Toward Accurate Reservoir Simulations on Unstructured Grids: Design of Simple Error Estimators and Critical Benchmarking of Consistent Discretization Methods for Practical Implementation

Summary In this paper, we aim to identify discretization errors caused by non-K-orthogonal grids upfront through simple preprocessing tools and perform a comparative study of a set of representative, state-of-the-art, consistent discretizations [multipoint flux approximation (MPFA-O), mimetic finite difference (MFD), nonlinear two-pointflux approximation (NTPFA, TPFA), and average multipoint flux approximation (AvgMPFA)] to select the method most suited for inclusion in a commercial reservoir simulator. To predict the potential impact of discretization errors, we propose two types of error indicators. Static indicators measure the degree of nonconsistency of the two-point method at a cell level, and dynamic indicators measure how local discretization errors affect flow paths. The latter are computed using a series of idealized tracer simulations. By changing monitoring and injection points, one can mimic the reservoir-development strategy and thus focus on the errors introduced on quantities of real interest. To assess the practical usability of various consistent methods and validate our new error indicators, we use a set of representative grid models generated by contemporary commercial tools, for which we discuss static error indicators and compare tracer responses for the various discretization methods. We also compare degrees of freedom, sparsity, and the condition number of the alternative methods and discuss challenges related to their practical implementation. Our results indicate that tracer simulations constitute an efficient tool to identify and classify discretization errors and quantify their potential impact. We observe distinctively different behavior with the inconsistent two-point method and the consistent methods, which agree closely in terms of accuracy of the response. We also note a deficiency in the commercial realization of so-called Depogrids, which can result in unnecessarily complicated polytopal cells with hundreds of faces. Our overall conclusion is that NTPFA and AvgMPFA are the most viable solutions for integration into a commercial simulator, with the linear AvgMPFA method being the least invasive.

Development of a large-eddy simulation subgrid model based on artificial neural networks: a case study of turbulent channel flow

Atmospheric boundary layers and other wall-bounded flows are often simulated with the large-eddy simulation (LES) technique, which relies on subgrid-scale (SGS) models to parameterize the smallest scales. These SGS models often make strong simplifying assumptions. Also, they tend to interact with the discretization errors introduced by the popular LES approach where a staggered finite-volume grid acts as an implicit filter. We therefore developed an alternative LES SGS model based on artificial neural networks (ANNs) for the computational fluid dynamics MicroHH code (v2.0). We used a turbulent channel flow (with friction Reynolds number Reτ=590) as a test case. The developed SGS model has been designed to compensate for both the unresolved physics and instantaneous spatial discretization errors introduced by the staggered finite-volume grid. We trained the ANNs based on instantaneous flow fields from a direct numerical simulation (DNS) of the selected channel flow. In general, we found excellent agreement between the ANN-predicted SGS fluxes and the SGS fluxes derived from DNS for flow fields not used during training. In addition, we demonstrate that our ANN SGS model generalizes well towards other coarse horizontal resolutions, especially when these resolutions are located within the range of the training data. This shows that ANNs have potential to construct highly accurate SGS models that compensate for spatial discretization errors. We do highlight and discuss one important challenge still remaining before this potential can be successfully leveraged in actual LES simulations: we observed an artificial buildup of turbulence kinetic energy when we directly incorporated our ANN SGS model into a LES simulation of the selected channel flow, eventually resulting in numeric instability. We hypothesize that error accumulation and aliasing errors are both important contributors to the observed instability. We finally make several suggestions for future research that may alleviate the observed instability.

On the Role of Discretization Errors in the Quantification of Parameter Uncertainties

The independence of numerical and parameter uncertainties is investigated for the flow around the KVLCC2 tanker at Re = 4.6 × 106 using the time-averaged RANS equations supplemented by the k–ω two-equation SST model. The uncertain input parameter is the inlet velocity that varies ±0.25% and ±0.50% for the determination of sensitivity coefficients using finite-difference approximations. The quantities of interest are the friction and pressure coefficients of the ship and the Cartesian velocity components and turbulence kinetic energy at the propeller plane. A grid refinement study is performed for the nominal conditions to allow the estimation of the discretization error with power series expansions. However, for grids between 6 × 106 and 47.6 × 106 cells, not all the selected quantities of interest exhibit monotonic convergence. Therefore, the estimates of the sensitivity coefficients of the selected quantities of interest using the local sensitivity method and finite-differences performed for refinement levels that correspond to 0.764 × 106, 6 × 106 and 47.6 × 106 cells lead to significantly different values. Nonetheless, for a given grid, negligible differences are obtained for the sensitivity coefficients obtained with two different intervals in the finite-differences approximation. Discrepancies between sensitivity coefficients are compared with the estimated numerical uncertainties. Results obtained in the study suggest that uncertainty quantification performed in coarse grids may be significantly affected by discretization errors.

h-adaptive topology optimization considering variations of material properties and energy error density recovery

Purpose The purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity. Design/methodology/approach The new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level. Findings Two-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure. Originality/value This paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Numerical Errors in Unsteady Flow Simulations

This article discusses numerical errors in unsteady flow simulations, which may include round-off, statistical, iterative, and time and space discretization errors. The estimation of iterative and discretization errors and the influence of the initial condition on unsteady flows that become periodic are discussed. In this latter case, the goal is to determine the simulation time required to reduce the influence of the initial condition to negligible levels. Two one-dimensional, unsteady manufactured solutions are used to illustrate the interference between the different types of numerical errors. One solution is periodic and the other includes a transient region before it reaches a steady-state. The results show that for a selected grid and time-step, statistical convergence of the periodic solution may be achieved at significant lower error levels than those of iterative and discretization errors. However, statistical convergence deteriorates when iterative convergence criteria become less demanding, grids are refined, and Courant number increased.For statistically converged solutions of the periodic flow and for the transient solution, iterative convergence criteria required to obtain a negligible influence of the iterative error when compared to the discretization error are more strict than typical values found in the open literature. More demanding criteria are required when the grid is refined and/or the Courant number is increased. When the numerical error is dominated by the iterative error, it is pointless to refine the grid and/or reduce the time-step. For solutions with a numerical error dominated by the discretization error, three different techniques are applied to illustrate how the discretization uncertainty can be estimated, using grid/time refinement studies: three data points at a fixed Courant number; five data points involving three time steps for the same grid and three grids for the same time-step; five data points including at least two grids and two time steps. The latter two techniques distinguish between space and time convergence, whereas the first one combines the effect of the two discretization errors.

The influence of numerical parameters in the finite-volume method on the Wigley hull resistance

The equations discretization errors are often overlooked compared to the spatial discretization errors. This article presents the results of the influence of various equations discretization schemes in computational fluid dynamics on the prediction of the ship resistance and wave elevation on the hull. For the analysis, steady flow around a model of the Wigley hull is numerically predicted by employing a finite-volume method solver based on Reynolds-averaged Navier–Stokes equations and the volume-of-fluid method. Six momentum discretization schemes, three multi-fluid discretization schemes and three gradient schemes, are used in the analysis. The results show that the choice of discretization schemes has a significant influence on the results of wave elevation and resistance force in the case of the flow around the Wigley hull. In conclusion, second-order discretization schemes should be used for the resistance evaluation, in order to properly capture the non-linear effects.