Interior boundary-aligned unstructured grid generation and cell-centered versus vertex-centered CVD-MPFA performance

Abstract

Grid generation for reservoir simulation must honor classical key constraints and be boundary aligned such that control-volume boundaries are aligned with geological features such as layers, shale barriers, fractures, faults, pinch-outs, and multilateral wells. An unstructured grid generation procedure is proposed that automates control-volume and/or control point boundary alignment and yields a PEBI-mesh both with respect to primal and dual (essentially PEBI) cells. In order to honor geological features in the primal configuration, we introduce the idea of protection circles, and to generate a dual-cell feature based grid, we construct halos around key geological features. The grids generated are employed to study comparative performance of cell-centred versus cell-vertex control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent degrees of freedom. The formulation of CVD-MPFA schemes in cellcentred and cell-vertex modes is analogous and requires switching control volume from primal to dual or vice versa together with appropriate data structures and boundaryThe original version of this article was revised: Due to an oversight, some author's corrections were not carried out during Performing proof corrections stage. The Publisher apologizes for these mistakes. conditions. The relative benefits of both types of approximation, i.e., cell-centred versus vertex-centred, are made clear in terms of flow resolution and degrees of freedom required.

Three-dimensional unstructured gridding for complex wells and geological features in subsurface reservoirs, with CVD-MPFA discretization performance

“…In this work geological feature based primal and dual grid generation is presented for two groups of features [55,56,57,59]. The first group involves domains that may include geological layers, pinch-outs, fractures and/or faults, and the second group involves well distributions.…”

“…Consequently the cellcentered formulation involves many more degrees of freedom, and is thus more computationally expensive, but might be expected to resolve flow fields more accurately. Secondly for the cell-centred method, the local flux molecule depends on the number of cells attached to a vertex, while the vertex-centred local flux molecule only depends upon the vertices of the primal cell, and consequently the vertex-centred method is relatively compact, with much smaller bandwidth when the discretization matrix is assembled, this is illustrated in two-dimensions in [57].…”

“…Cell- CVD-MPFA schemes work directly with the integral form of the flow equations and are optimal in the sense that they employ a single primal discrete pressure per control-volume, and provide consistent flux-continuous locally conservative approximations of the pressure equation for any permeability tensor and grid type, while satisfying local pressure and normal flux continuity conditions. The continuity conditions imposed around every cluster point, e.g., [26,57], leads to an increased pressure support with wider matrix bandwidth compared to the standard TPFA scheme, but crucially retains the same number of unknown discrete pressures or degrees of freedom. However the TPFA scheme has O(1) error in flux and is generally inconsistent unless the grid is K-orthogonal [6,16].…”

“…max(a i =j )/a ii for a ij > 0 are computed [57]. L ∞ and L 2 norms together with arithmetic mean(x) of all row violations are used as representative measures of M-matrix violation.…”

Unstructured PEBI Grids Conforming to Lower-Dimensional Objects

“…The main purpose of this chapter is to give a basic introduction to these functions, briefly explain some of the underlying theory, and give several code-centric examples of how the functionality in the module can be used to generate complex grids that conform to geological objects and well paths. For a more comprehensive overview of alternative methods and previous research, the reader can consult [2,6,8,[13][14][15]18]. We emphasize that the upr module is a research tool for constrained gridding and not a robust industry-grade geomodeling tool.…”