## Abstract

Mathematical formulations describing flow in porous media typically entail highly heterogeneous coefficients, changing over several orders of magnitude through the entirety of the domain. In addition, many of the target geological formations are fractured. Fractures are lower dimensional manifolds with properties that differ greatly from those of the surrounding porous rock. Therefore, given their significant role in establishing the patterns of the flow regime, accurate representation of fractures within flow and transport models is crucial for many geoscientific applications, including groundwater flow and geothermal energy exploitations.

Embedded Discrete Fracture Model (EDFM) employs independent grids for matrix and fractures. This results in efficient computations, specially for complex fracture geometries, and cases with dynamic fracture creations (and closures). Even though small-scale fractures are homogenized within the matrix rock, the remaining explicit fractures (bigger than fine-scale grid resolution) along with heterogeneous matrix, for realistic cases, lead to linear systems which are beyond the scope of classical simulation methods.

Multiscale finite element and volume (MSFE and MSFV, respectively) methods have been developed mainly for heterogeneous, but non-fractured, porous media. In order to extend them to account for flow in heterogeneous fractured formations, here, F-AMS is introduced as a novel Algebraic Multiscale Solver. It operates by defining coarse grids for both the porous matrix and the embedded discrete fractures. Then, by computing local basis functions, a general map between the fine- and coarse-scale systems, i.e. the prolongation operator, is obtained. These basis functions form a partition of unity and, in their present formulation, they allow for four degrees of fracture-matrix coupling: (1) Decoupled-AMS, in which the two media are completely decoupled, (2) Frac-AMS allows one-way coupling, where the fracture coarse solutions also affect the matrix fine-scale pressure, (3) Rock-AMS is the counterpart of Frac-AMS, where the matrix coarse solution is also employed to find the fracture fine-scale pressure, and (4) Coupled-AMS, in which matrix and fracture interpolators are fully coupled. If only one coarse degree of freedom (DOF) is considered for each fracture network, the Frac-AMS strategy becomes equivalent to the earlier method proposed by Hajibeygi et al. However, in order to maintain efficiency for general cases, the F-AMS framework permits full flexibility in terms of the definition of the fracture coarse grids and the level of matrix-fracture prolongation coupling. Moreover, by using the Finite Volume restriction operator after any iteration, a mass conservative velocity can be reconstructed and be used to solve the transport equations.

Systematic numerical experiments for 3D heterogeneous fractured domains (from $10^5$ to $10^7$ grid cells, and fracture-matrix transmissibility contrasts of $10^1$ to $10^8$) are presented and discussed. In addition, the F-AMS is benchmarked agains SAMG, a commercial Algebraic Multigrid solver. These results illustrate that F-AMS is an efficient multiscale procedure for large-scale fractured reservoirs. It is important to note that for multiphase flow scenarios, only a few F-AMS iterations are sufficient to obtain good quality pressure solutions. These lead to the conclusion that F-AMS is an important multiscale development for the efficient simulation of flow in naturally fractured porous media.

Embedded Discrete Fracture Model (EDFM) employs independent grids for matrix and fractures. This results in efficient computations, specially for complex fracture geometries, and cases with dynamic fracture creations (and closures). Even though small-scale fractures are homogenized within the matrix rock, the remaining explicit fractures (bigger than fine-scale grid resolution) along with heterogeneous matrix, for realistic cases, lead to linear systems which are beyond the scope of classical simulation methods.

Multiscale finite element and volume (MSFE and MSFV, respectively) methods have been developed mainly for heterogeneous, but non-fractured, porous media. In order to extend them to account for flow in heterogeneous fractured formations, here, F-AMS is introduced as a novel Algebraic Multiscale Solver. It operates by defining coarse grids for both the porous matrix and the embedded discrete fractures. Then, by computing local basis functions, a general map between the fine- and coarse-scale systems, i.e. the prolongation operator, is obtained. These basis functions form a partition of unity and, in their present formulation, they allow for four degrees of fracture-matrix coupling: (1) Decoupled-AMS, in which the two media are completely decoupled, (2) Frac-AMS allows one-way coupling, where the fracture coarse solutions also affect the matrix fine-scale pressure, (3) Rock-AMS is the counterpart of Frac-AMS, where the matrix coarse solution is also employed to find the fracture fine-scale pressure, and (4) Coupled-AMS, in which matrix and fracture interpolators are fully coupled. If only one coarse degree of freedom (DOF) is considered for each fracture network, the Frac-AMS strategy becomes equivalent to the earlier method proposed by Hajibeygi et al. However, in order to maintain efficiency for general cases, the F-AMS framework permits full flexibility in terms of the definition of the fracture coarse grids and the level of matrix-fracture prolongation coupling. Moreover, by using the Finite Volume restriction operator after any iteration, a mass conservative velocity can be reconstructed and be used to solve the transport equations.

Systematic numerical experiments for 3D heterogeneous fractured domains (from $10^5$ to $10^7$ grid cells, and fracture-matrix transmissibility contrasts of $10^1$ to $10^8$) are presented and discussed. In addition, the F-AMS is benchmarked agains SAMG, a commercial Algebraic Multigrid solver. These results illustrate that F-AMS is an efficient multiscale procedure for large-scale fractured reservoirs. It is important to note that for multiphase flow scenarios, only a few F-AMS iterations are sufficient to obtain good quality pressure solutions. These lead to the conclusion that F-AMS is an important multiscale development for the efficient simulation of flow in naturally fractured porous media.

Original language | English |
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Publication status | Published - 2016 |

Event | 11th International Conference on Computational Methods in Water Resources - University of Toronto, Toronto, Canada Duration: 20 Jun 2016 → 24 Jun 2016 Conference number: 11 |

### Conference

Conference | 11th International Conference on Computational Methods in Water Resources |
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Abbreviated title | CMWR 2016 |

Country/Territory | Canada |

City | Toronto |

Period | 20/06/16 → 24/06/16 |

## Keywords

- Algebraic multiscale methods
- Multiphase flow
- Naturally fractured porous rock
- Heterogeneous geological properties
- Scalable linear solvers