Bernd Flemisch

Abstract

Vertical equilibrium models have proven to be well suited for simulating fluid flow in subsurface porous media such as saline aquifers with caprocks. However, in most cases the dimensionally reduced model lacks the accuracy to capture the dynamics of a system. While conventional full-dimensional models have the ability to represent dynamics, they come at the cost of high computational effort. We aim to combine the efficiency of the vertical equilibrium model and the accuracy of the full-dimensional model by coupling the two models adaptively in a unified framework and solving the emerging system of equations in a monolithic, fully-implicit approach. The model domains are coupled via mass-conserving fluxes while the model adaptivity is ruled by adaptive criteria. Overall, the adaptive model shows an excellent behaviour both in terms of accuracy as well as efficiency, especially for elongated geometries of storage systems with large aspect ratios.

Abstract

Underground hydrogen storage (UHS) has emerged as a significant focus for the energy transition from fossil fuels to renewable energy. Saline aquifers, akin to those used in geological CO2 storage, are proposed due to their potential as reservoirs for hydrogen injection and withdrawal over extended periods. Modeling the drainage and imbibition processes at the reservoir scale is critical for efficient UHS implementation. To address the excessive computational costs of three-dimensional simulations over hundreds of cycles, low-order models such as the Vertical Equilibrium (VE) models offer practicality. This study centers on the VE model’s applicability to predict hydrogen plume shapes during a single injection and withdrawal cycle, considering varying viscous, capillary, and buoyancy forces, captured by Capillary number (Ca) and Bond number (Bo). Examining Ca values from 1.5 × 10−9 to 1.5 × 10−7 and Bo numbers from 0.1 to 0.3, we compared VE results to a 2D two-phase simulation, serving as the reference solution. For small Ca, the VE method accurately predicted both vertical and horizontal hydrogen plume distributions, although it occasionally overestimated lateral spread at high injection rates. As the vertical permeability and Bo increase, the VE predictions become more aligned with the reference solution. This is because greater density-driven flow and improved vertical connectivity make the phase segregation easier. Our study also demonstrated the VE model’s promising performance in hydrogen recovery during withdrawal at small Ca and large Bo.

Abstract

Modeling and prediction of reactive transport in porous media are challenging due to the complexity of the characterization of reactions across spatial and time scales. The application of reaction rates obtained from batch experiments to porous media is not valid because it disregards the influences of flow pathways, distribution of residence time, transport limitations within the porous media, as well as neglecting the impact of surface mineralogy heterogeneity. In this study, we presented how the pore-scale modeling can help to improve the Darcy-scale simulations. First, the reactive transport was simulated in a pore-network model and was validated against the experiments. Then, the pore-network results were averaged to obtain the effective reaction rates as a function of pore velocity, the inlet concentration, and the resident concentration. The upscaled function was incorporated into the Darcy-scale model and showed a significant improvement in predicting the concentration profiles. Notably, the use of batch reaction rates in the Darcy model (ADRE-batch) led to significant underestimation, up to 93%, of the required pore volume injection for complete contaminant removal from the simulation domain.

Abstract

Existing model validation studies in geoscience often disregard or partly account for uncertainties in observations, model choices, and input parameters. In this work, we develop a statistical framework that incorporates a probabilistic modeling technique using a fully Bayesian approach to perform a quantitative uncertainty-aware validation. A Bayesian perspective on a validation task yields an optimal bias-variance trade-off against the reference data. It provides an integrative metric for model validation that incorporates parameter and conceptual uncertainty. Additionally, a surrogate modeling technique, namely Bayesian Sparse Polynomial Chaos Expansion, is employed to accelerate the computationally demanding Bayesian calibration and validation. We apply this validation framework to perform a comparative evaluation of models for coupling a free flow with a porous-medium flow. The correct choice of interface conditions and proper model parameters for such coupled flow systems is crucial for physically consistent modeling and accurate numerical simulations of applications. We develop a benchmark scenario that uses the Stokes equations to describe the free flow and considers different models for the porous-medium compartment and the coupling at the fluid–porous interface. These models include a porous-medium model using Darcy’s law at the representative elementary volume scale with classical or generalized interface conditions and a pore-network model with its related coupling approach. We study the coupled flow problems’ behaviors considering a benchmark case, where a pore-scale resolved model provides the reference solution. With the suggested framework, we perform sensitivity analysis, quantify the parametric uncertainties, demonstrate each model’s predictive capabilities, and make a probabilistic model comparison.

Abstract

A new discretization approach for coupled free and porous-medium flows is introduced, which uses a finite-volume staggered-grid method for the discretization of the Navier-Stokes equations in the free-flow subdomain, while a vertex-centered finite-volume method is employed in the porous medium. The latter allows for the use of unstructured grids to discretize the porous medium, and the presented method is capable of handling non-matching grids at the interface. Moreover, the basis functions of the vertex-centered finite-volume method, with their degrees of freedom located at the interface, allow for an accurate evaluation of the coupling terms and of additional nonlinear velocity-dependent terms in the porous medium. The available advantages of this coupling method are investigated in a series of tests: a convergence test for various grid types, an evaluation of the implementation of the coupling conditions, and an example using the velocity-dependent Forchheimer term in the porous-medium subdomain.

Abstract

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.

Abstract

Abstract Efficient compositional models are required to simulate underground gas storage in porous formations where, for example, gas quality (such as purity) and loss of gas due to dissolution are of interest. We first extend the concept of vertical equilibrium (VE) to compositional flow, and derive a compositional VE model by vertical integration. Second, we present a hybrid model that couples the efficient compositional VE model to a compositional full-dimensional model. Subdomains, where the compositional VE model is valid, are identified during simulation based on a VE criterion that compares the vertical profiles of relative permeability at equilibrium to the ones simulated by the full-dimensional model. We demonstrate the applicability of the hybrid model by simulating hydrogen storage in a radially symmetric, heterogeneous porous aquifer. The hybrid model shows excellent adaptivity over space and time for different permeability values in the heterogeneous region, and compares well to the full-dimensional model while being computationally efficient, resulting in a runtime of roughly one-third of the full-dimensional model. Based on the results, we assume that for larger simulation scales, the efficiency of this new model will increase even more.

Abstract

Flow in fractured porous media is of high relevance in a variety of geotechnical applications, given the fact that they ubiquitously occur in nature and that they can have a substantial impact on the hydraulic properties of rock. As a response to this, an active field of research has developed over the past decades, focusing on the development of mathematical models and numerical methods to describe and simulate the flow processes in fractured rock. In this work, we present a comparison of different finite volume-based numerical schemes for the simulation of flow in fractured porous media by means of numerical experiments. Two novel vertex-centered approaches are presented and compared to well-established numerical schemes in terms of convergence behavior and their performance on benchmark studies taken from the literature. The new schemes show to produce results that are in good agreement with those of established methods while leading to less degrees of freedom on unstructured simplex grids.

Abstract

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.

Abstract

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.

Abstract

Understanding the influence of injection water composition on the displacement efficiency has been a long-standing issue in reservoir engineering; a reduction of the injection water salinity can lead to additional oil production, which is considered a low-cost and environmentally friendly method for enhanced oil recovery. Several underlying chemical mechanisms have been identified; however, the identification of the governing mechanisms remains difficult and specific to the chemical setting of the reservoir and injection water composition. In this work, we implement two potential mechanisms, namely, double layer expansion and multicomponent ion exchange, to achieve an explicit description of low-salinity effects in the open-source continuum-scale flow simulator DuMuX, with the goal of designing and interpreting experiments, and upscaling the results. By assuming sets of input parameters, we show that there is a minimum required core length dependent on the dispersion of the chemical flooding front dominated by sample heterogeneity. Above this length scale, the saturation profile is fully developed, and the chemical effluent analysis is conclusive, which both is required to calibrate continuum scale models. The change in the chemical water composition shows a characteristic fingerprint for both investigated mechanisms and therefore allows the identification of the leading low-salinity mechanism. The model is publicly available.

Abstract

We present version 3 of the open-source simulator for flow and transport processes in porous media DuMux. DuMux is based on the modular C++ framework Dune (Distributed and Unified Numerics Environment) and is developed as a research code with a focus on modularity and reusability. We describe recent efforts in improving the transparency and efficiency of the development process and community-building, as well as efforts towards quality assurance and reproducible research. In addition to a major redesign of many simulation components in order to facilitate setting up complex simulations in DuMux, version 3 introduces a more consistent abstraction of finite volume schemes. Finally, the new framework for multi-domain simulations is described, and three numerical examples demonstrate its flexibility.

Abstract

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.

Abstract

In this work we present an abstract finite volume discretization framework for incompressible immiscible two-phase flow through porous media. A priori error estimates are derived that allow us to prove the existence of discrete solutions and to establish the proof of convergence for schemes belonging to this framework. In contrast to existing publications the proof is not restricted to a specific scheme and it assumes neither symmetry nor linearity of the flux approximations. Two nonlinear schemes, namely a nonlinear two-point flux approximation and a nonlinear multipoint flux approximation, are presented, and some properties of these schemes, e.g. saturation bounds, are proven. Furthermore, the numerical behavior of these schemes (e.g. accuracy, coercivity, efficiency or saturation bounds) is investigated for different test cases for which the coercivity is checked numerically.

Abstract

Subsurface flow and geomechanics are often modeled with sequential approaches. This can be computationally beneficial compared with fully coupled schemes, while it requires usually compromises in numerical accuracy, at least when the sequential scheme is non-iterative. We discuss the influence of the choice of scheme on the numerical accuracy and the expected computational effort based on a comparison of a fully coupled scheme, a scheme employing a one-way coupling, and an iterative scheme using a fixed-stress split for two subsurface injection scenarios. All these schemes were implemented in the numerical simulator DuMux. This study identifies conditions of problem settings where differences due to the choice of the model approach are as important as differences in geologic features. It is shown that in particular transient and multiphase flow, effects can be causing significant deviations between non-iterative and iterative sequential schemes, which might be in the same order of magnitude as geologic uncertainty. An iterated fixed-stress split has the same numerical accuracy as a fully coupled scheme but only for a certain number of iterations which might use up the computational advantage of solving two smaller systems of equations rather than a big monolithical one.

Abstract

Abstract We propose a new mathematical model to learn capillary leakage coefficients from dynamic susceptibility contrast MRI data. To this end, we derive an embedded mixed-dimension flow and transport model for brain tissue perfusion on a subvoxel scale. This model is used to obtain the contrast agent concentration distribution in a single MRI voxel during a perfusion MRI sequence. We further present a magnetic resonance signal model for the considered sequence including a model for local susceptibility effects. This allows modeling MR signal-time curves that can be compared with clinical MRI data. The proposed model can be used as a forward model in the inverse modeling problem of inferring model parameters such as the diffusive capillary wall conductivity. Acute multiple sclerosis lesions are associated with a breach in the integrity of the blood-brain barrier. Applying the model to perfusion MR data of a patient with acute multiple sclerosis lesions, we conclude that diffusive capillary wall conductivity is a good indicator for characterizing activity of lesions, even if other patient-specific model parameters are not well-known.