The MAST-edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media

A novel methodology is proposed for the solution of the flow equation in a variably saturated heterogeneous porous medium. The computational domain is descretized using triangular meshes and the governing PDEs are discretized using a lumped in the edge centres numerical technique. The dependent unknown variable of the problem is the piezometric head. A fractional time step methodology is applied for the solution of the original system, solving consecutively a prediction and a correction problem. A scalar potential of the flow field exists and in the prediction step a MArching in Space and Time (MAST) formulation is applied for the sequential solution of the Ordinary Differential Equation of the cells, ordered according to their potential value computed at the beginning of the time step. In the correction step, the solution of a large linear system with order equal to the number of edges is required. A semi-analytical procedure is also proposed for the solution of the prediction step. The computational performance, the order of convergence and the mass balance error have been estimated in several tests and compared with the results of other literature models.

[1]  B. Philippe,et al.  Numerical Reliability for Mixed Methods Applied to Flow Problems in Porous Media , 2002 .

[2]  Cass T. Miller,et al.  Robust solution of Richards' equation for nonuniform porous media , 1998 .

[3]  E. Labolle,et al.  Commentary on Russo [1991], Serrano [1990, 1998], and other applications of the water‐content‐based form of Richards' Equation to heterogeneous soils , 1999 .

[4]  Tullio Tucciarelli,et al.  The MAST FV/FE scheme for the simulation of two-dimensional thermohaline processes in variable-density saturated porous media , 2009, J. Comput. Phys..

[5]  Emil O. Frind,et al.  Three‐dimensional modeling of groundwater flow systems , 1978 .

[6]  Yih-Chi Tan,et al.  A novel hysteresis model in unsaturated soil , 2005 .

[7]  Peter A. Forsyth,et al.  Robust numerical methods for saturated-unsaturated flow with dry initial conditions in heterogeneous media , 1996 .

[8]  Katsushi Ohmori,et al.  A technique of upstream type applied to a linear nonconforming finite element approximation of convective diffusion equations , 1984 .

[9]  J. Erhel,et al.  The maximum principle violations of the mixed‐hybrid finite‐element method applied to diffusion equations , 2002 .

[10]  Pierre Perrochet,et al.  On the primary variable switching technique for simulating unsaturated–saturated flows , 1999 .

[11]  Mario Putti,et al.  Finite Element Approximation of the Diffusion Operator on Tetrahedra , 1998, SIAM J. Sci. Comput..

[12]  R. H. Brooks,et al.  Hydraulic properties of porous media , 1963 .

[13]  Thomas J. R. Hughes,et al.  The Continuous Galerkin Method Is Locally Conservative , 2000 .

[14]  So-Hsiang Chou,et al.  Conservative P1 Conforming and Nonconforming Galerkin FEMs: Effective Flux Evaluation via a Nonmixed Method Approach , 2000, SIAM J. Numer. Anal..

[15]  L. A. Richards.  Capillary conduction of liquids through porous mediums , 1931 .

[16]  Tullio Tucciarelli,et al.  MAST solution of advection problems in irrotational flow fields , 2007 .

[17]  Jack C. Parker,et al.  Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties , 1987 .

[18]  Mario Putti,et al.  A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems , 1994 .

[19]  Costanza Aricò,et al.  A MARCHING IN SPACE AND TIME (MAST) SOLVER OF THE SHALLOW WATER EQUATIONS , 2006 .

[20]  P. Ackerer,et al.  A new mass lumping scheme for the mixed hybrid finite element method , 2006 .

[21]  Paola Pietra,et al.  Two-dimensional exponential fitting and applications to drift-diffusion models , 1989 .

[22]  Mary Catherine A. Kropinski,et al.  Monotonicity Considerations for Saturated-Unsaturated Subsurface Flow , 1997, SIAM J. Sci. Comput..

[23]  M. Celia,et al.  A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation , 1990 .

[24]  Randel Haverkamp,et al.  A Comparison of Numerical Simulation Models For One-Dimensional Infiltration1 , 1977 .

[25]  Van Genuchten,et al.  A closed-form equation for predicting the hydraulic conductivity of unsaturated soils , 1980 .

[26]  Gianmarco Manzini,et al.  Mass-conservative finite volume methods on 2-D unstructured grids for the Richards’ equation , 2004 .

[27]  Luca Bergamaschi,et al.  MIXED FINITE ELEMENTS AND NEWTON-TYPE LINEARIZATIONS FOR THE SOLUTION OF RICHARDS' EQUATION , 1999 .

[28]  Armin Iske,et al.  ADER schemes on adaptive triangular meshes for scalar conservation laws , 2005 .

[29]  C. Aricò,et al.  MAST-2D diffusive model for flood prediction on domains with triangular Delaunay unstructured meshes , 2011 .

[30]  R. G. Hills,et al.  Algorithms for solving Richards' equation for variably saturated soils , 1992 .

[31]  Bruno Brunone,et al.  Numerical analysis of one-dimensional unsaturated flow in layered soils , 1998 .

[32]  Cass T. Miller,et al.  Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines , 1997 .

[33]  J. Bear.  Hydraulics of Groundwater , 1979 .

[34]  P. Huyakorn,et al.  Techniques for Making Finite Elements Competitve in Modeling Flow in Variably Saturated Porous Media , 1984 .

[35]  Tullio Tucciarelli,et al.  An explicit unconditionally stable numerical solution of the advection problem in irrotational flow fields , 2004 .

[36]  Curtis M. Oldenburg,et al.  On numerical modeling of capillary barriers , 1993 .

[37]  F. Sartoretto,et al.  Linear Galerkin vs mixed finite element 2D flow fields , 2009 .