•Presents four benchmark cases for single-phase flow in three-dimensional fractured porous media.•Investigates the capabilities of DFM methods in handling complexities common to fracture ...networks.•Compares results of 17 different numerical methods, submitted by 11 participating groups.•Is supplemented by a public Git repository containing result data and reproducing Jupyter notebooks.
Flow in fractured porous media occurs in the earth’s subsurface, in biological tissues, and in man-made materials. Fractures have a dominating influence on flow processes, and the last decade has seen an extensive development of models and numerical methods that explicitly account for their presence. To support these developments, four benchmark cases for single-phase flow in three-dimensional fractured porous media are presented. The cases are specifically designed to test the methods’ capabilities in handling various complexities common to the geometrical structures of fracture networks. Based on an open call for participation, results obtained with 17 numerical methods were collected. This paper presents the underlying mathematical model, an overview of the features of the participating numerical methods, and their performance in solving the benchmark cases.
We present an abstract discretization framework and demonstrate that various cell-centered and hybrid finite-volume schemes fit into it. The different schemes considered in this work are then ...analyzed numerically for an elliptic model problem with respect to the properties consistency, coercivity, extremum principles, and sparsity. The test cases presented comprise of two- and three-dimensional setups, mildly and highly anisotropic tensors and grids of different complexities. The results show that all schemes show a similar convergence behavior, except for the two-point flux approximation scheme, and seem to be coercive. Furthermore, they confirm that linear schemes, in contrast to nonlinear schemes, are in general neither positivity-preserving nor satisfy discrete minimum or maximum principles.
Flow through heterogeneous landfills that include macropores may occur under Reynolds numbers higher than those where Darcy’s law is valid. Extensions, such as a Forchheimer approach, may be required ...to include inertial effects. Our aim is developing predictive models for such landfills that are built from the low-level radioactive waste and debris of dismantled nuclear power plants. It consists of different materials, which after crushing result in a spatially heterogeneous distribution of porous-media properties in the landfills. Rain events or leakage, for example, may wash out radionuclides and transport them with the water flow. We investigate here the water flow and consider an inclusion of macropores. To deal with possibly high velocities, we choose the Forchheimer model and, taking different Forchheimer coefficients into account, compare it to the Darcy model. The focal points of the study are (i) the influence of the macropores on the flow field and (ii) the impact of the choice of the Forchheimer coefficient both on the solution and the computational effort. The results show that dependent on their size, macropores can dominate the flow field. Furthermore, Forchheimer coefficients introducing more inertial effects are associated with considerably higher runtimes.
We investigate reactive flow and transport in evolving porous media. Solute species that are transported within the fluid phase are taking part in mineral precipitation and dissolution reactions for ...two competing mineral phases. The evolution of the three phases is not known a-priori but depends on the concentration of the dissolved solute species. To model the coupled behavior, phase-field and level-set models are formulated. These formulations are compared in three increasingly challenging setups including significant mineral overgrowth. Simulation outcomes are examined with respect to mineral volumes and surface areas as well as derived effective quantities such as diffusion and permeability tensors. In doing so, we extend the results of current benchmarks for mineral dissolution/precipitation at the pore-scale to the multiphasic solid case. Both approaches are found to be able to simulate the evolution of the three-phase system, but the phase-field model is influenced by curvature-driven motion.
In various research areas such as engineering, physics, and mathematics, numerical simulations play an important role. A number of research software simulation frameworks have been established, for ...instance, Dune (Bastian et al., 2008, 2021), Dumux (Flemisch et al., 2011; Koch et al., 2021), Deal.II (Arndt et al., 2022), FEniCS (A. Logg, 2012; FEniCS, 2023), and VirtualFluids (Kutscher et al., 2022). Numerical software typically has a high inherent complexity as it aims at solving complex physical model equations by using advanced mathematical methods for solving partial differential equations. Beyond this, the model equations often involve parameters that are described by means of empirical constitutive relationships. Thus, a numerical simulation usually brings together various software components: for the domain discretization, the discretization method for the equations, the physics, and a non-linear and/or linear solver to obtain a solution for the discretized equations.
While each of these components can be unit tested, it is important to have system tests that verify that a particular type of simulation can be carried out successfully. By successful we mean here that the simulation produces the correct results. As sufficiently complex problems often lack analytical solutions, determining correctness of numerical simulations poses a significant challenge. In the absence of an analytical solution, a common strategy is to use a trusted reference for comparison (e.g., data measured in experiments or results from previous publications). From the perspective of software quality assurance, it suffices to define a reference result as the correct one and continuously verify that the code still reproduces it. In numerical software, such regression tests play a vital role at the level of system tests (Kempf & Koch, 2017). They make sure that developers notice when a certain change to the code affects the results produced by the simulations. Whether the new results are better or worse has to be decided by the developers, and in the case of the former, the reference results may be updated.
In order to carry out regression tests, one must be able to detect significant deviations between newly-computed and reference results. What a significant deviation is has to be decided by the developers as well, and adequate tolerances have to be chosen that are big enough to avoid false negatives from machine precision issues, but small enough to ensure that physically relevant deviations in the results are detected. Some numerical software packages as, for instance, DUNE and DuMux (Flemisch et al., 2011), provide mechanisms to detect such deviations. However, the functionality is not provided independent of the frameworks themselves and is therefore only available to their users. Besides this, only those mesh file formats that are used by the frameworks are supported. Very recently, DuMux incorporated fieldcompare into its test suite in place of its in-house solutions.
Geochemical processes in subsurface reservoirs affected by microbial activity change the material properties of porous media. This is a complex biogeochemical process in subsurface reservoirs that ...currently contains strong conceptual uncertainty. This means, several modeling approaches describing the biogeochemical process are plausible and modelers face the uncertainty of choosing the most appropriate one. The considered models differ in the underlying hypotheses about the process structure. Once observation data become available, a rigorous Bayesian model selection accompanied by a Bayesian model justifiability analysis could be employed to choose the most appropriate model, i.e. the one that describes the underlying physical processes best in the light of the available data. However, biogeochemical modeling is computationally very demanding because it conceptualizes different phases, biomass dynamics, geochemistry, precipitation and dissolution in porous media. Therefore, the Bayesian framework cannot be based directly on the full computational models as this would require too many expensive model evaluations. To circumvent this problem, we suggest to perform both Bayesian model selection and justifiability analysis after constructing surrogates for the competing biogeochemical models. Here, we will use the arbitrary polynomial chaos expansion. Considering that surrogate representations are only approximations of the analyzed original models, we account for the approximation error in the Bayesian analysis by introducing novel correction factors for the resulting model weights. Thereby, we extend the Bayesian model justifiability analysis and assess model similarities for computationally expensive models. We demonstrate the method on a representative scenario for microbially induced calcite precipitation in a porous medium. Our extension of the justifiability analysis provides a suitable approach for the comparison of computationally demanding models and gives an insight on the necessary amount of data for a reliable model performance.
We propose a computational simulation framework for describing cancer-therapeutic transport in the lung. A discrete vascular graph model (VGM) is coupled to a double-continuum model (DCM) to ...determine the amount of administered therapeutic agent that will reach the cancer cells. An alveolar cell carcinoma is considered. The processes in the bigger blood vessels (arteries, arterioles, venules and veins) are described by the VGM. The processes in the alveolar capillaries and the surrounding tissue are represented by a continuum approach for porous media. The system of equations of the coupled discrete/continuum model contains terms that account for degradation processes of the therapeutic agent, the reduction of the number of drug molecules by the lymphatic system and the interaction of the drug with the tissue cells. The functionality of the coupled discrete/continuum model is demonstrated in example simulations using simplified pulmonary vascular networks, which are designed to show-off the capabilities of the model rather than being physiologically accurate.
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DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, SIK, UILJ, UKNU, UL, UM, UPUK
Research Data Management (RDM) has gained significant traction in recent years, being essential to allowing research data to be, e.g., findable, accessible, interoperable, and reproducible (FAIR), ...thereby fostering collaboration or accelerating scientific findings. We present solutions for RDM developed within the DFG-Funded Cluster of Excellence EXC2075 Data-Integrated Simulation Science (SimTech). After an introduction to the scientific context and challenges faced by simulation scientists, we outline the general data management infrastructure and present tools that address these challenges. Exemplary domain applications demonstrate the use and benefits of the proposed data management software solutions. These are complemented by additional measures for enablement and dissemination to foster the adoption of these techniques.
•Four benchmark cases for single-phase flow in fractured porous media.•Comparison of seven state-of-the-art discrete-fracture-matrix methods.•Public access to all mesh and solution data.
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 two cell-centred finite volume methods, 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 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.
A discrete fracture model on the basis of a cell-centered finite volume scheme with multi-point flux approximation (MPFA) is presented. The fractures are included in a d-dimensional computational ...domain as (d−1)-dimensional entities living on the element facets, which requires the grid to have the element facets aligned with the fracture geometries. However, the approach overcomes the problem of small cells inside the fractures when compared to equi-dimensional models. The system of equations considered is solved on both the matrix and the fracture domain, where on the prior the fractures are treated as interior boundaries and on the latter the exchange term between fracture and matrix appears as an additional source/sink. This exchange term is represented by the matrix-fracture fluxes, computed as functions of the unknowns in both domains by applying adequate modifications to the MPFA scheme. The method is applicable to both low-permeable as well as highly conductive fractures. The quality of the results obtained by the discrete fracture model is studied by comparison to an equi-dimensional discretization on a simple geometry for both single- and two-phase flow. For the case of two-phase flow in a highly conductive fracture, good agreement in the solution and in the matrix-fracture transfer fluxes could be observed, while for a low-permeable fracture the discrepancies were more pronounced. The method is then applied two-phase flow through a realistic fracture network in two and three dimensions.
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