Spatially-varying Compact Multi-point Flux Approximations for 3-d Adapted Grids with Guaranteed Monotonicity

We propose a new single-phase local transmissibility upscaling method for adapted grids in 3-D domains that uses spatially varying and compact multi-point flux approximations (MPFA). The multi-point stencils used to calculate the fluxes across coarse grid cell faces are required to honor three generic flow problems as closely as possible while maximizing compactness. We also present a corrector method that adapts the stencils locally to guarantee that the resulting pressure matrix is an Mmatrix. Finally, we show how the computed MPFA can be used to guide adaptivity of the grid. Introduction In this work we are concerned with transmissibility upscaling for coarse-scale modeling of subsurface formations with complex heterogeneity. Designing a transmissibility upscaling method that is both accurate and computationally efficient is challenging. Upscaling methods that use two-point flux approximations (TPFA) may not give sufficiently accurate upscaling results in formations with large scale connective paths that introduce full-tensor anisotropy at coarse scales [1, 2]. On the other hand, MPFA methods, which involve additional cells, add computational costs and may suffer from non-monotonicity [3]. Our goal is to construct a local multi-point flux approximation that accommodates full-tensor anisotropy and guarantees a monotone pressure solution. To achieve this we allow the MPFA ∗Address all correspondence to this author. stencil to vary spatially, we minimize the number of points involved in the stencil, and we let the stencil revert to a TPFA stencil where this is sufficiently accurate. We exploit the added freedom in allowing the stencils to vary from grid cell to grid cell in order to guarantee monotonicity. Our method, introduced in [4], is called the Variable Compact Multi-Point method, or VCMP. VCMP has been implemented in two spatial dimensions on various types of grids with cell-centered finite volume discretizations. In this paper, we describe an extension to 3-D problems, and how VCMP can guide adaptive mesh refinement. Fine and Coarse Grid Equations The upscaling strategy is based on single phase, steady and incompressible flow. The governing dimensionless pressure equation is ∇ · (k ·∇p) = 0. Here p is the pressure and k the permeability tensor.We follow the common practice of using this same equation for the coarse pressure, which is justified in [5]. Construction of VCMP for Cartesian Grids We aim to construct a multi-point finite-volume scheme that is “close” to a two-point scheme, for efficiency and robustness. However, it should be very accurate for smooth pressure fields, even in cases of full-tensor anisotropy. If the pressure field is not smooth, improved accuracy can be achieved by local grid refinement. Therefore, the scheme should be applicable to adaptive grid strategies, such as the CCAR strategy developed in [1, 6]. To create a MPFA with these properties, we allow the stencil 1 Copyright c © 2008 by ASME to vary per cell face. Our MPFA uses a subset of the ten pressure values p j, j = 1, . . . ,10 surrounding the face. Odd-numbered and even-numbered points are on opposite sides of the face, with points 1 and 2 being the points used in a TPFA, and the remaining points chosen to be the centers of neighboring cells. For each j, we let t j denote the weight that will be assigned to point j in the flux approximation, which has the general form f = −tT p, where t = [ t1 · · · t10 ]T , and p = [ p1 · · · p10 ]T . We solve the pressure equation on the local region of the fine grid containing the ten points with three generic Dirichlet boundary conditions. We let pi(x,y), i = 1,2,3, be the solutions of these local problems, and pj denote the value of pi(x,y) at point j. The pressure field p1 is computed using boundary values chosen so that the pressure gradient is across the face, and p2 and p3 are obtained from boundary values chosen so that the pressure gradient is parallel to the face. For i = 1,2,3, we let fi denote the coarse-scale flux (sum of fine-scale fluxes) across the face obtained from the local solution pi(x,y). To compute the weights t j, we solve the constrained optimization problem min t 3 ∑ i=1 αi ∣∣tT pi + fi∣∣2 + 10 ∑ j=3 βjt 2 j , (1) 10 ∑ j=1 t j = 0, t2 j−1 ≤ 0, t2 j ≥ 0, j = 1, . . . ,5. (2) In the current implementation, the weights αi are chosen to be | fi| and the weights β j are chosen to be equal to (| f1|+ | f2|+ | f3|)/M, where M is a tuning parameter. The larger the value of M, the more closely the flows are honored. The M-fix for Guaranteeing Monotonicity While VCMP is robust, it does not guarantee an M-matrix. This property is highly desirable, as it ensures monotonicity of the pressure solution, and improves solver efficiency [7]. We therefore employ the M-fix, a corrector method introduced in [8] for the 2-D case. It entails identifying matrix entries with the wrong sign, and recomputing the corresponding MPFAs, with additional constraints chosen in such a way as to guarantee an M-matrix. If the modified optimization problem is infeasible, which rarely occurs in practice, a two-point flux is used instead. Application to Adaptive Mesh Refinement In [1], a grid adaptation strategy was introduced in which cells surrounding a face are refined if, in global coarse-scale flow simulations, a sufficiently large fraction of the total flow passes through the face. However, this causes unnecessary refinement when flow is oriented with the grid. We solve this problem by refining only if the weights t3-t10 are sufficiently large, which only occurs in the presence of full-tensor anisotropy. In addition, as an alternative to applying the M-fix, we can refine where elements of the matrix have the wrong sign, or where no stencil that satisfies the sign constraints (2) can be computed by VCMP. Summary and ConclusionsWe have generalized VCMP to 3-D domains. VCMP ac-commodates full-tensor anisotropy, which is generally present incoarse-scale flow problems. The stencil adapts to the orientationof the underlying fine permeability distribution, and can be usedas an indicator for adaptivity.Lack of monotonicity may occur in cases where strong per-meability contrasts are not aligned with the grid. As in the 2-Dcase, the M-fix can be used as a corrector step in the VCMPmethod to guarantee the pressure matrix is an M-matrix. REFERENCES[1] Gerritsen, M., and Lambers, J., 2008. “Integration of local-global upscaling and grid adaptivity for simulation of sub-surface flow in heterogeneous formations”. ComputationalGeosciences, 12, pp. 193–208.[2] Chen, Y., Mallison, B., and Durlofsky, L., 2008. NonlinearTwo-point Flux Approximation for Modeling Full-tensor Ef-fects in Subsurface Flow Simulations. in press.[3] Nordbotten, J., Aavatsmark, I., and Eigestad, G., 2007.“Monotonicity of Control Volume Methods”. NumerischeMathematik, 106, pp. 255–288.[4] Lambers, J., Gerritsen, M., and Mallison, B., 2008. AccurateLocal Upscaling with Variable Compact Multi-point Trans-missibility Calculations. in press.[5] Durlofsky, L., 1991. “Numerical calculation of equivalentgrid block permeability tensors for heterogeneous porousmedia”. Water Resources Research, 27, pp. 699–708.[6] Nilsson, J., Gerritsen, M., and Younis, R., 2005. “A NovelAdaptive Anisotropic Grid Framework for Efficient Reser-voir Simulation”. In Proc. of the SPE Reservoir SimulationSymposium, SPE. SPE 93243. [7] Stuben, K., 1983. “Algebraic multigrid (amg): experiencesand comparisons”. Appl. Mathematics and Computation, 13,pp. 419–452.[8] Gerritsen, M., Lambers, J., and Mallison, B., 2006. “AVariable and Compact MPFA for Transmissibility Upscalingwith Guaranteed Monotonicity”. In Proceedings of the 10thEuropean Conference on the Mathematics of Oil Recovery,EAGE. 2Copyright c© 2008 by ASME