Fast, accurate, and robust solution of advection dominated transport equations

• time-of-flight and single-phase tracer flow
• multiphase and multicomponent flow

A grid-based alternative to streamline simulation that is mass-conservative and avoids problems with mapping and choice of representative streamline distribution.

Basic Ideas

• reordering of equations for efficient element-wise or blockwise solution of the resulting (non)linear discrete systems
• local control of nonlinear iterations reduces runtime dramatically and increases the range of feasible time-steps
• discontinuous Galerkin spatial discretisation for compact higher-order discretisations of purely advective transport equations

Features

The reordering procedure is applicable to any grid that can be mapped to a directed graph (with directions given by inter-cell fluxes). Based on the reordering, one can formulate a highly efficient Gauss-Seidel type (non)linear solver.  Applications studied so far: 

• fast computation of time-of-flight in grid cells; isocontours of time-of-flight are the natural time-lines in the reservoir and thus ideal for fast visualisation of flow patterns
• fast computation of tracer flow; isocontours of tracer concentrations can be used for delineation of reservoir volumes and computation of drainage/flooded volumes
• fast simulation of multiphase flow in the absence of gravity and capillary forces. Example: For the SPE10 model with 1,1 million cells,  runtime for 100 time steps with the implicit single-point upwind method was about 2 minutes on a standard PC.

The discontinuous Galerkin discretisation:

• gives increased spatial accuracy through higher-order stencils localised to a single grid cell
• allows for local hp-refinement
• reduces grid-orientation problems for miscible flows

Advantages

A fast grid-based alternative to streamline simulation that

• avoids mapping and choice of representative streamline distribution
• avoids problems associated with calculation of production curves
• is guaranteed to be mass conservative

References:

1. J. R. Natvig and K.-A. Lie. On efficient implicit upwind schemes.  Proceedings of ECMOR XI, Bergen, Norway, 8-11 September 2008. 
2. J. R. Natvig and K.-A. Lie. Fast computation of multiphase flow in porous media by implicit discontinuous Galerkin schemes with optimal ordering of elements. J. Comput. Phys, Volume 227, Issue 24, pp. 10108-10124, 2008. DOI: doi:10.1016/j.jcp.2008.08.024.
3. J. R. Natvig, K.-A. Lie, B. Eikemo, and I. Berre. A discontinous Galerkin method for single-phase flow in porous media. Advances in Water Resources, Vol. 30, Issue 12, December 2007, pp. 2424-2438. DOI: 10.1016/j.advwatres.2007.05.015.