Ivar Stefansson

Eirik Keilegavlen, Alessio Fumagalli, Runar Berge et al.

Fractures are ubiquitous in the subsurface and strongly affect flow and deformation. The physical shape of the fractures, they are long and thin objects, puts strong limitations on how the effect of this dynamics can be incorporated into standard reservoir simulation tools. This paper reports the development of an open-source software framework, termed PorePy, which is aimed at simulation of flow and transport in three-dimensional fractured reservoirs, as well as deformation of the reservoir due to shearing along fracture and fault planes. Starting from a description of fractures as polygons embedded in a 3D domain, PorePy provides semi-automatic gridding to construct a discrete-fracture-matrix model, which forms the basis for subsequent simulations. PorePy allows for flow and transport in all lower-dimensional objects, including planes (2D) representing fractures, and lines (1D) and points (0D), representing fracture intersections. Interaction between processes in neighboring domains of different dimension is implemented as a sequence of couplings of objects one dimension apart. This readily allows for handling of complex fracture geometries compared to capabilities of existing software. In addition to flow and transport, PorePy provides models for rock mechanics, poro-elasticity and coupling with fracture deformation models. The software is fully open, and can serve as a framework for transparency and reproducibility of simulations. We describe the design principles of PorePy from a user perspective, with focus on possibilities within gridding, covered physical processes and available discretizations. The power of the framework is illustrated with two sets of simulations; involving respectively coupled flow and transport in a fractured porous medium, and low-pressure stimulation of a geothermal reservoir.

Bernd Flemisch, Inga Berre, Wietse Boon et al.

This paper presents several test cases intended to be benchmarks for numerical schemes for single-phase fluid flow in fractured porous media. A number of solution strategies are compared, including a vertex and a cell-centered finite volume method, a non-conforming embedded discrete fracture model, a primal and a dual extended finite element formulation, and a mortar discrete fracture model. The proposed benchmarks test the schemes by increasing the difficulties in terms of network geometry, e.g. intersecting fractures, and physical parameters, e.g. low and high fracture-matrix permeability ratio as well as heterogeneous fracture permeabilities. For each problem, the results presented by the participants are the number of unknowns, the approximation errors in the porous matrix and in the fractures with respect to a reference solution, and the sparsity and condition number of the discretized linear system. All data and meshes used in this study are publicly available for further comparisons.

Eirik Keilegavlen, Alessio Fumagalli, Runar Berge et al.

Models that involve coupled dynamics in a mixed-dimensional geometry are of increasing interest in several applications. Here, we describe the development of a simulation model for flow in fractured porous media, where the fractures and their intersections form a hierarchy of interacting subdomains. We discuss the implementation of a simulation framework, with an emphasis on reuse of existing discretization tools for mono-dimensional problems. The key ingredients are the representation of the mixed-dimensional geometry as a graph, which allows for convenient discretization and data storage, and a non-intrusive coupling of dimensions via boundary conditions and source terms. This approach is applicable for a wide class of mixed-dimensional problems. We show simulation results for a flow problem in a three-dimensional fracture geometry, applying both finite volume and virtual finite element discretizations.

Ivar Stefansson, Inga Berre, Eirik Keilegavlen

Over the last decade, finite volume discretizations for flow in porous media have been extended to handle situations where fractures dominate the flow. These discretizations have successfully been combined with the discrete fracture-matrix models to yield mass conservative methods capable of explicitly incorporating the impact of fractures and their geometry. When combined with a hybrid-dimensional formulation, two central concerns are the restrictions arising from small cell sizes at fracture intersections and the coupling between fractures and matrix. Focusing on these aspects, we demonstrate how finite volume methods effectively can be extended to handle fractures, providing generalizations of previous work. We address the finite volume methods applying a general hierarchical formulation, facilitating implementation with extensive code reuse and providing a natural framework for coupling of different subdomains. Furthermore, we demonstrate how a Schur complement technique may be used to obtain a robust and versatile method for fracture intersection cell elimination. We investigate the accuracy of the proposed elimination method through a series of numerical simulations in 3D and 2D. The simulations, performed on fractured domains containing permeability heterogeneity and anisotropy, also demonstrate the flexibility of the hierarchical framework.

Inga Berre, Wietse Boon, Bernd Flemisch et al.

This call for participation proposes four benchmark tests to verify and compare numerical schemes to solve single-phase flow in fractured porous media. With this, the two-dimensional suite of benchmark tests presented by Flemisch et al. 2018 is extended to include three-dimensional problems. Moreover, transport simulations are included as a means to compare discretization methods for flow. With this publication, we invite researchers to contribute to the study by providing results to the test cases based on their applied discretization methods.