Florin A. Radu

A study on iterative methods for solving Richards’ equation

This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards’ equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards’ equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L-scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L-scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L-scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

A priori error estimates for a mixed finite element discretization of the Richards' equation

Summary.A mixed finite element discretization is applied to Richards’ equation, a nonlinear, possibly degenerate parabolic partial differential equation modeling water flow through porous medium. The equation is considered in its pressure formulation and includes both variably and fully saturated flow regime. Characteristic for such problems is the lack in regularity of the solution. To handle this we use a time-integrated scheme. We analyze the scheme and present error estimates showing its convergence.

A Mixed Hybrid Finite Element Discretization Scheme for Reactive Transport in Porous Media

We present a model to describe the simultaneous reactive transport in porous media of an arbitrary number of mobile and immobile species. The model includes the effects of advection, dispersion, sorption and degradation catalysed by microbial populations. The locally mass-conservative mixed hybrid finite element method (MHFEM) to discretize this system of coupled convection-diffusion-reaction equations and the algorithmic solution of the resulting nonlinear algebraic equations are described in detail. Further, new ideas regarding the discretization of the convective term are discussed. Finally a comparative numerical study (MHFEM versus conforming FEM) is presented.

Upscaling of Non-isothermal Reactive Porous Media Flow with Changing Porosity

Motivated by rock–fluid interactions occurring in a geothermal reservoir, we present a two-dimensional pore scale model of a periodic porous medium consisting of void space and grains, with fluid flow through the void space. The ions in the fluid are allowed to precipitate onto the grains, while minerals in the grains are allowed to dissolve into the fluid, and we take into account the possible change in pore geometry that these two processes cause, resulting in a problem with a free boundary at the pore scale. We include temperature dependence and possible effects of the temperature both in fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we perform a formal homogenization procedure to obtain upscaled equations. A pore scale model consisting of circular grains is presented as a special case of the porous medium.

A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity

Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Peclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM-GRW is tens of times faster than the full MFEM flow and transport simulation.

Space–time finite element approximation of the Biot poroelasticity system with iterative coupling

Abstract We analyze an optimized artificial fixed-stress iterative scheme for a space–time finite element approximation of the Biot system modeling fluid flow in deformable porous media. The iteration is based on a prescribed constant artificial volumetric mean total stress in the first half step. The optimization comes through the adaptation of a numerical stabilization or tuning parameter and aims at an acceleration of the iterations. The separated subproblems of fluid flow, written as a mixed first order in space system, and mechanical deformation are discretized by space–time finite element methods of arbitrary order. Continuous and discontinuous Galerkin discretizations of the time variable are encountered. The convergence of the iterative schemes is proved for the continuous and fully discrete case. The choice of the optimization parameter is identified in the proofs of convergence of the iterations. The analyses are illustrated and confirmed by numerical experiments.

Analysis of an Upwind-Mixed Hybrid Finite Element Method for Transport Problems

We prove optimal order convergence of an upwind-mixed hybrid finite element scheme for linear parabolic advection-diffusion-reaction problems. It was introduced in [Radu et al., Adv. Water Resources, 34 (2011), pp. 47--61] and is based on an Euler-implicit mixed hybrid finite element discretization of the problem in fully mass conservative form using the Raviart--Thomas mixed finite element of lowest order on triangular meshes. Optimal order convergence in time and space is obtained for the fully discrete formulation. The scheme provides the same order of convergence as the standard upwind-mixed method, while it is more efficient since a local elimination of variables is possible with our choice of the upwind weights. The theoretical findings are sustained by a numerical experiment.

Newton—Type Methods for the Mixed Finite Element Discretization of Some Degenerate Parabolic Equations

In this paper we discuss some iterative approaches for solving the nonlinear algebraic systems encountered as fully discrete counterparts of some degenerate (fast diffusion) parabolic problems. After regularization, we combine a mixed finite element discretization with the Euler implicit scheme. For the resulting systems we discuss three iterative methods and give sufficient conditions for convergence.

Robust fixed stress splitting for Biot’s equations in heterogeneous media

Abstract We study the iterative solution of coupled flow and geomechanics in heterogeneous porous media, modeled by a three-field formulation of the linearized Biot’s equations. We propose and analyze a variant of the widely used Fixed Stress Splitting method applied to heterogeneous media. As spatial discretization, we employ linear Galerkin finite elements for mechanics and mixed finite elements (lowest order Raviart–Thomas elements) for flow. Additionally, we use implicit Euler time discretization. The proposed scheme is shown to be globally convergent with optimal theoretical convergence rates. The convergence is rigorously shown in energy norms employing a new technique. Furthermore, numerical results demonstrate robust iteration counts with respect to the full range of Lame parameters for homogeneous and heterogeneous media. Being in accordance with the theoretical results, the iteration count is hardly influenced by the degree of heterogeneities.

A model for non-isothermal flow and mineral precipitation and dissolution in a thin strip

Motivated by porosity changes due to chemical reactions caused by injection of cold water in a geothermal reservoir, we propose a two-dimensional pore scale model of a thin strip. The pore scale model includes fluid flow, heat transport and reactive transport where changes in aperture is taken into account. The thin strip consists of void space and grains, where ions are transported in the fluid in the void space. At the interface between void and grain, ions are allowed to precipitate and become part of the grain, or conversely, minerals in the grain can dissolve and become part of the fluid flow, and we honor the possible change in aperture these two processes cause. We include temperature dependence and possible effects of the temperature in both fluid properties and in the mineral precipitation and dissolution reactions. For the pore scale model equations, we investigate the limit as the width of the thin strip approaches zero, deriving upscaled one-dimensional effective equations.