Iterative multi-scale method for flow in porous media

A computer-implemented method for use in simulating fluid flow in a subsurface reservoir using a model, the method comprising:

• executing on one or more data processors:
• creating a fine grid defining a plurality of fine cells associated with a geological formation of the subsurface reservoir, a coarse grid defining a plurality of coarse cells having interfaces between the coarse cells, the coarse cells being aggregates of the fine cells, and a dual coarse grid defining a plurality of dual coarse control volumes, the dual coarse control volumes being aggregates of the fine cells and having boundaries bounding the dual coarse control volumes;
• computing the model using a finite volume method, on a computer system, in a plurality of timesteps;
• wherein:
• the model comprises one or more variables representative of fluid flow in the subsurface reservoir, wherein at least one of the one or more variables representative of fluid flow is responsive to calculated basis functions;
• the computing comprises:
• calculating the basis functions on the dual coarse control volumes by solving local elliptic problems;
• calculating correction functions on the dual coarse control volumes by solving the local elliptic problems with fine grid source terms;
• integrating the basis functions over each coarse cell to solve for pressure values at coarse nodes;
• interpolating the pressure values at the coarse nodes into the fine cells using the basis functions and the correction functions to obtain an interpolated fine grid pressure;
• and
• for at least one timestep of the plurality of timesteps, updating the fine grid pressure using an iterative multi-scale method, the iterative multi-scale method comprising, for each iteration:
• (i) applying a smoothing scheme to the fine grid pressure;
• (ii) using the fine grid pressure that has been smoothed in step (i) to recalculate the correction functions;
• (iii) solving for the pressure on the coarse grid using the correction functions from step (ii) wherein the solving comprises:
• calculating the right-hand side of the linear system for the pressure over the coarse grid using the correction functions from step (ii);
• and
• solving for the pressure on the coarse grid using the calculated right-hand side of the linear system for the pressure over the coarse grid;
• and
• (iv) reconstructing a solution for the pressure over the fine grid using the result from step (iv);
• and
• results from the computed model, comprising the pressure calculated using the iterative multi-scale method in the at least one timestep, simulate fluid flow in the subsurface reservoir.

A computer-implemented system for use in simulating fluid flow in a geological formation of a subsurface reservoir using a model, the system comprising:

• a memory comprising one or more data structures for storing data representing a fine grid defining a plurality of fine cells, a coarse grid defining a plurality of coarse cells, a dual coarse grid defining a plurality of dual coarse control volumes, and dual basis functions calculated on the dual coarse control volumes by solving local elliptic problems;
• and
• one or more data processors for executing software instructions to compute the model using a finite volume method in at least two timesteps;
• wherein:
• the model comprises one or more variables representative of fluid flow in the subsurface reservoir, wherein at least one of the one or more variables representative of fluid flow is responsive to calculated basis functions;
• the computing comprises:
• calculating the basis functions on the dual coarse control volumes by solving local elliptic problems;
• calculating correction functions on the dual coarse control volumes by solving the local elliptic problems with fine grid source terms;
• integrating the basis functions over each coarse cell to solve for pressure values at coarse nodes;
• interpolating the pressure values at the coarse nodes into the fine cells using the basis functions and the correction functions to obtain an interpolated fine grid pressure;
• and
• for at least one timestep of the at least two timesteps, updating the fine grid pressure using an iterative multi-scale method, the iterative multi-scale method comprising, for each iteration:
• (i) applying a smoothing scheme to the fine grid pressure;
• (ii) using the fine grid pressure that has been smoothed in step (i) to recalculate the correction functions;
• (iii) solving for the pressure on the coarse grid using the correction functions from step (ii) wherein the solving comprises:
• calculating the right-hand side of the linear system for the pressure over the coarse grid using the correction functions from step (ii);
• and
• solving for the pressure on the coarse grid using the calculated right-hand side of the linear system for the pressure over the coarse grid;
• and
• (iv) reconstructing a solution for the pressure over the fine grid using the result from step (iii);
• and
• a visual display for displaying fluid flow in the geological formation of the subsurface reservoir using the computed model, comprising the pressure calculated using the iterative multi-scale method in the at least two timesteps.