An Optimal Fourth-Order Finite Difference Scheme for the Helmholtz Equation Based on the Technique of Matched Interface Boundary

In this paper, a new optimal fourth-order 21-point finite difference scheme is proposed to solve the 2D Helmholtz equation numerically, with the technique of matched interface boundary (MIB) utilized to treat boundary problems. For the approximation of Laplacian, two sets of fourth-order difference schemes are derived firstly based on the Taylor formula, with a total of 21 grid points involved. Then, a weighted combination of the two schemes is employed in order to reduce the numerical dispersion, and the weights are determined by minimizing the dispersion. Similarly, for the discretization of the zeroth-order derivative term, a weighted average of all the 21 points is implemented to obtain the fourth-order accuracy. The new scheme is noncompact; hence, it encounters great difficulties in dealing with the boundary conditions, which is crucial to the order of convergence. To tackle this issue, the matched interface boundary (MIB) method is employed and developed, which is originally used to accommodate free edges in the discrete singular convolution analysis. Convergence analysis and dispersion analysis are performed. Numerical examples are given for various boundary conditions, which show that new scheme delivers a fourth order of accuracy and is efficient in reducing the numerical dispersion as well.

Modelling the volcanic ash plume from Eyjafjallajökull eruption (May 2010) over Europe: evaluation of the benefit of source term improvements and of the assimilation of aerosol measurements

Numerical dispersion models are used operationally worldwide to mitigate the effect of volcanic ash on aviation. In order to improve the representation of the horizontal dispersion of ash plumes and of the 3D concentration of ash, a study was conducted using the MOCAGE model during the European Natural Airborne Disaster Information and Coordination System for Aviation (EUNADICS-AV) project. Source term modelling and assimilation of different data were investigated. A sensitivity study of source term formulation showed that a resolved source term, using the FPLUME plume rise model in MOCAGE, instead of a parameterised source term, induces a more realistic representation of the horizontal dispersion of the ash plume. The FPLUME simulation provides more concentrated and focused ash concentrations in the horizontal and the vertical dimensions than the other source term. The assimilation of Moderate Resolution Imaging Spectroradiometer (MODIS) aerosol optical depth has an impact on the horizontal dispersion of the plume, but this effect is rather low and local compared to source term improvement. More promising results are obtained with the continuous assimilation of ground-based lidar profiles, which improves the vertical distribution of ash and helps in reaching realistic values of ash concentrations. Using this configuration, the effect of assimilation may last for several hours and it may propagate several hundred kilometres downstream of the lidar profiles.

High-Order Adaptive Scheme for Reactive Transport in Heterogeneous Porous Media

Subsurface sequestration of carbon dioxide, contaminant transport, and enhanced oil recovery processes often involve complex reaction dynamics. The rock-fluid interactions span a very wide range of length and time scales, and it is important for the numerical solutions to resolve these scales properly. To address these challenges, we extend the adaptive transport scheme for the simulation of reactive transport in heterogeneous porous media developed previously (Deucher and Tchelepi, 2021) to account for (a) higher-order approximation of the convective fluxes and (b) coupling with a chemical solver connected to geochemical databases. The numerical results demonstrate that adaptivity is more effective when a higher-order approximation of the fluxes is used. This is because of lower levels of numerical dispersion compared with low-order approximations, which helps resolve the displacement fronts more accurately. As a result, the regions that experience significant concentration and saturation gradients are more confined, and that leads to improvements in the computational efficiency of the adaptive scheme. The robustness of the approach is demonstrated using a highly heterogeneous two-phase case with multiple wells and a variable total liquid-rate. Due to the modularity of the adaptive scheme, coupling with a chemical solver module is straightforward. The scheme is tested for a three-dimensional case that considers injection of carbonated water in a reservoir matrix of calcite. The results show that the adaptive scheme leads to an accurate representation of the reference concentration distributions of the six reactive components throughout the simulation and leads to a large reduction in the number of cell updates required to achieve the solution.

Numerical Simulations of Surfactant Flooding in Carbonate Reservoirs: The Impact of Geological Heterogeneities Across Scales

We demonstrate how geological heterogeneity impacts the effectiveness of surfactant-based enhanced oil recovery (EOR) at larger (inter-well and sector) scales when upscaling small (core) scale heterogeneity and physicochemical processes. We used two experimental datasets of surfactant-based EOR where spontaneous imbibition and viscous displacement, respectively dominate recovery. We built 3D core-scale simulation models to match the data and parameterize surfactant models. The results were deployed in high-resolution models that preserve the complexity and heterogeneity of carbonate formations in the inter-well and sector scale. These larger-scale models were based on two outcrop analogues from France and Morroco, respectively, which capture the reservoir architectures inherent to the productive carbonate reservoir systems in the Middle East. We then assessed and quantified the error in production forecast that arises due to upscaling, upgridding, and simplification of geological heterogeneity. Simulation results showed a broad range of recovery predictions. The variability arises from the choice of surfactant model parameterization (i.e., spontaneous imbibition vs viscous displacement) and the way the heterogeneity in the inter-well and sector models was upscaled and simplified. We found that the parameterization of surfactant models has a significant impact on recovery predictions. Oil recovery at the larger scale was observed to be higher when using the parametrization derived from viscous displacement experiments compared to parameterization from spontaneous imbibition experiments. This observation clearly demonstrated how core-scale processes impact recovery predictions at the larger scales. Also, the variability in recovery prediction due to the choice of surfactant model was as large as the variability arising from upscaling and upgridding. Upscaled and upgridded models overestimated recovery because of the simplified geology. Grid coarsening exacerbated this effect because of the increased numerical dispersion. These results emphasize the need to use correctly configured surfactant models, appropriate grid resolution that minimizes numerical dispersion, and properly upscaled reservoir models to accurately forecast surfactant floods. Our findings present new insights into how the uncertainty in production forecasts during surfactant flooding depends on the way surfactant models are parameterized, how the reservoir geology is upscaled, and how numerical dispersion is impacted by grid coarsening.

Numerical effects on the simulation of surfactant flooding for enhanced oil recovery

Numerical simulation of surfactant flooding using conventional reservoir simulation models can lead to unreliable forecasts and bad decisions due to the appearance of numerical effects. The simulations give approximate solutions to systems of nonlinear partial differential equations describing the physical behavior of surfactant flooding by combining multiphase flow in porous media with surfactant transport. The approximations are made by discretization of time and space which can lead to spurious pulses or deviations in the model outcome. In this work, the black oil model was simulated using the decoupled implicit method for various conditions of reservoir scale models to investigate behaviour in comparison with the analytical solution obtained from fractional flow theory. We investigated changes to cell size and time step as well as the properties of the surfactant and how it affects miscibility and flow. The main aim of this study was to understand pulse like behavior that has been observed in the water bank to identify cause and associated conditions. We report for the first time that the pulses occur in association with the simulated surfactant water flood front and are induced by a sharp change in relative permeability as the interfacial tension changes. Pulses are diminished when the adsorption rate was within the value of 0.0002kg/kg to 0.0005kg/kg. The pulses are absent for high resolution model of 5000 cells in x direction with a typical cell size as used in well-scale models. The growth or damping of these pulses may vary from case to case but in this instance was a result of the combined impact of relative mobility, numerical dispersion, interfacial tension and miscibility. Oil recovery under the numerical problems reduced the performance of the flood, due to large amounts of pulses produced. Thus, it is important to improve existing models and use appropriate guidelines to stop oscillations and remove errors.

Improved wave dispersion properties in 1D and 2D bond-based peridynamic media

In this study, a novel method for improving the simulation of wave propagation in Peridynamic (PD) media is investigated. Initially, the dispersion properties of the nonlocal Bond-Based Peridynamic model are computed for 1-D and 2-D uniform grids. The optimization problem, developed through inverse analysis, is set up by comparing exact and numerical dispersion and minimizing the error. Various weighted residual techniques, i.e., point collocation, sub-domain collocation, least square approximation and the Galerkin method, are adopted and the modification of the wave dispersion is then proposed. It is found that the proposed methods are able to significantly improve the description of wave dispersion phenomena in both 1-D and 2-D PD models.