Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media

Sánchez-Vargas, J.; Valdés‐Parada, Francisco J.; Lasseux, Didier

doi:10.1016/j.jnnfm.2022.104840

Cited by 8 publications

(5 citation statements)

References 61 publications

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“…This new closure problem can be combined with the flow problem in a unit cell (namely (2.1 a ), (2.1 c ), (2.5), (2.1 e ) and (2.1 f )) by means of a Green's formula. Taking two arbitrary scalar fields, and , and two arbitrary vector fields, and , all of them defined in the -phase and having sufficient regularity, this Green's formula can be written as (see the derivation in Appendix A of Sánchez-Vargas, Valdés-Parada & Lasseux (2022)) …”

Section: Derivation Of the Average Pressure Differencementioning

confidence: 99%

Upscaled dynamic capillary pressure for two-phase flow in porous media

Lasseux

Valdés‐Parada

2023

J. Fluid Mech.

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A closed expression for the average pressure difference (often called the macroscopic dynamic capillary pressure in the literature) is proposed for two-phase, Newtonian, incompressible, isothermal and creeping flow in homogeneous porous media. This upscaled equation complements the average equations for mass and momentum transport derived in a previous article. Consistently with this work, the expression is derived employing a simplified version of the volume-averaging method that makes use of elements of the adjoint method and Green's formula. The resulting equation for the average pressure difference is novel, as it shows that this quantity is controlled by the pressure gradient (and body forces) in each phase, as well as interfacial effects, and is applicable to situations in which the fluid–fluid interface is not necessarily at its steady position. The effective-medium quantities associated with the sources are all obtained from the solution of a single adjoint (or closure) problem to be solved on a (periodic) unit cell representative of the process. The average pressure difference predicted by the derived expression is validated through excellent comparisons with direct numerical simulations performed in a model porous structure.

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Section: Derivation Of the Average Pressure Differencementioning

confidence: 99%

Upscaled dynamic capillary pressure for two-phase flow in porous media

Lasseux

Valdés‐Parada

2023

J. Fluid Mech.

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“…The next step in the derivations consists of relating the associated closure problems with the flow problem with the help of a Green's formula. Considering a constant scalar and arbitrary and sufficiently regular scalar fields and , vector fields , and and a second-order tensor field , together with the following definitions: this formula is written (see the proof in Appendix A in Sánchez-Vargas, Valdés-Parada & Lasseux (2022)) Here, denotes the surfaces bounding .…”

Section: Mass and Momentum Macroscopic Equationsmentioning

confidence: 99%

“…In order to derive the relationship between the new adjoint closure variables and the pore-scale velocities and pressures, the following Green's formula is employed (see the derivation in Sánchez-Vargas et al. 2022): with defined in (3.7 a ) and In these two equations, is a constant scalar, whereas , , and , , are respectively three arbitrary scalar and three arbitrary vector fields having sufficient regularity.…”

Section: Macroscopic Pressure Differencementioning

confidence: 99%

“…Airiau & Bottaro 2020; Sánchez-Vargas et al. 2022), it is assumed that . Thus, the remaining degrees of freedom in the Carreau model are (), and .…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…For this purpose, the following Green's formula is of interest: the proof of which can be found in Appendix A in Sánchez-Vargas et al. (2022). Here, , , , , , and have the same properties as for (3.8) and (4.2), whereas is a second-order tensor having appropriate regularities and …”

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confidence: 99%

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Macroscopic model for generalised Newtonian inertial two-phase flow in porous media

Sánchez-Vargas,

Valdés-Parada,

Trujillo-Roldán

et al. 2023

J. Fluid Mech.

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A closed macroscopic model for quasi-steady, inertial, incompressible, two-phase generalised Newtonian flow in rigid and homogeneous porous media is formally derived. The model consists of macroscopic equations for mass and momentum balance as well as an expression for the macroscopic pressure difference between the two fluid phases. The model is obtained by upscaling the pore-scale equations, employing a methodology based on volume averaging, the adjoint method and Green's formulation, only assuming the existence of a representative elementary volume and the separation of scales between the microscale and the macroscale. The average mass equations coincide with those for Newtonian flow. The macroscopic momentum balance equation in each phase expresses the seepage velocity in terms of a dominant and a coupling Darcy-like term, a contribution from interfacial tension effects and another one from interfacial inertia. Finally, the expression of the macroscopic pressure difference is obtained in terms of the macroscopic pressure gradient and body force in each phase, and interfacial terms that account for capillary effects and inertia, if present when the interface is not stationary. All terms involved in the macroscale equations are predicted from the solution of adjoint closure problems in periodic representative domains. Numerical predictions from the upscaled models are compared with direct numerical simulations for two-dimensional configurations, considering flow of a Newtonian non-wetting fluid and a Carreau wetting fluid. Excellent agreement between the two approaches confirms the pertinence of the macroscopic models derived here.

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Least squares finite element simulation of local transfer for a generalized Newtonian fluid in 2D periodic porous media

Dimitrienko

2023

Journal of Non-Newtonian Fluid Mechanics

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Macroscopic model for unsteady generalized Newtonian fluid flow in homogeneous porous media

Cited by 8 publications

References 61 publications

Upscaled dynamic capillary pressure for two-phase flow in porous media

Upscaled dynamic capillary pressure for two-phase flow in porous media

Macroscopic model for generalised Newtonian inertial two-phase flow in porous media

Least squares finite element simulation of local transfer for a generalized Newtonian fluid in 2D periodic porous media

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