Standard finite-difference reservoir simulation considers single-point upstream weighting to approximate the components of the convective flow term at the interfaces between blocks. This procedure stabilizes the numerical solution, but introduces high levels of numerical dispersion, making difficult the correct interpretation of the simulated results.

Exponential schemes are reasonable when diffusion dominates, but, as we show in this work, reduce to the single-point uspstream scheme when flow is too convective. A new and simple expression for the finite-difference form of linear conservation equations is derived for the exponential schemes.

Higher-order methods, like Leonard scheme, are able to reduce the numerical dispersion, but may produce non-physical solutions when the conservation equation assumes a hyperbolic form. We demonstrate, through the numerical solution of some classical non-linear equations, that this occurs because the entropy condition is violated. The numerical solutions also show that Total Variation Diminishing (TVD) methods, recently introduced in the reservoir simulation area, have the remarkable property of producing high resolution solutions which respect the entropy condition, and, consequently, are physically consistent.

The main mathematical principles for conservation laws are summarized and a screening procedure to be followed by new numerical methods is detailed. The performance of various numerical methods is analyzed for some standard reservoir engineering problems: 1D convection-diffusion equation, Buckley-Leverett equation and 2D single-phase tracer flow. Stability is analyzed. TVD schemes were also implemented on a two-phase black-oil model, considering IMPES, semi-implicit and fully-implicit formulations. We also included options of tracer injection in the water phase, considering full dispersion tensor and adsorption with the rock. Comparisons are made with refined grid single-point upstream solutions and with semi-analytical solutions. Grid orientation effect is also investigated.

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