On the term and concepts of numerical model validation in geoscientific applications

Modeling and numerical simulation of the coupled physical and chemical processes observed in the subsurface are the only options for long-term analyses of complex geological systems. This contribution discusses some more general aspects of the (dynamic) process modeling for geoscientific applications including reflections about the slightly different understanding of the terms model and model validation in different scientific communities, and about the term and methods of model calibration in the geoscientifc context. Starting from the analysis of observations of a certain part of the perceived reality, the process of model development comprises the establishment of the physical model characterizing relevant processes in a problem-oriented manner, and subsequently the mathematical and numerical models. Considering the steps of idealization and approximation in the course of model development, Oreskes et al. [1] state that process and numerical models can neither be verified nor validated in general. Rather the adequacy of models with specific assumptions and parameterizations made during model set-up can be confirmed. If the adequacy of process models with observations can be confirmed using lab as well as field tests and process monitoring, the adequacy of numerical models can be confirmed using numerical benchmarking and code comparison. Model parameters are intrinsic elements of process and numerical models, in particular constitutive parameters. As they are often not directly measurable, they have to be established by solving inverse problems based on an optimal numerical adaptation of observation results. In addition, numerical uncertainty analyses should be an obligatory part of numerical studies for critical real world applications

On the necessity and a generalized conceptual model for the consideration of large strains in rock mechanics

This contribution presents a generalized conceptual model for the finite element solution of quasi-static isothermal hydro-mechanical processes in (fractured) porous media at large strains. A frequently used averaging procedure, known as Theory of Porous Media, serves as background for the complex multifield approach presented here. Within this context, a consistent representation of the weak formulation of the governing equations (i.e., overall balance equations for mass and momentum) in the reference configuration of the solid skeleton is preferred. The time discretization and the linearization are performed for the individual variables and nonlinear functions representing the integrands of the weak formulation instead of applying these conceptual steps to the overall nonlinear system of weighted residuals. Constitutive equations for the solid phase deformation are based on the multiplicative split of the deformation gradient allowing the adaptation of existing approaches for technical materials and biological tissues to rock materials in order to describe various inelastic effects, growth and remodeling in a thermodynamically consistent manner. The presented models will be a feature of the next version of the scientific open-source finite element code OpenGeoSys developed by an international developer and user group, and coordinated by the authors

Zur Numerik der inversen Aufgabe für gemischte (u/p) Formulierungen am Beispiel der nahezu inkompressiblen Elastizität bei großen Verzerrungen

In dieser Publikation werden ein numerisches Verfahren zur Kalibrierung von Materialmodellen für die Simulation großer, nahezu inkompressibler hyperelastischer Verzerrungen sowie dessen numerische Realiserung im Rahmen einer gemischten Finite Elemente Formulierung vorgestellt. Dabei werden die Parameter der konstitutiven Beziehungen auf der Grundlage experimentell erfasster Verschiebungsfelder (vorzugsweise inhomogener) bzw. globaler Informationen ermittelt. Dieses inkorrekte, inverse Problem wird mit Hilfe eines deterministischen Optimierungsverfahrens vom trust-region-Typ gelöst. Wesentlicher Bestandteil ist dabei die halbanalytische Sensitivitätsanalyse, die ein effizientes und hochgenaues Verfahren zur Ermittlung des Gradienten der Zielfunktion darstellt. Sie erfordert die einmalige Lösung eines zur direkten Aufgabe analogen Gleichungssystems pro Parameter und Lastschritt und basiert auf der impliziten Differentiation der schwachen Formulierung des gemischten Randwertproblems nach den Materialparametern. Genauigkeit und Konvergenzverhalten der numerischen Algorithmen werden an illustrativen Beispielen mit synthetischen Messwerten demonstriert. Im Mittelpunkt stehen dabei Untersuchungen zur Abhängigkeit des Optimierungsergebnisses von den Startwerten für unterschiedliche konstitutive Ansätze der kompressiblen und nahezu inkompressiblen Elastizität

Towards the generic conceptual and numerical framework for the simulation of CO2 sequestration in different types of georeservoirs

In this paper, conceptual and numerical modeling of coupled thermo-hydromechanical (THM) processes during CO2 injection and storage is presented. The commonly used averaging procedure combining the Theory of Mixtures and the Concept of Volume Fractions serves as background for the complex porous media approach presented here. Numerical models are based on a generalized formulation of the individual and overall balance equations for mass and momentum, as well as, in non-isothermal case, the energy balance equation. Within the framework of a standard Galerkin approach, the method of weighted residuals is applied to derive the weak forms of governing equations. After discretizing spatially these weak forms, a system of nonlinear algebraic equations can be obtained. For the required time discretization a generalized first order difference scheme is applied, linearization is performed using Picard or Newton-Raphson methods. The corresponding models are implemented within the scientific open source finite element code OpenGeoSys (OGS) developed by the authors, which is based on object oriented programming concepts. This assists the efficient treatment of different physical processes, whose mathematical models are of similar structure. Thus, the paper is mainly focused on a generic theoretical framework for the coupled processes under consideration. Within this context, CO2 sequestration in georeservoirs of different type can be simulated (e.g., saline aquifers, (nearly) depleted hydrocarbon reservoirs)

Numerical analysis of CO2 injection into deformable saline reservoirs: benchmarking and initial observations

A numerical scheme is presented for the solution of coupled multiphase hydromechanical problems in deformable porous media. Model verification is conducted against analytical solutions for multiphase flow with capillarity and coupled multiphase hydromechanical consolidation. A hybrid monolithic(flow)-staggered(mechanical) numerical solution scheme is verified to be stable for real materials, provided proper error control is placed on the hydraulic to mechanical iteration and the time-stepping scheme. Initial results of CO2 injection into an aquifer-caprock system do not show significant differences in CO2 migration rate between flow-only and hydro-mechanical simulations for conservative injection scenarios. However, the results highlight important regions in the reservoir with regard to potential mechanical failure and caprock integrity and suggest the need for further analysis

On the term and concepts of numerical model validation in geoscientific applications

Modeling and numerical simulation of the coupled physical and chemical processes observed in the subsurface are the only options for long-term analyses of complex geological systems. This contribution discusses some more general aspects of the (dynamic) process modeling for geoscientific applications including reflections about the slightly different understanding of the terms model and model validation in different scientific communities, and about the term and methods of model calibration in the geoscientifc context. Starting from the analysis of observations of a certain part of the perceived reality, the process of model development comprises the establishment of the physical model characterizing relevant processes in a problem-oriented manner, and subsequently the mathematical and numerical models. Considering the steps of idealization and approximation in the course of model development, Oreskes et al. [1] state that process and numerical models can neither be verified nor validated in general. Rather the adequacy of models with specific assumptions and parameterizations made during model set-up can be confirmed. If the adequacy of process models with observations can be confirmed using lab as well as field tests and process monitoring, the adequacy of numerical models can be confirmed using numerical benchmarking and code comparison. Model parameters are intrinsic elements of process and numerical models, in particular constitutive parameters. As they are often not directly measurable, they have to be established by solving inverse problems based on an optimal numerical adaptation of observation results. In addition, numerical uncertainty analyses should be an obligatory part of numerical studies for critical real world applications

Thermodynamisch konsistente Formulierung des gekoppelten Systems der Thermoelastoplastizität bei großen Verzerrungen auf der Basis eines Substrukturkonzepts

Non-negligible coupled thermal and mechanical effects occur in several physical and industrial procedures, e.g. warm for ming processes. The authors present the theoretical background of a phenomenological thermoelastoplastic material model at large strains as well as its numerical realization within the context of appropriate finite element formulations. As usual, the presented thermodynamical consistent constitutive approach is based on the multiplicative decomposition of the deformation gradient, and a corresponding additive decomposition of the free Helmholtz energy density. For the numerical treatment of thermoelastoplastic problems within a finite element approach, weak formulations of the balance equation of momentum and the heat conduction equation in material description are developed. For the solution of non-linear boundary value problems the linearization of the weak formulations is presented. Within the context of the mechanical problem the temperature dependence of material parameters as well as the thermal expansion are considered. The temperature evolution will be affected by non-thermal phenomena like the thermoelastic effect and plastic dissipation. Several numerical procedures for the solution of the coupled thermomechanical problem are discussed

On the necessity and a generalized conceptual model for the consideration of large strains in rock mechanics

This contribution presents a generalized conceptual model for the finite element solution of quasi-static isothermal hydro-mechanical processes in (fractured) porous media at large strains. A frequently used averaging procedure, known as Theory of Porous Media, serves as background for the complex multifield approach presented here. Within this context, a consistent representation of the weak formulation of the governing equations (i.e., overall balance equations for mass and momentum) in the reference configuration of the solid skeleton is preferred. The time discretization and the linearization are performed for the individual variables and nonlinear functions representing the integrands of the weak formulation instead of applying these conceptual steps to the overall nonlinear system of weighted residuals. Constitutive equations for the solid phase deformation are based on the multiplicative split of the deformation gradient allowing the adaptation of existing approaches for technical materials and biological tissues to rock materials in order to describe various inelastic effects, growth and remodeling in a thermodynamically consistent manner. The presented models will be a feature of the next version of the scientific open-source finite element code OpenGeoSys developed by an international developer and user group, and coordinated by the authors