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Advanced Numerical Methods in the Mathematical Sciences

Seong Lee, Chevron Energy Technology Company
Recent Advances in Numerical Methods Applied to Non-Linear Transport Equations in Reservoir Simulation


This paper discusses modeling and mathematical challenges in developing efficient, accurate numerical methods to solve multi-phase flow in heterogeneous porous media. The major mathematical difficulties originate from the uncertainties in governing equations and boundary geometries and scale-dependent complexity in physical parameters and measurements. For instance, the permeability of natural formations displays high variability levels and complex structures of spatial heterogeneity which spans a wide range of length scales. Relative permeabilities for multi-phase flow are measured in a laboratory by a core sample of 1-2 inches, and they are directly applied in a reservoir simulation grid of 10-100 feet that may include large and different heterogeneity. This paper reviews recent advances in numerical methods for non-linear transport equations in reservoir simulation:

  1. Multi-point flux approximation.
  2. Hierarchical approach to naturally fractured reservoir with multiple length scales.
  3. Multi-scale finite volume method.
  4. Dynamic upscaling/downscaling.
  5. Sequential fully implicit method.

The paper also reviews issues and challenges in practical simulation of field scale models.