Integration of local–global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations

Abstract

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...

A system-theoretical approach to selective grid coarsening of reservoir models

“…time-dependent, strategy for selective refinement and coarsening based on different criteria. In particular, adaptive gridding is often applied to maintain a fine grid in areas of high permeability or in areas where high saturation or concentration gradients occur; see, e.g., [5,13] and references therein.…”

“…An alternative technique is a local-global approach, in which the generic boundary conditions are only used to obtain an initial coarse-scale flow solution. The interpolation of the coarse-scale solution then gives more accurate local boundary conditions to find the new coarse-scale parameters and the process is iterated until the solution converges [7,13].…”

“…More accurate pressure calculations at the auxiliary points can be found in e.g. [12,13,15,22]. Spatial and temporal discretization of the flow equation over the non-uniform grid blocks leads to a system of equations that can be written in state-space form Eq.…”

Control-Relevant Upscaling

“…In particular, adaptive gridding is often applied to maintain a fine grid in areas of high permeability or in areas where high saturation or concentration gradients occur; see e.g. Berger & Colella (1989), or Gerritsen & Lambers (2008 and references therein.…”

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

“…Chen and Durlofsky [13] developed an adaptive local/global upscaling procedure that provides coarse-scale parameters that are adapted for more general types of global flows (such as, for example, those driven by a well, or a particular set of boundary conditions). Gerritsen et al [34] introduced a multilevel local/global method, which combines the local/global upscaling in order to improve the representation of the large-scale connectivities.…”