This work presents a new discrete fracture model, namely the Projection-based Embedded Discrete Fracture Model (pEDFM). Similar to the existing EDFM approach, pEDFM constructs independent grids for the matrix and fracture domains, and delivers strictly conservative velocity fields. However, as a significant step forward, it is able to accurately model the effect of fractures with general conductivity contrasts relative to the matrix, including impermeable flow barriers. This is achieved by automatically adjusting the matrix transmissibility field, in accordance to the conductivity of neighboring fracture networks, alongside the introduction of additional matrix-fracture connections. The performance of pEDFM is investigated extensively for two- and three-dimensional scenarios involving single-phase as well as multiphase flows. These numerical experiments are targeted at determining the sensitivity of the model towards the fracture position within the matrix control volume, grid resolution and the conductivity contrast towards the matrix. The pEDFM significantly outperforms the original EDFM and produces results comparable to those obtained when using DFM on unstructured grids, therefore proving to be a flexible model for field-scale simulation of flow in naturally fractured reservoirs.@en

Any endeavour to accurately model flow through fractured porous media at the field-scale must overcome two important challenges. First, the discretized representation of the medium needs to accommodate the complex geometry of intersecting small-scale fractures with various lengths, apertures and orientations. Second, the model formulation must ensure that the conductivity of these fractures, which can be orders of magnitude higher or lower than that of the host rock, is properly taken into account when computing the pressure map. The Embedded Discrete Fracture Model (EDFM) is well known in the literature for its flexibility in representing fractures. More specifically, in EDFM, fractures are lower-dimensional features, discretized independently from the matrix. Their effect on the flow is captured by defining fluxes between the fracture control volumes and the matrix grid cells they intersect. EDFM was proven effective in capturing the flow behaviour through porous media containing highly conductive fractures. However, its formulation fails to represent the effect of low-permeable features, such as embedded flow barriers. In this work, a novel projection-based Embedded Discrete Fracture Model (pEDFM) is introduced for flow simulation in fractured porous media with general conductivity contrasts. Similar to EDFM, pEDFM constructs independent grids for the fracture and matrix domains. As an additional step, the transmissibilities at matrix interfaces are automatically adjusted to account for the conductivity of neighbouring fracture networks, via a scaling factor proportional to their geometric projections. The performance of pEDFM is investigated extensively for two- and three-dimensional scenarios involving single- as well as multiphase flows. These numerical experiments are targeted at determining the sensitivity of the model towards the grid resolution, fracture position and orientation, as well as the conductivity contrast towards the matrix. The results of these studies support the conclusion that pEDFM significantly outperforms the original EDFM model and is a viable method for field-scale simulation of flow in naturally fractured reservoirs.@en

This work presents the formulation of a novel Projection-based Embedded Discrete Fracture Model (pEDFM), and its integration into an algebraic multiscale procedure. Similar to EDFM, pEDFM constructs independent grids for the matrix and fracture domains. However, as a significant step forward, it is able to accurately model the effect of fractures with general conductivity contrasts relative to the matrix, including impermeable flow barriers. This is achieved by automatically adjusting the matrix transmissibility field, in accordance to the conductivity of neighboring fracture networks. Then, in order to extend the pEDFM to real-field applications, F-AMS-pEDFM is introduced, which is an extension of the recently developed algebraic multiscale solver, F-AMS [Ţene et al., 2016], to include pEDFM. The performance (efficiency and scalability) of F-AMS-pEDFM is investigated extensively for challenging two- and three-dimensional scenarios with complex fracture geometries and a wide range of conductivity contrasts. Moreover, FAMS- pEDFM is benchmarked against the commercial SAMG solver, where CPU time is monitored during both the setup and solution phases. The results support the conclusions that (1) pEDFM significantly outperforms the original EDFM model, and (2) the F-AMS-pEDFM approach proposed in this work is an accurate and efficient method for field-scale simulation of flow in fractured reservoirs. @en

This paper introduces an Algebraic MultiScale method for simulation of flow in heteroge-neous porous media with embedded discrete Fractures (F-AMS). First, multiscale coarse grids are independently constructed for both porous matrix and fracture networks. Then, amap between coarse-and fine-scale is obtained by algebraically computing basis functions with local support. In order to extend the localization assumption to the fractured media, four types of basis functions are investigated: (1)Decoupled-AMS, in which the two media are completely decoupled, (2)Frac-AMS and (3)Rock-AMS, which take into account only one-way transmissibilities, and (4)Coupled-AMS, in which the matrix and fracture interpolators are fully coupled. In order to ensure scalability, the F-AMS framework permits full flexibility in terms of the resolution of the fracture coarse grids. Numerical results are presented for two-and three-dimensional heterogeneous test cases. During these experiments, the performance of F-AMS, paired with ILU(0) as second-stage smoother in a convergent iterative procedure, is studied by monitoring CPU times and convergence rates. Finally, in order to investigate the scalability of the method, an extensive benchmark study is conducted, where a commercial algebraic multigrid solver is used as reference. The results show that, given an appropriate coarsening strategy, F-AMS is insensitive to severe fracture and matrix conductivity contrasts, as well as the length of the fracture networks. Its unique feature is that a fine-scale mass conservative flux field can be reconstructed after any iteration, providing efficient approximate solutions in time-dependent simulations.@en

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. @en