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The last decade has seen a strong increase of research into flows in fractured porous media, mainly related to subsurface processes but also in materials science and biological applications, as connected fractures can totally dominate flow patterns. Due to the fractures’ characteristics as approximately planar discontinuities with an extreme size-to-width ratio, they challenge standard macroscale mathematical and numerical modeling of flow based on averaging. Thus, over the last decades, various, and also fundamentally different, modeling approaches have been developed. This paper reviews common conceptual models and discretization approaches for flow in fractured porous media, with an emphasis on the dominating effects the fractures have on flow processes. In this context, the paper discusses the tight connection between physical and mathematical modeling and simulation approaches. Extensions and research challenges related to transport, multi-phase flow and fluid-solid interaction are also commented on.

Development of models and dedicated numerical methods for dynamics in fractured rocks is an active research field, with research moving towards increasingly advanced process couplings and complex fracture networks. The inclusion of coupled processes in simulation models is challenged by the high aspect ratio of the fractures, the complex geometry of fracture networks, and the crucial impact of processes that completely change characteristics on the fracture-rock interface. This paper provides a general discussion of design principles for introducing fractures in simulators, and defines a framework for integrated modeling, discretization, and computer implementation. The framework is implemented in the open-source simulation software PorePy, which can serve as a flexible prototyping tool for multiphysics problems in fractured rocks. Based on a representation of the fractures and their intersections as lower-dimensional objects, we discuss data structures for mixed-dimensional grids, formulation of multiphysics problems, and discretizations that utilize existing software. We further present a Python implementation of these concepts in the PorePy open-source software tool, which is aimed at coupled simulation of flow and transport in three-dimensional fractured reservoirs as well as deformation of fractures and the reservoir in general. We present validation by benchmarks for flow, poroelasticity, and fracture deformation in porous media. The flexibility of the framework is then illustrated by simulations of non-linearly coupled flow and transport and of injection-driven deformation of fractures. All results can be reproduced by openly available simulation scripts.

Cooling magma plutons and intrusions are the heat sources for hydrothermal systems in volcanic settings. To explain system longevity and observed heat transfer at rates higher than those explained by pure conduction, the concept of fluid convection in fractures that deepen because of thermal rock contraction has been proposed as a heat‐source mechanism. While recent numerical studies have supported this half a century old hypothesis, understanding of the various regimes where convective downward migration of fractures can be an effective mechanism for heat transfer is lacking. Using a numerical model for fluid flow and fracture propagation in thermo‐poroelastic media, we investigate scenarios for which convective downward migration of fractures may occur. Our results support convective downward migration of fractures as a possible mechanism for development of hydrothermal systems, both for settings within active zones of volcanism and spreading and, under favorable conditions, in older crust away from such zones. Plain Language Summary Geothermal energy is transferred through and stored in the rock and fluids of the Earth's crust. If temperature increases sufficiently with depth and the crust provides sufficient pathways for water to flow through, colder water sinks and percolates downward, gets heated at depth and then rises due to its lower density at higher temperature. This creates a hydrothermal circulation system that transports heat from the deep crust to shallower depths from where it can be produced. Wells drilled into these systems produce hot water and/or steam for direct heat utilization or electricity production. To understand the renewability of hydrothermal systems, we need to understand how heat is transferred deep in the crust. A hypothesis has been proposed, suggesting that fractures, propagating downwards because of contraction of the water‐cooled surrounding rock, are central to maintaining the heat transfer from the deep crust. Our computer simulations corroborate this hypothesis. Based on settings found in Iceland, we show how fluid flow and propagation of fractures can be important for development of hydrothermal systems both in active regions of volcanism and, under favorable conditions, also in older crust away from such regions. The latter results are important for the identification of hidden geothermal systems. Key Points Numerical modeling supports convective downward migration of fractures as a source mechanism for hydrothermal systems Fluid flow, fracture opening and propagation in a thermo‐poroelastic rock mass are simulated in different geological settings in the crust Crustal stresses are key to understanding whether a hydrothermal system can evolve in regions away from active zones of volcanism

Over the last decades, finite-volume discretisations for flow in porous media have been extended to handle situations where fractures dominate the flow. Successful discretisations have been based on 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 can be efficiently extended to handle fractures, providing generalisations 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.

Co-production of geothermal energy and lithium is an emerging opportunity with the potential to enhance the economic potential of geothermal operations. The economic reward of extracting lithium from geothermal brine is determined by how the lithium concentration evolves during brine production. In the initial stage, production will target lithium contained in the brine resident close to the production well. While lithium recharge, in the form of rock dissolution and inflow from other parts of the reservoir, is possible, the efficiency of such recharge depends on the geology of the reservoir. In this work, we study how structural heterogeneities in the form of fractures impact the flow of lithium-carrying brine. Using a simulation tool that gives high resolution of flow and transport in fractures and the host rock, we study how the presence of fractures influences energy and lithium production. Our simulations show that, due to heat conduction and the lack of mineral recharge from the rock, differences in fracture network geometries have a much larger impact on lithium production than energy production. The simulations thus confirm that in addition to the geochemical characterisation of lithium in geothermal brines, understanding fracture characterisation and its impact on production is highly important for lithium production.

Convection-driven cooling in porous media influences thermo-poro-mechanical stresses, thereby causing deformation. These processes are strongly influenced by the presence of fractures, which dominate flow and heat transfer. At the same time, the fractures deform and propagate in response to changes in the stress state. Mathematically, the model governing the physics is tightly coupled and must account for the strong discontinuities introduced by the fractures. Over the last decade, and motivated by a number of porous media applications, research into such coupled models has advanced modelling of processes in porous media substantially. Building on this effort, this work presents a novel model that couples fracture flow and heat transfer and deformation and propagation of fractures with flow, heat transfer and thermo-poroelasticity in the matrix. The model is based on explicit representation of fractures in the porous medium and discretised using multi-point finite volume methods. Frictional contact and non-penetration conditions for the fractures are handled through active set methods, while a propagation criterion based on stress intensity factors governs fracture extension. Considering both forced and natural convection processes, numerical results show the intricate nature of thermo-poromechanical fracture deformation and propagation.

Motivated by rock–fluid interactions occurring in a geothermal reservoir, we present a two-dimensional pore scale model of a periodic porous medium consisting of void space and grains, with fluid flow through the void space. The ions in the fluid are allowed to precipitate onto the grains, while minerals in the grains are allowed to dissolve into the fluid, and we take into account the possible change in pore geometry that these two processes cause, resulting in a problem with a free boundary at the pore scale. We include temperature dependence and possible effects of the temperature both in fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we perform a formal homogenization procedure to obtain upscaled equations. A pore scale model consisting of circular grains is presented as a special case of the porous medium.

We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.

This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent, and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve-fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape, and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provides the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behavior. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.

Numerical simulations of reactive transport in fractured porous media require the solution of coupled physical and chemical processes that depend on the fractures. Such coupled processes are described by a system of nonlinear partial differential-algebraic equations, while strong heterogeneities characterise fractures. This paper presents an approach to simulate single-phase flow and non-isothermal reactive transport with mineral dissolution and precipitation in fractured porous media. Our numerical solution strategy is based on two ingredients. First, the model equations consist of coupled partial differential equations for the fluid flow, heat transfer and solute transport and nonlinear algebraic equations representing the chemical reactions. Second, fractures are explicitly represented and treated as lower-dimensional objects. The partial differential equations are discretised using finite-volume methods, and at each time step, we solve a nonlinear system of equations using Newton’s method. With numerical simulations, we illustrate our model’s ability to accurately describe the two-way interaction between coupled multi-physical processes and two- and three-dimensional porous media with intersecting fractures.