A fully implicit WENO scheme on stratigraphic and unstructured polyhedral grids

Abstract

Many authors have used higher-order spatial discretizations to reduce numerical diffusion, which can be particularly pronounced when simulating EOR processes involving active chemical substances that are transported by linear or weakly nonlinear waves. Most high-resolution methods reported in the literature are based on explicit temporal discretizations. This imposes severe time-step restrictions when applied to the type of grids seen in industry-standard simulation models of real assets, which usually have ordersof-magnitude variations in porosities and Darcy velocities that necessitate the use of implicit discretization. Herein, we propose a second-order WENO discretization suitable for complex grids with polyhedral cell geometries, unstructured topologies, large aspect ratios, and large variations in interface areas. The WENO scheme is developed as part of a standard, fully implicit formulation that solves for pressure and transported quantities simultaneously. We investigate the accuracy and utility of the WENO scheme for a series of test cases that involve corner-point and 2D/3D Voronoi grids and black-oil and compositional flow models

Better AD Simulators with Flexible State Functions and Accurate Discretizations

“…Let us return to the mass densities from (5.13). If a model has a density implementation, we can easily plot the function and all of its dependencies by calling 4 :…”

“…A nonzero weight w j, is assigned to each interpolated value, with j w j, = 1. These are nonlinear functions that depend on the cell-averaged values, and small weights are assigned to cell triplets (i,j, ) giving large gradients to avoid spurious oscillations near discontinuities in the solution (see Lie et al [4] for more details). The scheme is second-order accurate away from discontinuities and applicable to fairly general grids.…”

Using State Functions and MRST’ls AD-OO Framework to Implement Simulators for Chemical EOR

“…To improve the resolution we will use the implicit, second-order WENO scheme developed in [23,24], which is explained briefly in Chapter 5. Instead of computing fluxes from cell-averaged quantities, one uses a nonlinear procedure to reconstruct a piecewise linear approximation to the mobilities inside each cell and then uses reconstructed point values on opposite sides of each cell interface to evaluate the numerical fluxes.…”

Implicit Discontinuous Galerkin Methods for Transport Equations in Porous Media

“…. ,u d ipq ) by building planar reconstructions using triplets of values (u 0 i ,u 0 p ,u 0 q ) from i and two of its neighbors, p and q , for p,q ∈ N (i); see [10] for more details. The kth degree of freedom ûk i is then taken as the minmod of the original value u k i and all of the derivative approximations.…”