Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems

Gray, William G.; Miller, Cass T.

doi:10.1007/978-3-319-04010-3

Cited by 131 publications

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“…Problems where gas-phase dynamics are important usually demand a full two-phase formulation (Helmig, 2011;Smits et al, 2013), while more recent work on thermodynamically constrained averaging theory has focused on developing a consistent upscaling framework for deriving extensions to Richards' equation (Bronson, 2014;Gray and Miller, 2014). These approaches are primarily aimed at extending the operational range of validity for the mathematical system as a representation for unsaturated flow .…”

Section: Discussion and Alternativesmentioning

confidence: 99%

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

Farthing

Ogden

2017

Soil Science Soc of Amer J

223

131

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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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Section: Discussion and Alternativesmentioning

confidence: 99%

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

Farthing

Ogden

2017

Soil Science Soc of Amer J

223

131

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“…TCAT models are derived from first principles starting from the microscale. At the macroscale, important quantities such as phase pressures, specific interfacial areas, curvatures, and other averaged quantities are defined unambiguously based on the microscale state (e.g., Gray and Miller, 2014). For the twofluid-phase flow, we consider the wetting phase (w), the nonwetting phase (n), and the solid phase (s) within a domain .…”

Section: Theorymentioning

confidence: 99%

“…(8) applies even when the system is not at equilibrium. Additionally, if the mass per area of the interface is non-zero, Laplace's law must be modified to account for gravitational effects (Gray and Miller, 2014). Care must be taken when extending this relationship to the macroscale, as is shown below.…”

Section: Theorymentioning

confidence: 99%

“…Traditionally, these closure relations have been posited empirically and parameter estimation has been accomplished based upon relatively simple laboratory experiments. In general, traditional macroscale models, while the dominant class of model, suffer from several limitations related to the way in which such models are formulated and closed (Gray and Miller, 2014). A precise coupling between these disparate length scales has usually been ignored.…”

Section: Introductionmentioning

confidence: 99%

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On the consistency of scale among experiments, theory, and simulation

McClure

Dye

Miller

et al. 2017

Hydrol. Earth Syst. Sci.

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Abstract. As a tool for addressing problems of scale, we consider an evolving approach known as the thermodynamically constrained averaging theory (TCAT), which has broad applicability to hydrology. We consider the case of modeling of two-fluid-phase flow in porous media, and we focus on issues of scale as they relate to various measures of pressure, capillary pressure, and state equations needed to produce solvable models. We apply TCAT to perform physicsbased data assimilation to understand how the internal behavior influences the macroscale state of two-fluid porous medium systems. A microfluidic experimental method and a lattice Boltzmann simulation method are used to examine a key deficiency associated with standard approaches. In a hydrologic process such as evaporation, the water content will ultimately be reduced below the irreducible wetting-phase saturation determined from experiments. This is problematic since the derived closure relationships cannot predict the associated capillary pressures for these states. We demonstrate that the irreducible wetting-phase saturation is an artifact of the experimental design, caused by the fact that the boundary pressure difference does not approximate the true capillary pressure. Using averaging methods, we compute the true capillary pressure for fluid configurations at and below the irreducible wetting-phase saturation. Results of our analysis include a state function for the capillary pressure expressed as a function of fluid saturation and interfacial area.

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“…This is, in turn, an example of a hierarchical dynamic property, which may be applied at several different scales, including large scales, as shown in the thermodynamically constrained averaging [28,29].…”

Section: Introductionmentioning

confidence: 99%

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

et al. 2016

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Abstract:The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher's equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions.

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Introduction to the Thermodynamically Constrained Averaging Theory for Porous Medium Systems

Cited by 131 publications

References 0 publications

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

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

On the consistency of scale among experiments, theory, and simulation

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

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