Adaptive local discontinuous Galerkin approximation to Richards’ equation

Li, H.; Farthing, Matthew W.; Miller, Cass T.

doi:10.1016/j.advwatres.2007.02.007

Cited by 36 publications

(30 citation statements)

References 98 publications

(188 reference statements)

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“…Mass-conservative finite element methods have been used successfully for solving Richards' equation in heterogeneous porous media, including Galerkin methods [4,11], discontinuous Galerkin methods [20] and mixed finite element approaches [7,8]. Finite volume methods have also been used successfully due to their local mass conservation properties [10,16,22].…”

Section: Introductionmentioning

confidence: 99%

“…It is a parabolic equation derived by coupling a mass conservation equation with Darcy's Law [1]. The nonlinear formulation of saturation and permeability in terms of the wetting phase pressure describes sharp increases over small variations in pressure, which leads to the formation of sharp fronts under certain auxiliary conditions such as infiltration into very dry soil [20]. These nonlinearities present significant challenges to computing efficient and accurate solutions.…”

Section: Introductionmentioning

confidence: 99%

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A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

2011

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We present a mass-conservative vertex-centred finite volume method for efficiently solving the mixed form of Richards' equation in heterogeneous porous media. The spatial discretisation is particularly well-suited to heterogeneous media because it produces consistent flux approximations at quadrature points where material properties are continuous. Combined with the method of lines, the spatial discretisation gives a set of differential algebraic equations amenable to solution using higher-order implicit solvers. We investigate the solution of the mixed form using a Jacobian-free inexact Newton solver, which requires the solution of an extra variable for each node in the mesh compared to the pressure-head form. By exploiting the structure of the Jacobian for the mixed form, the size of the preconditioner is reduced to that for the pressure-head form, and there is minimal computational overhead for solving the mixed form.The proposed formulation is tested on two challenging test problems. The solutions from the new formulation offer conservation of mass at least one order of magnitude more accurate than a pressure head formulation, and the higher-order temporal integration significantly improves both the mass balance and computational efficiency of the solution.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

2011

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“…3C. Many researchers have shown that adaptive resolution can greatly improve accuracy and robustness (Bause and Knabner, 2004;Li et al, 2007b). Indeed even one-dimensional approximations have shown that they can pose a significant computational burden and benefit from adaptive resolution, depending on the combination of soil properties, initial conditions, and boundary forcing .…”

Section: Spatial Discretizationmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

257

138

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The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

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“…These applications include single-phase flow in heterogeneous porous media (Cao and Kitanidis, 1999), unsaturated flow (Li et al, 2007), multispecies reactive transport (Wolfsberg and Freyberg, 1994), coupled density-dependent flow and transport (Trompert, 1993;van Esch, 2002), and multiphase flow and multicomponent transport (Hornung and Trangenstein, 1997;Pau et al, 2008). All these applications are limited to capability demonstration.…”

Section: Actual Phase I Accomplishmentsmentioning

confidence: 99%

COUPLING STATE-OF-THE-SCIENCE SUBSURFACE SIMULATION WITH ADVANCED USER INTERFACE AND PARALLEL VISUALIZATION: SBIR Phase I Final Report

Hardeman¹,

Swenson²,

Finsterle³

et al. 2008

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Executive SummaryThis is a Phase I report on a project to significantly enhance existing subsurface simulation software using leadership-class computing resources, allowing researchers to solve problems with greater speed and accuracy. Subsurface computer simulation is used for monitoring the behavior of contaminants around nuclear waste disposal and storage areas, groundwater flow, environmental remediation, carbon sequestration, methane hydrate production, and geothermal energy reservoir analysis. The Phase I project was a collaborative effort between Thunderhead Engineering (project lead and developers of a commercial pre-and post-processor for the TOUGH2 simulator) and Lawrence Berkeley National Laboratory (developers of the TOUGH2 simulator for subsurface flow). The Phase I project successfully identified the technical approaches to be implemented in Phase II.

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Adaptive local discontinuous Galerkin approximation to Richards’ equation

Cited by 36 publications

References 98 publications

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

A mass-conservative control volume-finite element method for solving Richards’ equation in heterogeneous porous media

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

COUPLING STATE-OF-THE-SCIENCE SUBSURFACE SIMULATION WITH ADVANCED USER INTERFACE AND PARALLEL VISUALIZATION: SBIR Phase I Final Report

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