Higher-Order Implicit Flux Limiting Schemes for Black Oil Simulation

Abstract

Abstract Conventional Black Oil models normally use single point upstream weighting for mobility terms. This is known to give rise to significant levels of numerical dispersion. Several higher order schemes including Godunov's scheme, Flux Corrected Transport, and two-point upstream mobility weighting have been suggested for use in explicit or IMPES simulators to improve accuracy. FCT and Godunov's scheme cannot easily be applied to fully implicit models, while the unconstrained two-point upstream method is unstable for purely hyperbolic systems and is best suited for IMPES models. The use of higher order schemes in reservoir modelling has thus been extremely limited. In order to remedy this situation, we derive a total variation diminishing (TVD) higher order scheme which can be applied to both fully implicit and IMPES Black-Oil models. We develop IMPES and fully implicit analogues of these schemes for use in standard Black Oil models and discuss their relationship to commonly used mobility weighting schemes (mid-point, single-point upstream, two-point upstream and single-point downstream). The TVD mid-point scheme is a temporally first order version of the second order TVD flux limited Lax-Wendroff scheme. The TVD criterion ensures that oscillations which would have been produced by unconstrained second order schemes never arise. The unconditional stability of the fully implicit TVD mid-point scheme is guaranteed, since the limiter is evaluated along with the fully implicit flux terms. The IMPES implementation has a stability limit which is half that of the first order single-point upstream scheme. Results using the IMPES and implicit TVD scheme in a modified commercial Black Oil code (VIP) are presented for three test problems. The first test, which is used to determine the order of the scheme, is the one dimensioned waterflooding of a slightly compressible oil reservoir (fully implicit formulation). The second test, for scheme accuracy, is the two dimensional water flooding of one-quarter of a 5 spot pattern (IMPES and fully implicit formulations) and the third test, again for accuracy, is the coning of water in a two dimensional radial (R-Z) coordinate system. The results of the above examples are compared with standard first order calculations. In the examples presented, the TVD scheme produced more accurate results from models using one-seventh (and one-fourth) the number of blocks, resulting in a better than four-fold (and three-fold) saving of CRAY XMP CPU time.