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Adaptive homotopy continuation for relative permeability models in reservoir simulation

Peter Moritz von Schultzendorff: Monday, 13th January 2025 at 10:30 Accurate modeling of physical processes requires an appropriate selection of constitutive laws. In physics-based reservoir simulation, constitutive laws, e.g., relative permeabilities are often chosen to be, mathematically speaking, simple functions, not necessarily adhering to physics. The paradigm of hybrid modeling allows the integration of machine learned (ML) constitutive laws. Trained on lab, field, and fine-scale simulation data, ML models represent the underlying physics with high fidelity.Strong nonlinearities in classic (i.e., non-ML) relative permeability have been identified as one of the main sources for convergence issues of nonlinear solvers in reservoir simulation. This issue grows in severance for ML relative permeability models, as their high fidelity to real-world data compromises the mathematically desirable properties of simpler models.In this work, we employ the homotopy continuation (HC) method to recover nonlinear solver robustness for classic relative permeability models. The HC method improves nonlinear solver robustness by first solving a problem with simpler relative permeabilities and then iteratively traversing a solution curve towards the original, more complex problem. To efficiently trace the solution curve, we leverage a posteriori error estimates to design an adaptive HC algorithm that minimizes the total number of solver iterations.We show the current status of our work, both on the theoretical and implementation side, and give an outlook into the application to ML relative permeabilities.

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Analyse du problème des oscillations parasites des méthodes de Volumes Finis pour des écoulements à faible nombre de Mach en mécanique des fluides.

Ibtissem Lannabi: Thursday, 23rd January 2025 at 11:00 Ce travail de recherche porte sur la simulation numérique d’écoulements de fluides à faibles nombres de Mach, modélisés par le système d’Euler compressible. Les solveurs fréquemment utilisés pour discrétiser ce modèle sont des solveurs de type Godunov. Cependant, ces solveurs se comportent très mal à bas nombre de Mach en termes d’efficacité et de précision.En effet, lorsque le nombre de Mach tend vers zéro, les ondes matérielles et acoustiques se propagent sur deux échelles de temps distinctes, rendant ainsi la discrétisation temporelle délicate.En particulier, un schéma explicite est stable sous critère CFL, qui dépend de la vitesse du son, rendant ainsi cette condition très contraignante. En ce qui concerne le problème de précision que l’on observe particulièrement dans le cas des grilles quadrangulaires, il s’agit du fait que la solution discrète ne converge pas vers la solution incompressible lorsque le nombre de Mach tend vers zéro.Pour s’affranchir de ce problème de précision, plusieurs correctifs ont été développés, consistant à modifier la diffusion numérique du schéma original. Ces correctifs permettent d’améliorer la précision des schémas compressibles lorsque le nombre de Mach tend vers zéro. Malheureusement, ils introduisent d’autres problèmes, tels que l’apparition de modes oscillants (mode en échiquier sur une grille cartésienne) dans la solution numérique ou l’extrême diffusion des ondes acoustiques à bas nombre de Mach. L’efficacité est également compromise car ces schémas sont stables sous une CFL encore plus restrictive que le schéma original. Dans cet exposé, nous proposons d’analyser le phénomène des oscillations qui affecte certains des correctifs proposés dans la littérature. Nous nous intéressons principalement aux correctifs basés sur le schéma de Roe, en particulier ceux qui réduisent la diffusion numérique sur la vitesse normale. L’analyse asymptotique de ces schémas conduit à une discrétisation du système des ondes pour…

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Iterative solvers and optimal complexity of adaptive finite element methods

Ani Miraci: Tuesday, 17th December at 11:00 Finite element methods (FEMs) are often used to discretize second-order elliptic partial differential equations (PDEs). While standard FEMs rely on underlying uniform meshes, adaptive FEMs (AFEMs) drive the local mesh-refinement to capture potential singularities of the (unknown) PDE solution (stemming, e.g., from the data or the domain geometry). Crucially, adaptivity is steered by reliable a posteriori error control, often encoded in the paradigm SOLVE — ESTIMATE — MARK — REFINE. AFEMs allow to obtain optimal rates of convergence with respect to the number of degrees of freedom (an improvement to standard FEMs). However, in terms of computational costs, an adaptive algorithm is inherently cumulative in nature: an initial coarse mesh is used as input and exact finite element solutions need to be computed on consecutively refined meshes before a desired accuracy can be ensured. Thus, in practice, one strives instead to achieve optimal complexity, i.e., optimal rate of convergence with respect to the overall computational cost. The core ingredient needed for optimal complexity consists in the use of appropriate iterative solvers to be integrated as the SOLVE module within the adaptive algorithm. More precisely, one requires:(i) a solver whose each iteration is: (a) of linear complexity and (b) contractive;(ii) a-posteriori-steered solver-stopping criterion which allows to discern and balance discretization and solver error;(iii) nested iteration, i.e., the last computed solver-iterate is used as initial guess in the newly-refined mesh. First, we develop an optimal local multigrid for the context of symmetric linear elliptic second order PDEs and a finite element discretization with a fixed polynomial degree p and a hierarchy of bisection-generated meshes with local size h. The solver contracts the algebraic error hp-robustly and comes with a built-in a posteriori estimator equivalent to the algebraic error.Second, the overall adaptive algorithm is then shown…

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A parameter-free HDG method for linear elasticity with strongly symmetric stress

Lina Zhao: Thursday, 12th December at 11:00 In this talk, we present a parameter-free hybridizable discontinuous Galerkin (HDG) method of arbitrary polynomial orders for the linear elasticity problem, where the symmetry of stress is strongly imposed. The $H(\tdiv;\Omega)$-conforming space is used for the approximation of the displacement and the standard polynomial space is used for the approximation of the stress. The tangential trace of displacement acts as the Lagrange multiplier. The quasi-optimal approximation (up to data-oscillation term) is established for the $L^2$-error of stress and discrete $H^1$-error of displacement with $\lambda$-independent constants without requiring additional regularity assumption.  To guide adaptive mesh refinement, $\lambda$-robust a posteriori error estimator is derived. Several numerical experiments will be reported to demonstrate the performance of the proposed scheme.

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𝐻² conforming virtual element discretization of nondivergence form elliptic equations

Guillaume Bonnet: Thursday, 21st November at 11:00 The numerical discretisation of elliptic equations in nondivergence form is notoriously challenging, due to the lack of a notion of weak solutions based on variational principles. In many cases, there still is a well-posed variational formulation for such equations, which has the particularity of being posed in 𝐻², and therefore leads to a strong solution. Galerkin discretizations based on this formulation have been studied in the literature. Since 𝐻² conforming finite elements tend to be considered impractical, most of these discretizations are of discontinuous Galerkin type. On the other hand, it has been observed in the virtual element literature that the virtual element method provides a practical way to build 𝐻² conforming discretizations of variational problems. In this talk, I will describe a virtual element discretization of equations in nondivergence form. I will start with a simple linear model problem, and show how the 𝐻² conformity of the method allows for a particularly simple well-posedness and error analysis. I will then discuss the extension to equations with lower-order terms and with Hamilton-Jacobi-Bellman type nonlinearities, and present some numerical results.

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