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
• applications in flow diagnostics
• fast simulation of multiphase and multicomponent flow. 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 in 2009.

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. K.-A. Lie, H. M. Nilsen, A. F. Rasmussen, and X. Raynaud. Fast simulation of polymer injection in heavy-oil reservoirs based on topological sorting and sequential splitting. Paper 163599 presented at the 2013 SPE Reservoir Simulation Symposium, The Woodlands, Texas, USA, 18-20 February 2013.
2. K.-A. Lie, H. M. Nilsen, A. F. Rasmussen, and X. Raynaud. An unconditionally stable splitting method using reordering for simulating polymer injection. Proceedings of ECMOR XIII, Biarritz, France, 10-13 September 2012.
3. K.-A. Lie, J. R. Natvig, and H. M. Nilsen. Discussion of dynamics and operator splitting techniques for two-phase flow with gravity. Int. J Numer. Anal. Mod. (Special issue in memory of Magne Espedal), Vol. 9, No. 3, pp. 684-700, 2012.
4. B. Eikemo, K.-A. Lie, H.K. Dahle, and G.T. Eigestad. A discontinuous Galerkin method for transport in fractured media using unstructured triangular grids. Adv. Water Resour. Vol. 32, Issue 4, pp. 493-506. 2009. DOI: 10.1016/j.advwatres.2008.12.010.
5. 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.
6. J. R. Natvig and K.-A. Lie. On efficient implicit upwind schemes.  Proceedings of ECMOR XI, Bergen, Norway, 8-11 September 2008.
7. 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.
8. B. Eikemo, I. Berre, H. K. Dahle, K.-A. Lie, and J. R. Natvig. A discontinuous Galerkin method for computing time-of-flight in discrete-fracture models. In "Proceedings of the XVI International Conference on Computational Methods in Water Resources, Copenhagen, Denmark, June, 2006", Eds., P.J. Binning et al.
9. J.R. Natvig, K.-A, Lie, and B. Eikemo. Fast solvers for flow in porous media based on discontinuous Galerkin methods and optimal reordering. In "Proceedings of the XVI International Conference on Computational Methods in Water Resources, Copenhagen, Denmark, June, 2006", Eds., P.J. Binning et al.