneous Ignition in Porous Media at Long Times
Authors J. Bruining and D. MarchesinIf oxygen is in intimate contact with fuel Arrhenius law says that reaction will occur, even at low temperatures. Heat losses can become equal to the small reaction heat generated, so that the system remains trapped in a slow reaction mode. Such a mode is indistinguishable from extinction. On the other hand, if heat losses remain smaller than the heat generated by the reaction the temperature increases and spontaneous ignition occurs. Heat losses are strongly dependent on the geometry of the heat generating region. We therefore distinguish three idealized geometries, viz. linear, cylindrical and spherical infinite domains. We analyze the long time behavior of a basic heat diffusion problem that incorporates an Arrhenius heat generation term; the resulting model is temperature controlled. Only in spherical geometry the model recovers the results from the conventional heat transfer concept. Indeed, we come to the following results. As reactant depletion is ignored, spontaneous ignition always occurs, in linear and cylindrical coordinates, even if it takes very long time. For the spherical case, the occurrence of ignition or extinction depends on the process conditions.



Study of Heavy Oil Recovery from a Fractured Carbonate Reservoir Using in Situ Combustion
Authors H. Fadaei, M. Quintard, G. Debenest, G. Renard and A. KampIn spite of its strategic importance, the topic of recovery of heavy crude oils from fractured carbonate reservoir has not been extensively addressed. Thermal methods seem well suited for this kind of problems, particularly in situ combustion has shown promising results in laboratory experiments. Extensive work has been done on development of thermal process simulator but for the insitu combustion applied specially in fracture reservoirs where one is dealing with multiscale multiprocess problem, many unknowns are still exist. The recovery mechanism, reservoir and operational conditions on which the combustion can propagate in fractured systems are not enough clear. Also due to safety issues, air injection required careful assessment of the reservoir displacement mechanisms in particular the magnitude and the kinetics of matrixfracture transfers. To understand the mechanism of heavy oil recovery from a fractured reservoir we propose the development of a numerical simulation strategy, starting from existing simulation tools that are adapted to this particular problem. This will allow firstly understanding the role of each driving mechanism and physical as well as operational parameters in recovery process and secondly choosing the suitable upscaling method. The study is based on the fine grid, single porosity, multiphase and multicomponent simulation of a core surrounded by two parallel air invaded fractures using the thermal simulator. Firstly the simulator is validated for different processes: one and two dimensional diffusion and onedimensional combustion are compared with the corresponding analytical solutions. The twodimensional combustion is validated using experimental results available in the literature. The simulation results predict the conditions on which the combustion is sustained in the fractured reservoir as a function of oxygen diffusion coefficient, injection rate, and the permeability of the matrix. Oil production via natural drainage, hot fluid injection and insitu combustion are compared to address the importance of different driving mechanisms. At the block scale the effect of fracture spacing, heterogeneity in the matrix and the grid size on the propagation of combustion and the oil production are studied and then a suitable upscaling procedure is proposed.



Modeling CO2 Sequestration Using a Sequentially Coupled 'IterativeIMPECtimesplitthermal' Compositional Simulator
Authors M. Delshad, S.G. Thomas and M.F. WheelerThis paper presents an efficient numerical scheme for multiphase compositional flow coupled with subsurface heat transfer. The flow equations are first presented followed by a brief discussion of the equation of state (EOS) and a description of the twophase flash algorithm. An implicitpressure, explicitconcentrations (IMPEC) sequential algorithm is then applied iteratively to enforce the nonlinear volume balance saturation) constraint. The pressure equation is solved using mixed FEM, while the concentrations are updated consistent with requiring local mass balance of every component. Thermal effects also play an important role in such problems since they effect the phase properties and hence, stable and accurate locally conservative methods are desirable to model the thermal energy balance equation. To this end, we present also a timesplit scheme for modeling the energy balance equation that is sequentially coupled to the flow solution rendering it cheap yet accurate for the complex problems being modeled. Results of benchmark problems in compositional flow modeling are presented and validated where possible. Some largescale parallel tests are performed on more challenging applications such as those with highly heterogeneous permeability fields on very fine grids. Efficient parallel scalability of the code on upto 512 processors is also demonstrated. Finally, some test cases simulating "real world" problems of CO2 sequestration in deep saline aquifers are presented. Initial results show reasonably good agreement of CO2 plume shapes, arrival times, leakage rates and production curves with semianalytic solutions, wherever available. All computations were carried out within the IPARSCO2 code base framework.



Densitydriven Natural Convection in Dual Layered and Anisotropic Porous Media with Application for CO2 Injection Project
Authors R. Farajzadeh, F. Farshbaf Zinati, P.L.J. Zitha and J. BruiningIn this paper we investigate the mass transfer of CO2 injected into a layered and anisotropic (sub)surface porous formation saturated with water. Solutions of carbon dioxide in water and oil are denser than pure water or oil. We perform our analysis to a rectangular part of the porous medium that is impermeable at the sides except at the top, which is exposed to CO2. For this configuration densitydriven natural convection enhances the mass transfer rate of CO2 into the initially stagnant liquid. The analysis is done numerically using mass and momentum conservation laws and diffusion of CO2 into the liquid. This configuration leads to an unstable flow process. Numerical computations do not show natural convection effects for homogeneous initial conditions. Therefore a sinusoidal perturbation is added for the initial top boundary condition. It is found that the development of fingers is fastest for mass transfer enhancement by natural convection is largest for large anisotropy ratio’s and smaller for small ratio's. It is found that the mass transfer increases and concentration front moves faster with increasing Rayleigh number if the high permeability layer is on top. Of particular interest is the case when the Rayleigh number for the high permeable layer is above the critical Rayleigh number (Racr = 40) and smaller than Racr for the low permeable layer. The results of this paper have implications in enhanced oil recovery and CO2 sequestration in aquifers.



Operator Splitting of Advection and Diffusion on Nonuniformly Coarsened Grids
Authors V.L. Hauge, J.E. Aarnes and K.A. LieHighresolution geological models of subsurface reservoirs typically contain much more details than can be used in conventional reservoir simulators. Geomodels are therefore upscaled before flow simulation is performed. Here, we present a nonuniform coarsening strategy to upscale geomodels, where the coarse grid is generated by grouping cells in the finescaled geomodel selectively into connected coarse blocks, with some minimum volume and with the total flow through each coarse block bounded above. Transport is then modeled on this simulation grid. For purely advective flow, the coarsening strategy has been shown to be robust, allowing for both accurate and fast transport simulations of highly heterogeneous and fractured reservoirs. Here, we consider advectiondominated twophase flow, where the capillary diffusion is discretized separately from the advective and gravity terms using operator splitting. In particular, we investigate different damping strategies of the diffusive term to counteract overestimation of the diffusion operator on the coarse grid, this to ensure accurate diffusion modelling in the transport solver.



Asynchronous Time Integration of Fluxconservative Transport
Authors B.T. Mallison, M.G. Gerritsen and G. KreissWe investigate the potential of a fluxconservative, asynchronous method for the explicit time integration of subsurface transport equations. This method updates each discrete unknown using a local time step, chosen either in accordance with local stability conditions, or based on predicted change of the solution, as in Omelchenko and Karimabadi (Selfadaptive time integration of fluxconservative equations with sources. J. Comput. Phys. 216 (2006)). We show that the scheme offers the advantage of avoiding the overlyrestrictive global CFL conditions. This makes it attractive for transport problems with localized time scales, such as those encountered in reservoir simulation where localized time scales can arise due to well singularities, spatial heterogeneities, moving fronts, or localized kinetics, amongst others. We conduct an analysis of the accuracy properties of the method for onedimensional linear transport and first order discretization in space. The method is found to be locally inconsistent when the temporal step sizes in two adjacent cells differ significantly. We show numerically, however, that these errors do not destroy the order of accuracy when localized. The asynchronous time stepping compares favorably with a traditional first order explicit method for both linear and nonlinear problems, giving similar accuracy for much reduced computational costs. The computational advantage is even more striking in two dimensions where we combine our integration strategy with an IMPES treatment of multiphase flow and transport. Global time steps between pressure updates are determined using a strategy often used for adaptive implicit methods. Numerical results are given for immiscible and miscible displacements using a structured grid with nested local refinements around wells.
