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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. 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 problems. The first test, which is used to determine the order of the scheme, is the one dimensional 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.

Introduction

Several investigators have shown that the use of singlepoint upstream inter-block transmissibility in Black Oil simulation results in unacceptable levels of numerical diffusion. Simple second order weighting schemes, such as mid-point and two-point upstream, have failed to become widely used alternatives, as their use causes either unphysical solutions (mid-point weighting) or spurious oscillations (unconstrained two-point upstream weighting). Even properly constrained two-point upstream schemes, do not reduce numerical diffusion in multi-component applications where miscible flow of components dominates the problem. In addition, second order schemes are difficult to incorporate into fully implicit codes

More complex second order schemes have been used for some time to produce accurate results for simple hyperbolic models. When applied to Black Oil equations, their mixed parabolichyperbolic nature complicates usage. Second order techniques parabolichyperbolic nature complicates usage. Second order techniques such as Godunov's method, Flux Limiter methods and Flux Corrected Transport all require that the parabolic and hyperbolic portions of the Black Oil equations are split and solved separately. It normally requires a great deal of code development to introduce these schemes into IMPES models. Further, these schemes cannot easily be introduced into Fully Implicit (eg. Coning) codes.

This work introduces a second order scheme which is simple to apply to both IMPES Black Oil models and Fully Implicit Black Oil models. The total variation diminishing (TVD) mid-point scheme is derived from TVD Flux Limiting schemes by reducing the latter scheme's temporal accuracy to first order. The resulting scheme is second order correct spatially and is easily implemented in existing simulators

In the following sections of this work, we derive the TVD criterion, and review the TVD mid-point flux scheme.

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