Massimiliano Ferronato

Importance of poroelastic coupling in dynamically active aquifers of the Po river basin, Italy.

Uncoupling between the flow field and the stress field in pumped aquifers is the basis of the classical groundwater hydrology. Recently, some authors have disputed the assumption of uncoupling with regard to both fluid dynamics and porous medium deformation. The issue is very important as it could undermine the traditional approach to simulate subsurface flow, analyze pumping tests, and predict land subsidence caused by fluid withdrawal. The present paper addresses the problem of coupling versus uncoupling in the Po river plain, a normally consolidated and normally pressurized basin which has experienced in the last 50 years a pronounced pore pressure drawdown because of water and gas removal and where a large hydromechanical database is available from the ground surface down to 4000 m depth. A numerical study is performed which shows that the matrix which relates flow to stress is very similar to the capacity matrix of the uncoupled flow equation. A comparison of results obtained with the finite element integration of the coupled and uncoupled models indicates that pore pressure is rather insensitive to coupling anywhere within the pumped formation while in the adjacent aquitard-aquifer units, coupling induces a slight overpressure which quickly dissipates in time with a small initial influence on medium deformation, and specifically on land subsidence. As a major consequence the uncoupled solutions to the fluid dynamic and the structural problems appear to be fully warranted on any timescale of practical interest in a typical normally consolidated and pressurized basin.

A fully coupled 3-D mixed finite element model of Biot consolidation

The numerical solution to the Biot equations of 3-D consolidation is still a challenging task because of the ill-conditioning of the resulting algebraic system and the instabilities that may affect the pore pressure solution. Recently new approaches have been advanced based on mixed formulations. In the present paper a fully coupled 3-D mixed finite element model is developed with the aim at alleviating the pore pressure numerical oscillations at the interface between materials with different permeabilities. A solution algorithm is implemented that takes advantage of the block structure of the discretized problem. The proposed model is verified against well-known analytical solutions and successfully experimented with in realistic applications of soil consolidation.

Ill-conditioning of finite element poroelasticity equations

Abstract The solution to Biots coupled consolidation theory is usually addressed by the finite element (FE) method thus obtaining a system of first-order differential equations which is integrated by the use of an appropriate time marching scheme. For small values of the time step the resulting linear system may be severely ill-conditioned and hence the solution can prove quite difficult to achieve. Under such conditions efficient and robust projection solvers based on Krylovs subspaces which are usually recommended for non-symmetric large size problems can exhibit a very slow convergence rate or even fail. The present paper investigates the correlation between the ill-conditioning of FE poroelasticity equations and the time integration step Δt. An empirical relation is provided for a lower bound Δtcrit of Δt below which ill-conditioning may suddenly occur. The critical time step is larger for soft and low permeable porous media discretized on coarser grids. A limiting value for the rock stiffness is found such that for stiffer systems there is no ill-conditioning irrespective of Δt however small, as is also shown by several numerical examples. Finally, the definition of a different Δtcrit as suggested by other authors is reviewed and discussed.

A Block FSAI-ILU Parallel Preconditioner for Symmetric Positive Definite Linear Systems

A novel parallel preconditioner for symmetric positive definite matrices is developed coupling a generalized factored sparse approximate inverse (FSAI) with an incomplete LU (ILU) factorization. The generalized FSAI, called block FSAI, is derived by requiring the preconditioned matrix to resemble a block-diagonal matrix in the sense of the minimal Frobenius norm. An incomplete block Jacobi algorithm is then effectively used to accelerate the convergence of a Krylov subspace method. The block FSAI-ILU preconditioner proves superior to both FSAI and the incomplete block Jacobi by themselves in a number of realistic finite element test cases and is fully scalable for a given number of blocks.

Adaptive Pattern Research for Block FSAI Preconditioning

An adaptive algorithm is presented to generate automatically the nonzero pattern of the block factored sparse approximate inverse (BFSAI) preconditioner. It is demonstrated that in symmetric positive definite (SPD) problems BFSAI minimizes an upper bound to the Kaporin number of the preconditioned matrix. The mathematical structure of this bound suggests an efficient and easily parallelizable strategy for improving the given nonzero pattern of BFSAI, thus providing a novel adaptive BFSAI (ABF) preconditioner. Numerical experiments performed on large sized finite element problems show that ABF coupled with a block incomplete Cholesky (IC) outperforms BFSAI-IC even by a factor of 4, preserving the same preconditioner density and exhibiting an excellent parallelization degree.

Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations

The Finite Element (FE) integration of the coupled consolidation equations requires the solution of linear symmetric systems with an indefinite saddle point coefficient matrix. Because of ill-conditioning, the repeated solution in time of the FE equations may be a major computational issue requiring ad hoc preconditioning strategies to guarantee the efficient convergence of Krylov subspace methods. In the present paper a Mixed Constraint Preconditioner (MCP) is developed combining implicit and explicit approximations of the inverse of the structural sub-matrix, with the performance investigated in some representative examples. An upper bound of the eigenvalue distance from unity is theoretically provided in order to give practical indications on how to improve the preconditioner. The MCP is efficiently implemented into a Krylov subspace method with the performance obtained in 2D and 3D examples compared to that of Inexact Constraint Preconditioners and Least Square Logarithm scaled ILUT preconditioners. Two variants of MCP (T-MCP and D-MCP), developed with the aim at reducing the cost of the preconditioner application, are also tested. The results show that the MCP variants constitute a reliable and robust approach for the efficient solution of realistic coupled consolidation FE models, and especially so in severely ill-conditioned problems.

A comparison of numerical integration rules for the meshless local Petrov–Galerkin method

The meshless local Petrov–Galerkin (MLPG) method is a mesh-free procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.

Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper.

Can CO2 help save Venice from the Sea

On 14 May this year, Italian Prime Minister Silvio Berlusconi cut the ribbon on a multi-billion-dollar project named MOSE that is aimed at solving the problem of “acqua alta,” the increasingly frequent floods that jeopardize the survival of Venice. Cost is estimated (a few say conservatively) at 3 billion euros and construction time (a few say optimistically) at 8 years. MOSE involves building mobile barriers at the Venice Lagoon inlets to prevent severe Adriatic Sea storms from flooding the city. Although the Italian government and the local administrations have given their final approval, MOSE still has several opponents who believe it will cause severe threats to the lagoon ecosystem, and will soon become obsolete because of the expected sea level rise due to global warming.

Scalable algorithms for three-field mixed finite element coupled poromechanics

Abstract We introduce a class of block preconditioners for accelerating the iterative solution of coupled poromechanics equations based on a three-field formulation. The use of a displacement/velocity/pressure mixed finite-element method combined with a first order backward difference formula for the approximation of time derivatives produces a sequence of linear systems with a 3 × 3 unsymmetric and indefinite block matrix. The preconditioners are obtained by approximating the two-level Schur complement with the aid of physically-based arguments that can be also generalized in a purely algebraic approach. A theoretical and experimental analysis is presented that provides evidence of the robustness, efficiency and scalability of the proposed algorithm. The performance is also assessed for a real-world challenging consolidation experiment of a shallow formation.