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Abstract. We propose a methodology, called multi-level local-global (MLLG) upscaling, for generating accurate upscaled models of permeabilities or transmissibilities for flow simulation on adapted grids in heterogeneous subsurface formations. The method generates an initial adapted grid based on the given fine-scale reservoir heterogeneity and potential flow paths. It then applies local-global (LG) upscaling for permeability or transmissibility [7], along with adaptivity, in an iterative manner. In each iteration of MLLG, the grid can be adapted where needed to reduce flow solver and upscaling errors. The adaptivity is controlled with a flow-based indicator. The iterative process is continued until consistency between the global solve on the adapted grid and the local solves is obtained. While each application of local-global upscaling is also an iterative process, this inner iteration generally takes only one or two iterations to converge. Furthermore, the number of outer iterations is bounded above, and hence the computational costs of this approach are low. We design a new flow-based weighting of transmissibility values in local-global upscaling that significantly improves the accuracy of LG and MLLG over traditional local transmissibility calculations.For highly heterogeneous (e.g., channelized) systems the integration of grid adaptivity and localglobal upscaling is shown to consistently provide more accurate coarse-scale models for global flow, relative to reference fine-scale results, than do existing upscaling techniques applied to uniform grids of similar densities. Another attractive property of the integration of upscaling and adaptivity is that process dependency is strongly reduced, that is, the approach computes accurate global flow results also for flows driven by boundary conditions different from the generic boundary conditions used to compute the upscaled parameters.The method is demonstrated on Cartesian Cell-based Anisotropic Refinement (CCAR) grids, but can be applied to other adaptation strategies for structured grids, and extended to unstructured grids.1. Introduction. The permeability of reservoir rocks is often highly variable and uncertain. However, it can greatly affect flow and transport in subsurface formations, and must therefore be adequately modeled. The effects of uncertain reservoir heterogeneity can be included either through stochastic modeling or by simulating a number of deterministic geostatistical realizations of the reservoir [9], which is the underlying assumption in this paper. Because the computational costs of solving for a large number of realizations are high, simulations are generally performed on grids that are coarse compared to the given geological models. The grid size is typically determined by computational considerations. These coarsened flow models should adequately represent key behaviors, such as the overall flow rate for given boundary conditions or critical connected flow paths.In this paper, we focus on single-phase upscaling to test the effectiveness o...

Abstract. This paper presents a modification of Krylov subspace spectral (KSS) methods, which build on the work of Golub, Meurant and others, pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to variable-coefficient time-dependent PDEs. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, our modification uses block Lanczos iteration in order to avoid the need to compute two quadrature rules for each component of the solution, or use perturbations of quadrature rules. It will be shown that, under reasonable assumptions on the coefficients of the problem, a 1-node KSS method is unconditionally stable, and methods with more than one node are shown to possess favorable stability properties as well. Numerical results suggest that block KSS methods are significantly more accurate than their non-block counterparts.

For scalar time-dependent variable-coefficient PDE, it has been demonstrated that Krylov subspace spectral (KSS) methods achieve high-order accuracy and also possess highly desirable stability properties, especially considering that they are explicit. In this paper, we examine the generalization of these methods to systems of variable-coefficient PDE by selection of appropriate bases of trial and test functions. Furthermore, we show that for certain special cases, even higher-order accuracy is possible, as has previously been established in the case of the scalar second-order wave equation. Finally, we consider the use of KSS methods to obtain high-order operator splittings for systems of equations.

We propose a new single-phase local upscaling method that uses spatially varying multipoint transmissibility calculations. The method is demonstrated on two-dimensional Cartesian and adaptive 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 two generic local flow problems. The multipoint stencils used to calculate the fluxes across coarse grid cell faces involve the six neighboring pressure values. They are required to honor the two generic flow problems. The remaining degrees of freedom are used to maximize compactness and to ensure that the flux approximation is as close as possible to being twopoint. The resulting multipoint flux approximations are spatially varying (a subset of the six neighbors is adaptively chosen) and reduce to two-point expressions in cases without full-tensor anisotropy. Numerical tests show that the method significantly improves upscaling accuracy as compared to commonly used local methods and also compares favorably with a local-global upscaling method.