An Efficient Lumped Mixed Hybrid Finite Element Formulation for Variably Saturated Groundwater Flow

Abstract

352P of accurate fl uid movement in porous media is an important issue for scientists and engineers who are interested in the management of water resources. Computational simulations have received much attention to achieve this predictive role. Even if its validity is still discussed, the Richards equation (RE) is a valuable model to predict water movement and solute transport in variably saturated media (Simunek and Bradford, 2008).From a mathematical point of view, the RE can be a highly nonlinear parabolic equation under unsaturated conditions, or a partial diff erential equation (PDE) of elliptic type for a fl uid-saturated incompressible porous media. Among the various numerical schemes that can be used to solve the RE, the mixed fi nite element method is well suited for the discretization of elliptic and parabolic PDEs on heterogeneous domains. Moreover, it is locally conservative, can handle general irregular grids, and allows simultaneous approximation of both pressure and velocity.Consequently, this method has been used extensively during the last few years (Mosé et al., 1994;Bergamaschi et al., 1998;Younes et al., 1999Younes et al., , 2006Ackerer et al., 1999; Chavent et al., 2003, among others). For practical applications, the lowest order mixed method of Raviart-Th omas (RT0) is frequently applied and was considered in this study. Th e RT0 uses a piecewise constant approximation for pressure (Brezzi and Fortin, 1991). Th e velocity space has three degrees of freedom for triangular elements and four for quadrangular elements. In their original form, the mixed methods require the resolution of algebraic equations that typically lead to indefi nite systems (Chavent and Jaff ré, 1986;Brezzi and Fortin, 1991).Th e most widely used approach to circumvent this mathematical diffi culty is the hybridization technique (Roberts and Th omas, 1990). It consists in introducing pressure Lagrange multipliers at the element edges. Th e MHFE method leads to a symmetric and positive defi nite matrix, which generally does not satisfy the M-matrix property (Raviart and Th omas, 1977;Th omas, 1977;Wheeler and Peszynska, 2002). Th is property (which requires a nonsingular matrix with m ii > 0 and m ij ≤ 0) has nonetheless a nice physical impact, as the scheme in this case satisfi es the discrete maximum principle, i.e., local maxima or minima will not appear in the numerical solution for a domain without local sources or sinks. Th erefore, the resulting numerical state variable and its related fl uxes are consistent with the physics.For elliptic problems, the matrix obtained with the MHFE method is an M-matrix in the case of a weakly acute triangulation (all angles are <π/2) (Brezzi and Fortin, 1991). Th is condition on the angles is not suffi cient for parabolic problems. A fi rst A : MHFE, mixed hybrid fi nite element; RE, Richards equation; RT0, lowest order Raviart-Th omas method.O R Accurate numerical simula on of infi ltra on in the vadose zone remains a challenge, especially when very sharp fronts are modeled. In th...

Sensitivity and identifiability of hydraulic and geophysical parameters from streaming potential signals in unsaturated porous media

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Gradient schemes for two‐phase flow in heterogeneous porous media and Richards equation

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Identification of groundwater flow parameters using reciprocal data from hydraulic interference tests

“…(1) is numerically solved by way of a Mixed Hybrid Finite Element method (MHFE), discretizing the domain in triangular elements, the method being well-known and very accurate (Younes et al, 2010). The numerical code uses an implicit discretization in time and mass lumping to avoid spurious oscillations (Belfort et al, 2009). Sensitivities of drawdowns to model parameters are derived by directly differentiating the discrete equations of the flow model with respect to parameters (see, e.g., Delay et al, 2007;Ackerer and Delay, 2010;Trottier et al, 2014).…”