Delta: An object-oriented finite element code architecture for massively parallel computers

An approximate Jacobian nonlinear solver for multiphase flow and transport

A 6M digital twin for modeling and simulation in subsurface reservoirs

A numerical model is presented for the computation of unsteady two‐fluid interfaces in nonlinear porous media flow. The nonlinear Forchheimer equation is included in the Navier‐Stokes equations for porous media flow. The model is based on capturing the interface on a fixed mesh domain. The zero level set of a pseudo‐concentration function, which defines the interface between the two fluids, is governed by a time‐dependent advection equation. The time‐dependent Navier‐Stokes equations and the advection equation are spatially discretized by the finite element (FE) method. The fully coupled implicit time integration scheme and the explicit forward Eulerian scheme are implemented for the advancement in time. The trapezoidal rule is applied to the fully implicit scheme, while the operator‐splitting algorithm is used for the velocity‐pressure segregation in the explicit scheme. The spatial and time discretizations are stabilized using FE stabilization techniques. Numerical examples of unsteady flow of two‐fluid interfaces in an earth dam are investigated.

The focus of the underlying research work is on the macroscopic modeling of unstable multiphase fluid flow in deformable porous media, where a lower‐viscous fluid is displaced by a more viscous fluid. This process leads to the formation of channel‐like networks, called viscous fingering. The instability effect is involved in a wide range of different fields in engineering. Some of the most common applications include carbon sequestration to store carbon dioxide (CO2) in underground reservoirs, contaminant transport in geostructures, in industrial processes, such as filtration, catalytic reactions, and in the operation of polymer electrolyte membrane fuel cells with multiphase flow in the gas diffusion layers. In this work, ideal miscible water–glycerin fluids are considered. It is assumed that the interacting fluids in the deformable porous media are incompressible. In addition, a dispersion–diffusion law is applied to capture the fluid–fluid interactions. The presence of a deformable porous material adds additional effects to the problem of the multiphase flow in porous media. The stresses in the porous solid matrix lead to changes in the porosity and influence the flow velocity of the fluids. To couple the deformable porous media with the multiphase flow, a macroscopic approach is used that relies on the theory of porous media. For the porous solid matrix, a linear elastic material model within small strains assumption is applied. The influence of the deformation‐dependent porosity on the instability is studied for 2D simulations, such as a multilayered geometry with different elastic parameters. The presented coupled nonlinear system of differential equations is simulated with the finite element method. Furthermore, a stabilization technique based on the quasi‐compressibility method is used.

Purpose – This paper aims to present a computationally efficient finite element model for the simulation of isothermal immiscible two-phase flow in a rigid porous media with a particular application to CO2 sequestration in underground formations. Focus is placed on developing a numerical procedure, which is effectively mesh-independent and suitable to problems at regional scales. Design/methodology/approach – The averaging theory is utilized to describe the governing equations of the involved unsaturated multiphase flow. The level-set (LS) method and the extended finite element method (XFEM) are utilized to simulate flow of the CO2 plume. The LS is employed to trace the plume front. A streamline upwind Petrov-Galerkin method is adopted to stabilize possible occurrence of spurious oscillations due to advection. The XFEM is utilized to model the high gradient in the saturation field front, where the LS function is used for enhancing the weighting and the shape functions. Findings – The capability of the proposed model and its features are evaluated by numerical examples, demonstrating its accuracy, stability and convergence, as well as its advantages over standard and upwind techniques. The study showed that a good combination between a mathematical model and a numerical model enables the simulation of complicated processes occurring in complicated and large geometry using minimal computational efforts. Originality/value – A new computational model for two-phase flow in porous media is introduced with basic requirements for accuracy, stability, and convergence, which are met using relatively coarse meshes.

Accurate modelling of the fate of injected CO 2 is necessary if geological storage is to be used at a large scale. In one form of geological storage, CO2 is injected into an aquifer that has a sealing caprock, forming a CO2 cap beneath the caprock. The diffusion of CO2 into underlying formation waters increases the density of water near the top of the aquifer, bringing the system to a hydro-dynamically unstable state. Instabilities can arise from the combination of an unstable density profile and inherent perturbations within the system, e.g., formation heterogeneity. If created, this instability causes convective mixing and greatly accelerates the dissolution of CO2 into the aquifer. Accurate estimation of the rate of dissolution is important for risk assessments because the timescale for dissolution is the timescale over which the CO2 has a chance to leak through the caprock or any imperfectly sealed wells. A new 2D numerical model which has been developed to study the diffusive and convective mixing in geological storage of CO2 is described. Effects of different formation parameters are investigated in this paper. Results reveal that there are two different timescales involved. The first timescale is the time to onset the instability and the second one is the time to achieve ultimate dissolution. Depending on system Rayleigh number and the formation heterogeneity, convective mixing can greatly accelerate the dissolution of CO2 in an aquifer. Two field scale problems were studied. In the first, based on the Nisku aquifer, more than 60% of the ultimate dissolution was achieved after 800 years, while the computed timescale for dissolution in the same aquifer in the absence of convection was orders of magnitude larger. In the case of the Glauconitic sandstone aquifer, there was no convective instability. Results suggest that the presence and strength of convective instability should play an important role in choosing aquifers for CO2 storage. risk of leakage of CO2 from a storage formation may need to analyze leakage mechanisms and their likelihood of occurrence during the full-time period over which mobile free-phase CO 2 is expected to remain in the reservoir. Once dissolved, risk assessments may well ignore the leakage pathways resulting from the very slow movement of CO2-saturated brines. An accurate assessment of the timescales for dissolution are therefore of the first order of importance. The CO2 injected into a saline reservoir is typically 40 – 60% less dense than the resident brines (4) . Driven by density contrasts, CO 2 will flow horizontally (in a horizontal aquifer) spreading under the caprock, and flow upwards, potentially leaking through any high permeability zones or artificial penetrations, such as abandoned wells. The free-phase CO2 (usually supercritical fluid) slowly dissolves in the brines. The resulting CO2-rich brines are slightly denser than undersaturated brines, making them negatively buoyant, and thus greatly reducing or eliminating the possibility of leakage. The rate of dissolution depends on the rate at which diffusion or convection brings undersaturated brine in contact with CO2. Convective mixing enhances the dissolution rate as compared to diffusion by distributing the CO2 into the aquifer (5) . Therefore, the role of convective mixing in CO2 sequestration and the timescales involved in the process are important. The dissolution time of the injected CO2 into brine is important because during this time the injected CO2 has a chance to leak into the atmosphere through the caprock and wellbores. Accurate modelling of the convective mixing in heterogeneous porous media plays a central role in predicting the fate of CO2 injected into aquifers. In this paper, geological CO2 storage is modelled by solving the convection-diffusion equation while considering the CO2-brine interface as a boundary condition. Geochemical reactions that can reduce the timescale of sequestration of CO2 are not included, since they generally occur on longer timescales (6) . The paper is organized as follows. First, the mathematical model for simulating density-driven flow through porous media is briefly presented. The model is validated with a benchmark problem for density-driven flow in porous media. Then, the geological CO2 sequestrations both in small and field scale are simulated using the model. Two important timescales, the effect of formation properties, as well as sensitivity to temporal and spatial discretisations, are discussed. Finally, the results are summarized and their relevance to geological storage of CO 2 in aquifers is discussed.

A new approach for phase transitions in miscible multi-phase flow in porous media

SummaryNumerical modeling of large deformations in fluid‐infiltrated porous media must accurately describe not only geometrically nonlinear kinematics but also fluid flow in heterogeneously deforming pore structure. Accurate simulation of fluid flow in heterogeneous porous media often requires a numerical method that features the local (elementwise) conservation property. Here, we introduce a new finite element framework for locally mass conservative solution of coupled poromechanical problems at large strains. At the core of our approach is the enriched Galerkin discretization of the fluid mass balance equation, whereby elementwise constant functions are augmented to the standard continuous Galerkin discretization. The resulting numerical method provides local mass conservation by construction with a usually affordable cost added to the continuous Galerkin counterpart. Two equivalent formulations are developed using total Lagrangian and updated Lagrangian approaches. The local mass conservation property of the proposed method is verified through numerical examples involving saturated and unsaturated flow in porous media at finite strains. The numerical examples also demonstrate that local mass conservation can be a critical element of accurate simulation of both fluid flow and large deformation in porous media.

Mixed finite element – discontinuous finite volume element discretization of a general class of multicontinuum models

FAST.Farm is a newly developed multiphysics, midfidelity engineering tool that can be used to predict turbine power and structural loads of wind turbines in a wind farm with minimal computational expense. Previous studies have shown similarities and differences in wind turbine performance between FAST.Farm and high-fidelity large-eddy simulations (LES) using both LES precursor-generated inflow and TurbSim-generated synthetic inflow. While conservative resolutions have been used to date, no formal spatial or temporal discretization study has been performed for the FAST.Farm model wind domains. This work aims to study the effects of varying the spatial and temporal discretization of the wind domains on wake meandering (low-resolution domain) and turbine structural response (high-resolution domains) and resulting wake and load calculations. The purpose of this study is to establish convergence criteria and recommendations for discretization values in terms of rotor diameter (D, expressed in meters) that will maximize computational efficiency. To ensure a percent error of ≤ 1% in standard deviation of wake center position relative to the finest included resolution, a low-resolution time step of 0.024D s, 0.016D s, and 0.0079D s and a low-resolution spatial discretization of 0.079D m, 0.16D m, and 0.24D m is recommended for 8 m/s, 12 m/s, and 18 m/s mean ambient wind speed at hub height, respectively. These guidelines are likely applicable to other implementations of the dynamic wake meandering model. To ensure a percent error of ≤ 1% in standard deviation of all considered structural response outputs, a high-resolution time step that captures the highest influential excitation and natural frequencies in the system and a high-resolution spatial discretization comparable to the maximum airfoil cord length is recommended. These guidelines are likely applicable to any aeroelastic analysis.

Petrov-Galerkin formulations with weighting functions dependent upon spatial and temporal discretization: Applications to transient convection-diffusion problems

Dispersion analysis of finite difference and discontinuous Galerkin schemes for Maxwell's equations in linear Lorentz media

Implicit Hybrid Upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity

<p>GPU architectures are characterized by the abundant computing capacity in relation to memory bandwich. This makes them very good for solving problems temporaly explicit and with compact spatial discretizations. Most works using GPU focuses on the parallelization of solvers of linear equations generated by the numerical methods. However, to obtain a good performance in numerical applications using GPU it is crucial to work preferably in codes based entirely on GPU. In this work we solve a 3D nonlinear diffusion equation, using finite volume method in cartesian meshes. Two different time schemes are compared, explicit and implicit, considering for the latter, the Newton method and Conjugate Gradient solver for the system of equations. An evaluation is performed in CPU and GPU of each scheme using different metrics to measure performance, accuracy, calculation speed and mesh size. To evaluate the convergence propierties of the different schemes in relation to spatial and temporal discretization, an arbitrary analytical solution is proposed, which satisfies the differential equation by chossing a source term chosen based on it.</p>

Considerable time and energy has been devoted to the design and development of second order, time-domain schemes for Maxwell's equations. Schemes, for example, such as the finite-difference time-domain (FDTD) method have been applied to many diverse problems ranging from radar cross-section analysis to ionospheric radio wave propagation studies. Yet, the second-order nature of the schemes prevents one from obtaining accurate data for geometries or propagation distances that span tens of wavelengths. This property is clearly manifested by frequency domain analyses, which demonstrate the high accumulation of phase and dissipation errors over many time steps. To mitigate the effect of phase and dissipation errors, two key options exist: (1) decrease the size of the discretization cell and the time step or (2) increase the order of accuracy. We consider only option two. In essence, the design of a time domain solver is subdivided into three essential tasks: (1) spatial discretization, (2) temporal discretization and (3) evaluation of the fully discrete system.