Inga Berre

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.

Mats Kirkesæther Brun, Elyes Ahmed, Inga Berre et al.

This paper concerns splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE's, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare three approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from mechanics; these approaches have in common a simultaneous application of the fixed-stress splitting scheme on both the non-linearity and the coupling structure of the problem. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and validate our results through several numerical examples.

Eren Ucar, Eirik Keilegavlen, Inga Berre et al.

Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multipoint stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (faces in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems for which analytical solutions are available and more complex benchmark problems, including comparison with a finite-element discretization.

Micheal B. Oguntola, Omar Duran, Eirik Keilegavlen et al.

Simulating high-enthalpy fractured geothermal reservoirs is challenging due to the complex coupled processes of non-isothermal, multiphase, multicomponent flow, strongly nonlinear thermodynamics, and the dominant role of fractures. These complexities are amplified by mineral scaling, such as halite precipitation, which can impair reservoir permeability and well productivity. To address this, we present a new compositional flow model based on a persistent set of primary variables (pressure, enthalpy, and overall salt mass fraction). The formulation naturally handles phase transitions without manual switching, enhancing numerical stability. The model integrates a discrete fracture-matrix approach and employs an efficient, robust correlation-based phase-behaviour linearisation of saltwater thermodynamics, replacing expensive on-the-fly phase separation calculations. It incorporates the Kozeny-Carman relation to dynamically model porosity and permeability reduction from halite precipitation. Implemented in the open-source PorePy framework, the model is verified through a 1D salt dissolution benchmark against the established closed-source simulator CSMP++, showing strong agreement across geothermal conditions involving transitions between single- and multi-phase regions. Application to a 2D halite-saturated fractured reservoir with injection and production demonstrates the model's capability to predict halite precipitation patterns and their impact on permeability damage and energy recovery. Numerical results further show the model's value in predicting operational challenges such as wellbore blockage and the role of fracture connectivity. The model thus provides an open-source numerical tool for analysing complex heat and mass transport with mineral scaling in high-enthalpy fractured geothermal systems.

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.

Marius Nevland, Kundan Kumar, Inga Berre et al.

We consider a mathematical model of a poro-visco-elastic medium subject to frictional contact with a rigid obstacle, and study its numerical approximation. This model couples the Biot equations and contact conditions in the form of normal compliance and Coulomb friction. The resulting variational problem consists of a linear partial differential equation coupled to a nonlinear variational inequality. We propose and analyze a fully discrete numerical scheme for this problem, using conformal finite elements in space and the implicit Euler method in time. Existence and uniqueness of the discrete solution is established, and stability and a priori error estimates are derived. A numerical experiment is performed in which numerical error estimates are computed and compared to the theoretical results.

Juan Michael Sargado, Eirik Keilegavlen, Inga Berre et al.

Numerical simulations of brittle fracture using phase-field approaches often employ a discrete approximation framework that applies the same order of interpolation for the displacement and phase-field variables. Most common is to use linear finite elements to discretize the linear momentum and phase-field equations. However the use of $P_1$ Lagrange shape functions to model the phase-field is not optimal, since the latter develops cusps for fully developed cracks that in turn occur at locations correspoding to Gauss points of the associated FE model for the mechanics. Such feature is challenging to reproduce accurately with low order elements, and consequently element sizes must be made very small relative to the phase-field regularization parameter in order to achieve convergence of results with respect to the mesh. In this paper, we combine the standard $P_1$ FE discretization of stress equilibrium with a cell-centered finite volume approximation of the phase-field evolution equation based on the two-point flux approximation that is constructed on the same simplex mesh. Compared to a pure FE formulation utilizing linear elements, the proposed framework results in looser restrictions on mesh refinement with respect to the phase-field length scale. Furthermore, initialization of the history field is straightforward and accomplished through a local procedure. The ability to employ a coarser mesh relative to the traditional implementation is shown for several numerical examples, demonstrating savings in computational cost on the order of 50 to 80 percent for the studied cases.

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.