Alessio Fumagalli

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.

Jan M. Nordbotten, Wietse M. Boon, Alessio Fumagalli et al.

In this paper, we introduce a mortar-based approach to discretizing flow in fractured porous media, which we term the mixed-dimensional flux coupling scheme. Our formulation is agnostic to the discretizations used to discretize the fluid flow equations in the porous medium and in the fractures, and as such it represents a unified approach to integrated fractured geometries into any existing discretization framework. In particular, several existing discretization approaches for fractured porous media can be seen as special instances of the approach proposed herein. We provide an abstract stability theory for our approach, which provides explicit guidance into the grids used to discretize the fractures and the porous medium, as dependent on discretization methods chosen for the respective domains. The theoretical results are sustained by numerical examples, wherein we utilize our framework to simulate flow in 2D and 3D fractured media using control volume methods (both two-point and multi-point flux), Lagrangian finite element methods, mixed finite element methods, and virtual element methods. As expected, regardless of the ambient methods chosen, our approach leads to stable and convergent discretizations for the fractured problems considered, within the limits of the discretization schemes.

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.

Alessio Fumagalli, Eirik Keilegavlen, Stefano Scialò

Simulations of fluid flow in naturally fractured rocks have implications for several subsurface applications, including energy storage and extraction, and waste storage. We are interested in flow in discrete fracture networks, which explicitly represent flow in fracture surfaces, but ignore the impact of the surrounding host rock. Fracture networks, generated from observations or stochastic simulations, will contain intersections of arbitrary length, and intersection lines can further cross, forming a highly complex geometry. As the flow exchange between fractures, thus in the network, takes place in these intersections, an adequate representation of the geometry is critical for simulation accuracy. In practice, the intersection dynamics must be handled by a combination of the simulation grid, which may or may not resolve the intersection lines, and the numerical methods applied on the grid. In this work, we review different classes of numerical approaches proposed in recent years, covering both methods that conform to the grid, and non-matching cases. Specific methods considered herein include finite element, mixed and virtual finite elements and control volume methods. We expose our methods to an extensive set of test cases, ranging from artificial geometries designed to test difficult configurations, to a network extruded from a real fracture outcrop. The main outcome is guidances for choice of simulation models and numerical discretization with a trade off on the computational cost and solution accuracy.

Alessio Fumagalli, Eirik Keilegavlen

The accurate description of fluid flow and transport in fractured porous media is of paramount importance to capture the macroscopic behaviour of an oil reservoir, a geothermal system, or a CO2 sequestration site, to name few applications. The construction of accurate simulation model for flow in fractures is challenging due to the high ratios between a fracture's length and width, which makes modeling by lower-dimensional manifolds a natural option. In this paper we present a mixed-dimensional Darcy problem able to describe pressure and Darcy velocity in all the dimensions, i.e. in the rock matrix, in the fractures, and in their intersections. Moreover, we present a mixed-dimensional transport problem which, given the Darcy velocity, describes coupled advection and diffusion of a passive scalar into the fractured porous media. The approach can handle both conducting and blocking fractures. Our computational grids are created by coarsening of simplex tessellations that conform to the fractures surfaces. An accurate choice of the discrete approximation of the previous model, by virtual finite element and finite volume, allows us to simulate complex problem with a good balance in term of accuracy and computational cost. We illustrate the performance of our method by comparing to benchmark studies for two-dimensional fractured porous media, as well as a complex three-dimensional fracture geometry.

Elyes Ahmed, Alessio Fumagalli, Ana Budiša

In this paper, a multiscale flux basis algorithm is developed to efficiently solve a flow problem in fractured porous media. Here, we take into account a mixed-dimensional setting of the discrete fracture matrix model, where the fracture network is represented as lower-dimensional object. We assume the linear Darcy model in the rock matrix and the non-linear Forchheimer model in the fractures. In our formulation, we are able to reformulate the matrix-fracture problem to only the fracture network problem and, therefore, significantly reduce the computational cost. The resulting problem is then a non-linear interface problem that can be solved using a fixed-point or Newton-Krylov methods, which in each iteration require several solves of Robin problems in the surrounding rock matrices. To achieve this, the flux exchange (a linear Robin-to-Neumann co-dimensional mapping) between the porous medium and the fracture network is done offline by pre-computing a multiscale flux basis that consists of the flux response from each degree of freedom on the fracture network. This delivers a conserve for the basis that handles the solutions in the rock matrices for each degree of freedom in the fractures pressure space. Then, any Robin sub-domain problems are replaced by linear combinations of the multiscale flux basis during the interface iteration. The proposed approach is, thus, agnostic to the physical model in the fracture network. Numerical experiments demonstrate the computational gains of pre-computing the flux exchange between the porous medium and the fracture network against standard non-linear domain decomposition approaches.

Alessio Fumagalli, Anna Scotti

In this work we present a new conceptual model to describe fluid flow in a porous media system in presence of a large fault. Geological faults are often modeled simply as interfaces in the rock matrix, but they are complex structure where the high strain core is surrounded by the so called damage zones, characterized by the presence of smaller fractures which enhance the permeability of the medium. To obtain reliable simulation outcomes these damage zone, as well as the fault, have to be accurately described. The new model proposed in this work considers both these two regions as lower dimensional and embedded in the rock matrix. The model is presented, analyzed, and tested in several configurations to prove its robustness and ability to capture many important features, such as hight contrast and heterogeneity of permeability.

Alessio Fumagalli, Isabelle Faille

In this work we are interested in dealing with single-phase flows in fractured porous media for underground processes. We focus our attention on domains where the presence of faults, with thickness several orders of magnitude smaller than other characteristic sizes, can allow one part of the domain to slide past to the other. We propose a mathematical scheme where a reduced model for the fault flows is employed yielding a problem of co-dimension one. The hybrid finite volume method is used to obtain the discretized problem, which employs two different meshes for each fault, one associated with the porous-medium domain on each side of the fault. These two meshes can move with the corresponding domain, resulting in non-matching grids between the two parts of the fault. In an earlier paper a mathematical scheme was proposed where the numerical discretization considers the hybrid finite volume method. In this paper we focus on the well-posedness of the continuous problem, the convergence of the discretized problem, and with several numerical tests we support the theoretical findings.

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.

Enrico Ballini, Marco Gambarini, Alessio Fumagalli et al.

We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low-dimensional one. We investigate the architecture of neural networks for constructing effective autoencoders that hold necessary properties to reconstruct the input manifold with exact representation capabilities. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach in different scenarios, highlighting its effectiveness in highly complex and nonlinear settings without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution.

Enrico Ballini, Luca Muscarnera, Alessio Fumagalli et al.

The unmatched ability of Deep Neural Networks in capturing complex patterns in large and noisy datasets is often associated with their large hypothesis space, and consequently to the vast amount of parameters that characterize model architectures. Pruning techniques affirmed themselves as valid tools to extract sparse representations of neural networks parameters, carefully balancing between compression and preservation of information. However, a fundamental assumption behind pruning is that expendable weights should have small impact on the error of the network, while highly important weights should tend to have a larger influence on the inference. We argue that this idea could be generalized; what if a weight is not simply removed but also compensated with a perturbation of the adjacent bias, which does not contribute to the network sparsity? Our work introduces a novel pruning method in which the importance measure of each weight is computed considering the output behavior after an optimal perturbation of its adjacent bias, efficiently computable by automatic differentiation. These perturbations can be then applied directly after the removal of each weight, independently of each other. After deriving analytical expressions for the aforementioned quantities, numerical experiments are conducted to benchmark this technique against some of the most popular pruning strategies, demonstrating an intrinsic efficiency of the proposed approach in very diverse machine learning scenarios. Finally, our findings are discussed and the theoretical implications of our results are presented.

Luca Muscarnera, Luigi Loreti, Giovanni Todeschini et al.

Randomness is ubiquitous in many applications across data science and machine learning. Remarkably, systems composed of random components often display emergent global behaviors that appear deterministic, manifesting a transition from microscopic disorder to macroscopic organization. In this work, we introduce a theoretical model for studying the emergence of collective behaviors in ensembles of random classifiers. We argue that, if the ensemble is weighted through the Gibbs measure defined by adopting the classification loss as an energy, then there exists a finite temperature parameter for the distribution such that the classification is optimal, with respect to the loss (or the energy). Interestingly, for the case in which samples are generated by a Gaussian distribution and labels are constructed by employing a teacher perceptron, we analytically prove and numerically confirm that such optimal temperature does not depend neither on the teacher classifier (which is, by construction of the learning problem, unknown), nor on the number of random classifiers, highlighting the universal nature of the observed behavior. Experiments on the MNIST dataset underline the relevance of this phenomenon in high-quality, noiseless, datasets. Finally, a physical analogy allows us to shed light on the self-organizing nature of the studied phenomenon.

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.

Alessio Fumagalli

We consider a Darcy problem for saturated porous media written in dual formulation in presence of a fully immersed inclusion. The lowest order virtual element method is employ to derive the discrete approximation. In the present work we study the effect of cells with cuts on the numerical solution, able to geometrically handle in a more natural way the inclusion tips. The numerical results show the validity of the proposed approach.

Alessio Fumagalli, Eirik Keilegavlen

Discrete fracture networks is a key ingredient in the simulation of physical processes which involve fluid flow in the underground, when the surrounding rock matrix is considered impervious. In this paper we present two different models to compute the pressure field and Darcy velocity in the system. The first allows a normal flow out of a fracture at the intersections, while the second grants also a tangential flow along the intersections. For the numerical discretization, we use the mixed virtual finite element method as it is known to handle grid elements of, almost, any arbitrary shape. The flexibility of the discretization allows us to loosen the requirements on grid construction, and thus significantly simplify the flow discretization compared to traditional discrete fracture network models. A coarsening algorithm, from the algebraic multigrid literature, is also considered to further speed up the computation. The performance of the method is validated by numerical experiments.