Fast Flow Simulations by Optimal Reordering and Discontinuous Galerkin Discretisations
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 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
Isocontours of time-of-flight define natural timelines in the reservoir, here visualized in a half slice of a 3D quarter five-spot with heterogeneous permeability.
Drainage volumes for the SPE 10 model defined by computing the steady-state solution of a continuous tracer injection.
Localization of nonlinear iteration by use of reordering. The color shows the number of iterations per cell during a typical transport step. Compared with a standard Newton-Raphson method this gives 100 times reduction of runtime for this model.
Published April 21, 2008
A portfolio of strategic research projects funded by the Research Council of Norway