Stable cell-centered finite volume discretization for Biot equations

In this paper we discuss a new discretization for the Biot equations. The discretization treats the coupled system of deformation and flow directly, as opposed to combining discretizations for the two separate sub-problems. The coupled discretization has the following key properties, the combination of which is novel: 1) The variables for the pressure and displacement are co-located, and are as sparse as possible (e.g. one displacement vector and one scalar pressure per cell center). 2) With locally computable restrictions on grid types, the discretization is stable with respect to the limits of incompressible fluid and small time-steps. 3) No artificial stabilization term has been introduced. Furthermore, due to the finite volume structure embedded in the discretization, explicit local expressions for both momentum-balancing forces as well as mass-conservative fluid fluxes are available. We prove stability of the proposed method with respect to all relevant limits. Together with consistency, this proves convergence of the method. Finally, we give numerical examples verifying both the analysis and convergence of the method

A FIC-based stabilized mixed finite element method with equal order interpolation for solid–pore fluid interaction problems

This is the peer reviewed version of the following article: [de-Pouplana, I., and Oñate, E. (2017) A FIC-based stabilized mixed finite element method with equal order interpolation for solid–pore fluid interaction problems. Int. J. Numer. Anal. Meth. Geomech., 41: 110–134. doi: 10.1002/nag.2550], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nag.2550/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving."A new mixed displacement-pressure element for solving solid–pore fluid interaction problems is presented. In the resulting coupled system of equations, the balance of momentum equation remains unaltered, while the mass balance equation for the pore fluid is stabilized with the inclusion of higher-order terms multiplied by arbitrary dimensions in space, following the finite calculus (FIC) procedure. The stabilized FIC-FEM formulation can be applied to any kind of interpolation for the displacements and the pressure, but in this work, we have used linear elements of equal order interpolation for both set of unknowns. Examples in 2D and 3D are presented to illustrate the accuracy of the stabilized formulation for solid–pore fluid interaction problems.Peer ReviewedPostprint (author's final draft

An algebraic multigrid method for Q2−Q1Q_2-Q_1 mixed discretizations of the Navier-Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa

An exploration of the IGA method for efficient reservoir simulation

Novel numerical methods present exciting opportunities to improve the efficiency of reservoir simulators. Because potentially significant gains to computational speed and accuracy may be obtained, it is worthwhile explore alternative computational algorithms for both general and case-by-case application to the discretization of the equations of porous media flow, fluid-structure interaction, and/or production. In the present work, the fairly new concept of isogeometric analysis (IGA) is evaluated for its suitability to reservoir simulation via direct comparison with the industry standard finite difference (FD) method and 1st order standard finite element method (SFEM). To this end, two main studies are carried out to observe IGA’s performance with regards to geometrical modeling and ability to capture steep saturation fronts. The first study explores IGA’s ability to model complex reservoir geometries, observing L2 error convergence rates under a variety of refinement schemes. The numerical experimental setup includes an 'S' shaped line sink of varying curvature from which water is produced in a 2D homogenous domain. The accompanying study simplifies the domain to 1D, but adds in multiphase physics that traditionally introduce difficulties associated with modeling of a moving saturation front. Results overall demonstrate promise for the IGA method to be a particularly effective tool in handling geometrically difficult features while also managing typically challenging numerical phenomena.Petroleum and Geosystems Engineerin

On the design of block preconditioners for maritime engineering

The iterative error can be an important part of the total numerical error of any Com- putational Fluid Dynamics simulation when the iterative convergence stagnates or when loose convergence criteria are used. In the quest for better iterative convergence of CFD simulations, we consider the design of iterative methods for the Reynolds-averaged Navier-Stokes equations, discretized by finite-volume methods with cell-centered, co-located variables. The central point is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation

New numerical approaches for modeling thermochemical convection in a compositionally stratified fluid

Seismic imaging of the mantle has revealed large and small scale heterogeneities in the lower mantle; specifically structures known as large low shear velocity provinces (LLSVP) below Africa and the South Pacific. Most interpretations propose that the heterogeneities are compositional in nature, differing in composition from the overlying mantle, an interpretation that would be consistent with chemical geodynamic models. Numerical modeling of persistent compositional interfaces presents challenges, even to state-of-the-art numerical methodology. For example, some numerical algorithms for advecting the compositional interface cannot maintain a sharp compositional boundary as the fluid migrates and distorts with time dependent fingering due to the numerical diffusion that has been added in order to maintain the upper and lower bounds on the composition variable and the stability of the advection method. In this work we present two new algorithms for maintaining a sharper computational boundary than the advection methods that are currently openly available to the computational mantle convection community; namely, a Discontinuous Galerkin method with a Bound Preserving limiter and a Volume-of-Fluid interface tracking algorithm. We compare these two new methods with two approaches commonly used for modeling the advection of two distinct, thermally driven, compositional fields in mantle convection problems; namely, an approach based on a high-order accurate finite element method advection algorithm that employs an artificial viscosity technique to maintain the upper and lower bounds on the composition variable as well as the stability of the advection algorithm and the advection of particles that carry a scalar quantity representing the location of each compositional field. All four of these algorithms are implemented in the open source FEM code ASPECT