Mixed Hybrid Finite Element Method Research Articles

A lattice Boltzmann approach to mathematical modeling of myocardial perfusion.

A mathematical model of myocardial perfusion based on the lattice Boltzmann method (LBM) is proposed and its applicability is investigated in both healthy and diseased cases. The myocardium is conceptualized as a porous material in which the transport and mass transfer of a contrast agent in blood flow is studied. The results of myocardial perfusion obtained using LBM in 1D and 2D are confronted with previously reported results in the literature and the results obtained using the mixed-hybrid finite element method. Since LBM is not suitable for simulating flow in heterogeneous porous media, a simplified and computationally efficient 1D-analog approach to 2D diseased case is proposed and its applicability discussed.

Numerical simulation of foam displacement impacted by kinetic and equilibrium surfactant adsorption

This study explores the effectiveness of foam in reducing gas mobility in porous media impacted by surfactant adsorption in highly heterogeneous porous media. The research focuses on assessing the influence of kinetic and equilibrium adsorption during foam displacement on reservoir sweep efficiency, considering co-injection and SAG (Surfactant Alternating Gas) injection strategies. Including the surfactant on the liquid phase and considering the surfactant adsorption effects fitted by experimental data, the mathematical modeling combines Darcy’s law, fluid phase conservation, surfactant transport equations, and a non-Newtonian foam model in local equilibrium. The simulations utilize FOSSIL (FOam diSplacement SImuLator), an in-house software composed of a stable and conservative numerical algorithm with reduced numerical diffusion. These properties are achieved by combining a hybrid mixed finite element method to solve the Darcy system with a central-upwind finite volume scheme to approximate the transport equations and adopting an implicit adaptive method in time. The results reveal that while omitting adsorption phenomena enhances sweep efficiency due to reduced surfactant loss, differences between equilibrium and kinetic adsorption models are minor (up to 0.3% in production). Consequently, these results suggest using simple adsorption models, such as equilibrium isotherms, rather than more complex models, such as kinetics. Additionally, the SAG approach outperforms co-injection (up to 19% in production), despite the fact that the adsorption is detrimental to both of them, aligning with literature results and previous findings.

Mathematical modeling of the multicomponent flow in porous media using higher-order methods

In this paper, we present a detailed numerical scheme for a single-phase compressible flow without diffusion of a multi-component mixture in porous media with the higher-order approximation in both space and time. The mathematical model consists of Darcy velocity, transport equations for each component of a mixture, pressure equation and associated relations for physical quantities such as viscosity or equation of state. The discrete problem is obtained using a combination of the discontinuous Galerkin method for the transport equations and the mixed-hybrid finite element method for the Darcy velocity and the pressure equation. In both methods the higher-order approximation is used. The resulting nonlinear problem for concentrations is solved with the fully mass-conservative iterative IMPEC method. Experimental order of convergence analysis (EOC) and some numerical experiments of a 2D flow are carried out.

Coupling mixed hybrid and extended finite element methods for the simulation of hydro-mechanical processes in fractured porous media

Simulation of coupled hydro-mechanical (HM) processes in fractured porous media requires specific numerical schemes to deal with nonlinearity, computational burden, and heterogeneity. The mixed hybrid finite element method (MHFEM) is superior to the standard finite element method in simulating flow in fractured domains. However, MHFEM formulation cannot be efficiently used for the discretization of solid mechanics equations. The extended finite element method (XFEM) has significant advantages in modeling mechanical processes in cracked domains. The main goal of this paper is to extend the application of the MHFEM to HM processes in fractured domains by combining it with XFEM. MHFEM and XFEM are applied to flow and mechanical equations, respectively. This coupling allows for an efficient extending of the hybrid dimensional approach to deal with coupled HM processes in fractured domains. The mass lumping technique is generalized to fractured domains and mechanical processes. We show how this technique can be implemented with the fixed-stress split scheme in order to improve the performance and stability of the numerical scheme. The new scheme (MHFEM-XFEM) is validated against analytical and numerical solutions. Comparison against the standard finite element method shows that the new scheme significantly reduces the computational overhead while providing a high accuracy.

A hybrid mixed finite element method for convection-diffusion-reaction equation with local exponential fitting technique

A new hybrid mixed finite element method is proposed for solving the convection-diffusion-reaction equation with local exponential fitting technique. In this method, the exponential fitting technique in [7] is used to discretize the convection-diffusion-reaction equation by introducing a new variable at element level; then the hybrid mixed discontinuous Galerkin finite element method is used to approximate the discretized problem. The convergence of the proposed method is analyzed, and the corresponding a priori error estimate is derived. The numerical results are presented to confirm our theoretical analysis.

A robust upwind mixed hybrid finite element method for transport in variably saturated porous media

Abstract. The mixed finite element (MFE) method is well adapted for the simulation of fluid flow in heterogeneous porous media. However, when employed for the transport equation, it can generate solutions with strong unphysical oscillations because of the hyperbolic nature of advection. In this work, a robust upwind MFE scheme is proposed to avoid such unphysical oscillations. The new scheme is a combination of the upwind edge/face centered finite volume method with the hybrid formulation of the MFE method. The scheme ensures continuity of both advective and dispersive fluxes between adjacent elements and allows to maintain the time derivative continuous, which permits employment of high-order time integration methods via the method of lines (MOL). Numerical simulations are performed in both saturated and unsaturated porous media to investigate the robustness of the new upwind MFE scheme. Results show that, contrarily to the standard scheme, the upwind MFE method generates stable solutions without under and overshoots. The simulation of contaminant transport into a variably saturated porous medium highlights the robustness of the proposed upwind scheme when combined with the MOL for solving nonlinear problems.

Configurable Open-source Data Structure for Distributed Conforming Unstructured Homogeneous Meshes with GPU Support

A general multi-purpose data structure for an efficient representation of conforming unstructured homogeneous meshes for scientific computations on CPU and GPU-based systems is presented. The data structure is provided as open-source software as part of the TNL library (https://tnl-project.org/). The abstract representation supports almost any cell shape and common 2D quadrilateral, 3D hexahedron and arbitrarily dimensional simplex shapes are currently built into the library. The implementation is highly configurable via templates of the C++ language, which allows avoiding the storage of unnecessary dynamic data. The internal memory layout is based on state-of-the-art sparse matrix storage formats, which are optimized for different hardware architectures in order to provide high-performance computations. The proposed data structure is also suitable for meshes decomposed into several subdomains and distributed computing using the Message Passing Interface (MPI). The efficiency of the implemented data structure on CPU and GPU hardware architectures is demonstrated on several benchmark problems and a comparison with another library. Its applicability to advanced numerical methods is demonstrated with an example problem of two-phase flow in porous media using a numerical scheme based on the mixed-hybrid finite element method (MHFEM). We show GPU speed-ups that rise above 20 in 2D and 50 in 3D when compared to sequential CPU computations, and above 2 in 2D and 9 in 3D when compared to 12-threaded CPU computations.

Fully implicit local time-stepping methods for advection-diffusion problems in mixed formulations

This paper is concerned with numerical solution of transport problems in heterogeneous porous media. A semi-discrete continuous-in-time formulation of the linear advection-diffusion equation is obtained by using a mixed hybrid finite element method, in which the flux variable represents both the advective and diffusive flux, and the Lagrange multiplier arising from the hybridization is used for the discretization of the advective term. Based on global-in-time and nonoverlapping domain decomposition, we propose two implicit local time-stepping methods to solve the semi-discrete problem. The first method uses the time-dependent Steklov-Poincaré type operator and the second uses the optimized Schwarz waveform relaxation (OSWR) with Robin transmission conditions. For each method, we formulate a space-time interface problem which is solved iteratively. Each iteration involves solving the subdomain problems independently and globally in time; thus, different time steps can be used in the subdomains. The convergence of the fully discrete OSWR algorithm with nonmatching time grids is proved. Numerical results for problems with various Peclét numbers and discontinuous coefficients, including a prototype for the simulation of the underground storage of nuclear waste, are presented to illustrate the performance of the proposed local time-stepping methods.

Pressure robust hybrid mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a pressure robust hybrid mixed finite element method for Darcy–Forchheimer model. The method uses piecewise polynomials of degrees for the velocity and k−1 for pressure in the interior of elements inside the domain, and piecewise polynomials of degree k for the numerical traces of the pressure on the inter-element boundaries. Our method can get the globally divergence-free approximation solution of velocity. Optimal error estimates are derived and the error estimates of velocity are pressure robust. Finally, numerical experiments confirm the theoretical results.

Conservative Finite Element Modeling of EEG and MEG on Unstructured Grids.

For interpretation of electroencephalography (EEG) and magnetoencephalography (MEG) data, multiple solutions of the respective forward problems are needed. In this paper, we assess performance of the mixed-hybrid finite element method (MHFEM) applied to EEG and MEG modeling. The method provides an approximate potential and induced currents and results in a system with a positive semi-definite matrix. The system thus can be solved with a variety of standard methods (e.g. the preconditioned conjugate gradient method). The induced currents satisfy discrete charge conservation law making the method conservative. We studied its performance on unstructured tetrahedral grids for a layered spherical head model as well as a realistic head model. We also compared its accuracy versus the conventional nodal finite element method ( P1 FEM). To avoid modeling singular sources, we completed our computations with a subtraction approach; the derived expression for the MEG response different from earlier published and involves integration of finite quantities only. We conclude that although the MHFEM is more computationally demanding than the P1 FEM, its use is justified for EEG and MEG modeling on low-resolution head models where P1 FEM loses accuracy.

A hybrid-mixed finite element method for single-phase Darcy flow in fractured porous media

We present a hybrid-mixed finite element method for a novel hybrid-dimensional model of single-phase Darcy flow in a fractured porous media. In this model, the fracture is treated as a (d−1)-dimensional interface within the d-dimensional fractured porous domain, for d=2,3. Two classes of fracture are distinguished based on the permeability magnitude ratio between the fracture and its surrounding medium: when the permeability in the fracture is (significantly) larger than in its surrounding medium, it is considered as a conductive fracture; when the permeability in the fracture is (significantly) smaller than in its surrounding medium, it is considered as a blocking fracture. The conductive fractures are treated using the classical hybrid-dimensional approach of the interface model where pressure is assumed to be continuous across the fracture interfaces, while the blocking fractures are treated using the recent Dirac-δ function approach where normal component of Darcy velocity is assumed to be continuous across the interface. Due to the use of Dirac-δ function approach for the blocking fractures, our numerical scheme allows for nonconforming meshes with respect to the blocking fractures. This is the major novelty of our model and numerical discretization. Moreover, our numerical scheme produces locally conservative velocity approximations and leads to a symmetric positive definite linear system involving pressure degrees of freedom on the mesh skeleton only. As an application, we extend the idea to a simple transport model. The performance of the proposed method is demonstrated by various benchmark test cases in both two- and three-dimensions. Numerical results indicate that the proposed scheme is highly competitive with existing methods in the literature

Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems

<p style='text-indent:20px;'>This paper is concerned with the optimized Schwarz waveform relaxation method and Ventcel transmission conditions for the linear advection-diffusion equation. A mixed formulation is considered in which the flux variable represents both diffusive and advective flux, and Lagrange multipliers are introduced on the interfaces between nonoverlapping subdomains to handle tangential derivatives in the Ventcel conditions. A space-time interface problem is formulated and is solved iteratively. Each iteration involves the solution of time-dependent problems with Ventcel boundary conditions in the subdomains. The subdomain problems are discretized in space by a mixed hybrid finite element method based on the lowest-order Raviart-Thomas space and in time by the backward Euler method. The proposed algorithm is fully implicit and enables different time steps in the subdomains. Numerical results with discontinuous coefficients and various Peclét numbers validate the accuracy of the method with nonconforming time grids and confirm the improved convergence properties of Ventcel conditions over Robin conditions.</p>

Multi-phase compositional modeling in porous media using iterative IMPEC scheme and constant volume–temperature flash

In this paper, we present a numerical solution of a multi-phase compressible Darcy’s flow of a multi-component mixture in a porous medium. The mathematical model consists of mass conservation equation for each component, extended Darcy’s law for each phase, and an appropriate set of initial and boundary conditions. The phase split is computed using the phase equilibrium computation in the VTN-specification (known as VTN-flash). The transport equations are solved numerically using the mixed-hybrid finite element method and a novel iterative IMPEC scheme [1]. We provide examples showing the performance of the numerical scheme. The convergence of the numerical scheme is verified using the experimental orders of convergence (EOC). This is an extended version of conference paper [2].

Modeling variable-density flow in saturated-unsaturated porous media: An advanced numerical model

Modeling variable-density flow in unconfined aquifers is a challenging task because of the nonlinear coupling between variably saturated flow and contaminant transport. This results in a highly nonlinear system since the strongly nonlinear Richards flow equation is, in addition, coupled to the advection-dispersion transport equation by viscosity and density variation. The solution of such a nonlinear system is often subject to convergence issues and can be very expansive in terms of computational time, especially for large-scale problems. Conventional numerical algorithms based on the sequential approach and the classical finite difference or finite element methods with the first-order backward Euler time integration scheme are generally inefficient and/or do not provide satisfactory results. In this work, we develop a new efficient and accurate 2D numerical model for the transport of dense contaminants in unsaturated porous media that allows for the simulation of large-scale problems. This research describes a new model that combines advanced spatial discretization methods (mixed hybrid finite element method, discontinuous Galerkin finite element method, and multipoint flux approximation method) with higher-order time integration techniques via the method of lines (MOL). The latter allows one to adapt the time step's size and the order of the time integration to improve the computational efficiency while maintaining accuracy. The robustness and accuracy of the new model are shown by comparison against a widely used commercial code based on the standard finite element method. The applicability of the developed model to a large-scale problem is then investigated by simulating saltwater intrusion under a climate change projection and long-term pumping regimes for the Akkar coastal aquifer in Lebanon using a simplified 2D conceptual model.

Phase‐Field Simulation of Hydraulic Fracturing by CO2, Water and Nitrogen in 2D and Comparison With Laboratory Data

AbstractHydraulic fracturing of tight subsurface formations has mainly been conducted using water. Waterless hydraulic fracturing by CO2 may have advantages over fracturing by water. The challenge has been numerical simulation of the process. In this work, we conduct a comprehensive study on 2D hydraulic fracturing simulation by water, CO2, and nitrogen. The simulations are based on the phase‐field; the results are compared with the lab data. We first present advances in simulation of fluid exchange between the fractures and the rock matrix. The conventional nodal‐based finite element method may give rise to unphysical negative pressure around the tip and fracture notch. We demonstrate that the mixed hybrid finite element method gives non‐negative pressure distribution. Next, we examine the effect of inertial term on fracture configuration and observe branching by CO2 under dynamic formulation with a low critical energy release rate. Lastly, we find from our simulations that in shale rocks, the critical energy release rate is the lowest for CO2, followed by nitrogen, and then water. Consequently, CO2 provides the highest fracture surface area and fracture intensity. In a sandstone sample we find nitrogen has a lower energy release rate than water. As a result, the fracture surface area is higher for nitrogen than water. Our simulation results confirm that the phase‐field method may incorporate fluid‐rock surface free energy without the need for various parameter adjustments.

BDDC for MHFEM discretization of unsteady two-phase flow in porous media

This work deals with the application of the Balancing Domain Decomposition by Constrains (BDDC) method to unsteady two-phase flow problems in porous media. We briefly describe the spatial discretization of the problem which is based on the mixed-hybrid finite element method (MHFEM) and semi-implicit time discretization. Then, we describe the BDDC method, in detail discuss the differences between the symmetric and nonsymmetric cases, and present the necessary modifications of the algorithm for the more complicated nonsymmetric case. We describe the parallel implementation of the method and highlight the critical steps of the algorithm that affect the performance and scalability. The parallel implementation is then tested on benchmark problems in 2D and 3D and its efficiency is investigated on various meshes. The numerical results indicate that the method preserves good computational efficiency for increasing number of processes and, therefore, allows solving problems on very fine meshes. In the case of unsteady problem, additional speed-up is achieved using the information from previous time steps for the solution in the current time step.

A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based Darcy flow simulations

In this work, we propose a novel block preconditioner, labeled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed hybrid finite element method for Darcy's equation with the Finite volume scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur complement and their inexact fully-parallel computation by means of restriction operators. We introduce two adaptive techniques aimed at building the restriction operators according to the properties of the system at hand. The proposed block preconditioner has been tested through extensive experimentation on both synthetic and real-case applications, pointing out its robustness and computational efficiency.

A multiscale mixed finite element method applied to the simulation of two-phase flows

The multiscale hybrid mixed finite element method (MHM-H(div)), previously developed for Darcy’s problems, is extended for coupled flow/pressure and transport system of two-phase flow equations on heterogeneous media under the effect of gravitational segregation. It is combined with an implicit transport solver in a sequential fully implicit (SFI) manner. The MHM-H(div) method is designed to cope with the complex geometry and inherent multiscale nature of the phenomena. The discretizations are based on a general domain partition formed by polyhedral subregions, where a hierarchy of meshes and approximation spaces are considered. The multiscale approach is applied to the flux/pressure kernel making use of coarse scale normal fluxes between subregions (trace variable). The fine-scale features inside each subregion are determined by resolving completely independent local Neumann problems, the boundary conditions being set by the trace variable, by the mixed finite element method using fine flux and pressure representations. These properties imply that the MHM-H(div) can be interpreted as a classical mixed formulation of the model problem in the whole domain, based on a H(div)-conforming space with normal components over the macro-partition interfaces constrained by the trace space, and showing divergence compatibility with the pressure space. Consequently, local mass conservation is observed at the micro-scale elements inside the subregions, an essential property for flows in heterogeneous media, and divergence-free constraint strongly enforced for incompressible flows. The efficient use of static condensation leads to a global system to be solved only in terms of primary degrees of freedom associated with the trace variable and of a piecewise constant pressure for each subregion. This procedure allows a substantial reduction of the dominant computational costs associated with the flux/pressure kernel embedded in the numerical model. An iterative coupling technique is adopted to solve the two-phase flow equations using a shared integration point memory implementation model. At each SFI time step, the efficiency iterative method for the transport equations is improved using a Quasi-Newton method with a simple but effective nonlinear acceleration. The numerical examples show that the proposed scheme is able to solve challenging coupled flow and transport problems.

COMPUTATIONAL METHODOLOGY TO ANALYZE THE EFFECT OF MASS TRANSFER RATE ON ATTENUATION OF LEAKED CARBON DIOXIDE IN SHALLOW AQUIFERS

Exsolution and re-dissolution of CO2 gas within heterogeneous porous media are investigated using experimental data and mathematical modeling. In a set of bench-scale experiments, water saturated with CO2 under a given pressure is injected into a 2-D water-saturated porous media system, causing CO2 gas to exsolve and migrate upwards. A layer of fine sand mimicking a heterogeneity within a shallow aquifer is present in the tank to study accumulation and trapping of exsolved CO2. Then, clean water is injected into the system and the accumulated CO2 dissolves back into the flowing water. Simulated exsolution and dissolution mass transfer processes are studied using both nearequilibrium and kinetic approaches and compared to experimental data under conditions that do and do not include lateral background water flow. The mathematical model is based on the mixed hybrid finite element method that allows for accurate simulation of both advection- and diffusion- dominated processes.

Hybrid finite-element-based numerical solution of the multi-group SP3 equations and its application on hexagonal reactor problems

The C-PORCA reactor physics code of the Paks Nuclear Power Plant performs three-dimensional, two-group diffusion calculations applying parametrized group constants generated by the HELIOS code. To improve the accuracy of the calculations, the C-PORCA code was extended with a simplified spherical harmonics module. This paper presents the applied finite-element-based solution algorithm of the SP3 equations in response matrix formalism. Radially primal mixed-hybrid finite element method is used for response matrix calculation with the inclusion of Lagrange multipliers to enforce nodal balance on each element. The axial solution is analytically performed connecting the axial and radial directions through radial leakage with transverse integration. Coupling between the adjacent volumetric elements is performed with node-to-node axial and radial partial current-like moment iterations, that enables the application of SP3 discontinuity factors and the parallelization of the calculation process. The accuracy of the SP3 solution algorithm is demonstrated on academic benchmark and VVER-440 core calculations.