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http://hdl.handle.net/10985/9724
Two-phase non-Darcy flow in heterogeneous porous media: A numerical investigation
AHMADI-SENICHAULT, Azita; ABBASIAN ARANI, Ali Akbar; LASSEUX, Didier
Significant inertial effects are observed for many applications such as flow in the near-wellbore region, in very permeable reservoirs or in packed-bed reactors. In these cases, the classical description of two-phase flow in porous media by the generalized Darcy's law is no longer valid. Due to the lack of a formalized theoretical model confirmed experimentally, this study is based on a generalized Darcy-Forchheimer approach for modelling two-phase incompressible non-stationary inertial flow in porous media. In this model, the momentum conservation equation for each phase, , has a quadratic correction to generalized Darcy’s law and is expressed as: (=”w” for water or “o” for oil): (1) This equation is completed with the mass conservation equation for each phase given by (2) and the capillary pressure and saturation relationships (3) (4) Using a finite volume formulation, an IMPES (IMplicit for Pressures, Explicit for Saturations) scheme and a Fixed Point method for the treatment of non-linearities caused by inertia, a 3D numerical tool has been developed. For clarity, results are presented in 1D and 2D configurations only. For 1D flow in a homogeneous porous medium, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the “Buckley-Leverett” model extended to take into account inertia. The influence of inertial effects on the saturation profiles and therefore on the breakthrough curves for homogeneous media is analysed for different Reynolds numbers, thus emphasizing the necessity of taking into account this additional energy loss when necessary. For 1D heterogeneous configurations, a thorough analysis of the saturation fronts as well as the saturation jumps at the interface between two media of contrasted properties highlights the influence of inertial effects for different Reynolds and capillary numbers. In 2D heterogeneous configurations, saturation distributions are strongly affected by inertial effects. In particular, capillary trapping of the displaced fluid observed for the Darcy regime in certain regions can completely disappears when inertial effects become dominant.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10985/97242010-01-01T00:00:00ZAHMADI-SENICHAULT, AzitaABBASIAN ARANI, Ali AkbarLASSEUX, DidierSignificant inertial effects are observed for many applications such as flow in the near-wellbore region, in very permeable reservoirs or in packed-bed reactors. In these cases, the classical description of two-phase flow in porous media by the generalized Darcy's law is no longer valid. Due to the lack of a formalized theoretical model confirmed experimentally, this study is based on a generalized Darcy-Forchheimer approach for modelling two-phase incompressible non-stationary inertial flow in porous media. In this model, the momentum conservation equation for each phase, , has a quadratic correction to generalized Darcy’s law and is expressed as: (=”w” for water or “o” for oil): (1) This equation is completed with the mass conservation equation for each phase given by (2) and the capillary pressure and saturation relationships (3) (4) Using a finite volume formulation, an IMPES (IMplicit for Pressures, Explicit for Saturations) scheme and a Fixed Point method for the treatment of non-linearities caused by inertia, a 3D numerical tool has been developed. For clarity, results are presented in 1D and 2D configurations only. For 1D flow in a homogeneous porous medium, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the “Buckley-Leverett” model extended to take into account inertia. The influence of inertial effects on the saturation profiles and therefore on the breakthrough curves for homogeneous media is analysed for different Reynolds numbers, thus emphasizing the necessity of taking into account this additional energy loss when necessary. For 1D heterogeneous configurations, a thorough analysis of the saturation fronts as well as the saturation jumps at the interface between two media of contrasted properties highlights the influence of inertial effects for different Reynolds and capillary numbers. In 2D heterogeneous configurations, saturation distributions are strongly affected by inertial effects. In particular, capillary trapping of the displaced fluid observed for the Darcy regime in certain regions can completely disappears when inertial effects become dominant.A numerical approach of two-phase non-Darcy flow in heterogeneous porous media
http://hdl.handle.net/10985/9982
A numerical approach of two-phase non-Darcy flow in heterogeneous porous media
ABBASIAN ARANI, Ali Akbar; AHMADI-SENICHAULT, Azita; LASSEUX, Didier
Significant inertial effects are observed for many applications such as flow in the near-wellbore region, in very permeable reservoirs or in packed-bed reactors. In these cases, the classical description of two-phase flow in porous media by the generalized Darcy's law is no longer valid. Due to the lack of a formalized theoretical model confirmed experimentally, our study is based on a generalized Darcy-Forchheimer approach for modelling two-phase incompressible inertial flow in porous media. Using a finite volume formulation, an IMPES (IMplicit for Pressures, Explicit for Saturations) scheme and a Fixed Point method for the treatment of non-linearities caused by inertia, a 3D numerical tool has been developed. For 1D flow in a homogeneous porous medium, comparison of saturation profiles obtained numerically at different times to those obtained semi-analytically using an “Inertial Buckley-Leverett model” allows a validation of the tool. The influence of inertial effects on the saturation profiles and therefore on the breakthrough curves for homogeneous media is analysed for different Reynolds numbers, thus emphasizing the necessity of taking into account this additional energy loss when necessary. For 1D heterogeneous configurations, a thorough analysis of the saturation fronts as well as the saturation jumps at the interface between two media of contrasted properties highlights the influence of inertial effects for different Reynolds and capillary numbers. In 2D heterogeneous configurations, saturation distributions are strongly affected by inertial effects. In particular, capillary trapping of the displaced fluid observed for the Darcy regime in certain regions can completely disappears when inertial effects become dominant.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10985/99822009-01-01T00:00:00ZABBASIAN ARANI, Ali AkbarAHMADI-SENICHAULT, AzitaLASSEUX, DidierSignificant inertial effects are observed for many applications such as flow in the near-wellbore region, in very permeable reservoirs or in packed-bed reactors. In these cases, the classical description of two-phase flow in porous media by the generalized Darcy's law is no longer valid. Due to the lack of a formalized theoretical model confirmed experimentally, our study is based on a generalized Darcy-Forchheimer approach for modelling two-phase incompressible inertial flow in porous media. Using a finite volume formulation, an IMPES (IMplicit for Pressures, Explicit for Saturations) scheme and a Fixed Point method for the treatment of non-linearities caused by inertia, a 3D numerical tool has been developed. For 1D flow in a homogeneous porous medium, comparison of saturation profiles obtained numerically at different times to those obtained semi-analytically using an “Inertial Buckley-Leverett model” allows a validation of the tool. The influence of inertial effects on the saturation profiles and therefore on the breakthrough curves for homogeneous media is analysed for different Reynolds numbers, thus emphasizing the necessity of taking into account this additional energy loss when necessary. For 1D heterogeneous configurations, a thorough analysis of the saturation fronts as well as the saturation jumps at the interface between two media of contrasted properties highlights the influence of inertial effects for different Reynolds and capillary numbers. In 2D heterogeneous configurations, saturation distributions are strongly affected by inertial effects. In particular, capillary trapping of the displaced fluid observed for the Darcy regime in certain regions can completely disappears when inertial effects become dominant.Derivation of a macroscopic model for two-phase non-Darcy flow in homogeneous porous media using volume averaging
http://hdl.handle.net/10985/9981
Derivation of a macroscopic model for two-phase non-Darcy flow in homogeneous porous media using volume averaging
ABBASIAN ARANI, Ali Akbar; LASSEUX, Didier; AHMADI-SENICHAULT, Azita
The purpose of this work is to propose a derivation of a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. The starting point of the procedure is the pore-scale boundary value problem given by the continuity and Navier–Stokes equations in each phase β and γ along with boundary conditions at interfaces. The method of volume averaging is employed subjected to a series of constraints for the development to hold. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The development also assumes that fluctuations of the curvature of the fluid–fluid interfaces are unimportant over the unit cell representing the porous medium. Under these circumstances, the resulting macroscopic momentum equation, for the -phase (=, ) relates the gradient of the phase-averaged pressure to the filtration or Darcy velocity in a coupled nonlinear form. All tensors appearing in the macroscopic equation can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. Some indications to compute these tensors are provided.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10985/99812009-01-01T00:00:00ZABBASIAN ARANI, Ali AkbarLASSEUX, DidierAHMADI-SENICHAULT, AzitaThe purpose of this work is to propose a derivation of a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. The starting point of the procedure is the pore-scale boundary value problem given by the continuity and Navier–Stokes equations in each phase β and γ along with boundary conditions at interfaces. The method of volume averaging is employed subjected to a series of constraints for the development to hold. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The development also assumes that fluctuations of the curvature of the fluid–fluid interfaces are unimportant over the unit cell representing the porous medium. Under these circumstances, the resulting macroscopic momentum equation, for the -phase (=, ) relates the gradient of the phase-averaged pressure to the filtration or Darcy velocity in a coupled nonlinear form. All tensors appearing in the macroscopic equation can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. Some indications to compute these tensors are provided.A numerical analysis of the inertial correction to Darcy's law
http://hdl.handle.net/10985/9980
A numerical analysis of the inertial correction to Darcy's law
ABBASIAN ARANI, Ali Akbar; LASSEUX, Didier; AHMADI-SENICHAULT, Azita
Our interest in this work is the stationary one-phase Newtonian flow in a class of homogeneous porous media at large enough flow rates so that the relationship between the filtration velocity and the pressure gradient is no longer linear. The non linear -inertial- correction to Darcy's law is investigated from a numerical point of view on model periodic structures made of regular arrays of cylinders. The starting point of the analysis is the macroscopic model resulting from the volume averaging of the mass and momentum (Navier-Stokes) equations at the pore scale. Identification of the macroscopic properties in this model is made by first solving the microscopic flow as well as the closure problem resulting from the upscaling. From these solutions, the inertial correction is computed and analyzed with respect to the Reynolds number and the pressure gradient orientation relative to the principal axes of the periodic unit cell.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10985/99802009-01-01T00:00:00ZABBASIAN ARANI, Ali AkbarLASSEUX, DidierAHMADI-SENICHAULT, AzitaOur interest in this work is the stationary one-phase Newtonian flow in a class of homogeneous porous media at large enough flow rates so that the relationship between the filtration velocity and the pressure gradient is no longer linear. The non linear -inertial- correction to Darcy's law is investigated from a numerical point of view on model periodic structures made of regular arrays of cylinders. The starting point of the analysis is the macroscopic model resulting from the volume averaging of the mass and momentum (Navier-Stokes) equations at the pore scale. Identification of the macroscopic properties in this model is made by first solving the microscopic flow as well as the closure problem resulting from the upscaling. From these solutions, the inertial correction is computed and analyzed with respect to the Reynolds number and the pressure gradient orientation relative to the principal axes of the periodic unit cell.Inertial flow in porous media: A numerical investigation on model structures
http://hdl.handle.net/10985/9888
Inertial flow in porous media: A numerical investigation on model structures
AGNAOU, Mehrez; LASSEUX, Didier; AHMADI-SENICHAULT, Azita
The aim of this work is to study the correction to Darcy's law for inertial flow in porous media. In many situations encountered in industrial applications such as flow in column reactors, gas flow near wells for hydrocarbon recovery and CO2 sequestration, flow in filters... , Reynolds numbers are large enough to lead to a non-linear relationship between the filtration velocity and the pressure gradient. In this work, a numerical analysis of the non linear -inertial- correction to Darcy's law is carried out for the stationary inertial flow of a one-phase Newtonian incompressible fluid on model 2D and 3D structures. Effective properties appearing in the macroscopic model resulting from the volume averaging of the mass and momentum (Navier-Stokes) equations at the pore scale are determined using the microscopic flow fields and solving the closure problems resulting from up-scaling. From the numerical simulations, the dependence of the correction to Darcy's law on the geometrical properties of the 3D structure is studied. These properties are the shape of the solid grains which may be cubic or spherical and the degree of disorder in their arrangement in the domain. Weak disorder corresponds to a random placement of the grains of identical shape and size within each cell of a regular 3D lattice, while for strong disorder, grain size is also randomly distributed.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/10985/98882013-01-01T00:00:00ZAGNAOU, MehrezLASSEUX, DidierAHMADI-SENICHAULT, AzitaThe aim of this work is to study the correction to Darcy's law for inertial flow in porous media. In many situations encountered in industrial applications such as flow in column reactors, gas flow near wells for hydrocarbon recovery and CO2 sequestration, flow in filters... , Reynolds numbers are large enough to lead to a non-linear relationship between the filtration velocity and the pressure gradient. In this work, a numerical analysis of the non linear -inertial- correction to Darcy's law is carried out for the stationary inertial flow of a one-phase Newtonian incompressible fluid on model 2D and 3D structures. Effective properties appearing in the macroscopic model resulting from the volume averaging of the mass and momentum (Navier-Stokes) equations at the pore scale are determined using the microscopic flow fields and solving the closure problems resulting from up-scaling. From the numerical simulations, the dependence of the correction to Darcy's law on the geometrical properties of the 3D structure is studied. These properties are the shape of the solid grains which may be cubic or spherical and the degree of disorder in their arrangement in the domain. Weak disorder corresponds to a random placement of the grains of identical shape and size within each cell of a regular 3D lattice, while for strong disorder, grain size is also randomly distributed.Numerical simulation of two-phase inertial flow in heterogeneous porous media
http://hdl.handle.net/10985/9727
Numerical simulation of two-phase inertial flow in heterogeneous porous media
AHMADI-SENICHAULT, Azita; ABBASIAN ARANI, Ali Akbar; LASSEUX, Didier
In this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.
We wish to thank our students, Y. Benarafa and S. Delau, who participated in this study at its early stage.
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/10985/97272010-01-01T00:00:00ZAHMADI-SENICHAULT, AzitaABBASIAN ARANI, Ali AkbarLASSEUX, DidierIn this study, non-Darcy inertial two-phase incompressible and non-stationary flow in heterogeneous porous media is analyzed using numerical simulations. For the purpose, a 3D numerical tool was fully developed using a finite volume formulation, although for clarity, results are presented in 1D and 2D configurations only. Since a formalized theoretical model confirmed by experimental data is still lacking, our study is based on the widely used generalized Darcy–Forchheimer model. First, a validation is performed by comparing numerical results of the saturation front kinetics with a semi-analytical solution inspired from the Buckley–Leverett model extended to take into account inertia. Second, we highlight the importance of inertial terms on the evolution of saturation fronts as a function of a suitable Reynolds number. Saturation fields are shown to have a structure markedly different from the classical case without inertia, especially for heterogeneous media, thereby, emphasizing the necessity of a more complete model than the classical generalized Darcy’s one when inertial effects are not negligible.Résolution numérique de l’écoulement diphasique en milieu poreux hétérogène incluant les effets inertiels
http://hdl.handle.net/10985/10029
Résolution numérique de l’écoulement diphasique en milieu poreux hétérogène incluant les effets inertiels
ABBASIAN ARANI, Ali Akbar; LASSEUX, Didier; AHMADI-SENICHAULT, Azita
La mise en place d'un outil numérique 3D de simulation d'écoulement diphasique hors régime de Darcy basé sur le modèle de Darcy-Forchheimer généralisé est présentée. L'outil est tout d’abord validé à l’aide d'une solution semi analytique 1D de type Buckley-Leverett. Des résultats obtenus dans différentes configurations homogène et hétérogènes 1D et 2D mettent en évidence l'importance des termes inertiels en fonction d'un nombre de Reynolds de l'écoulement.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10985/100292007-01-01T00:00:00ZABBASIAN ARANI, Ali AkbarLASSEUX, DidierAHMADI-SENICHAULT, AzitaLa mise en place d'un outil numérique 3D de simulation d'écoulement diphasique hors régime de Darcy basé sur le modèle de Darcy-Forchheimer généralisé est présentée. L'outil est tout d’abord validé à l’aide d'une solution semi analytique 1D de type Buckley-Leverett. Des résultats obtenus dans différentes configurations homogène et hétérogènes 1D et 2D mettent en évidence l'importance des termes inertiels en fonction d'un nombre de Reynolds de l'écoulement.In vitro cartilage culture: flow, transport and reaction in fibrous porous media
http://hdl.handle.net/10985/9753
In vitro cartilage culture: flow, transport and reaction in fibrous porous media
AHMADI-SENICHAULT, Azita; LASSEUX, Didier; LETELLIER, Samuel
Flow and transport in fibrous media are encountered in a wide variety of domains ranging from biotechnology to filtration in chemical engineering. The context of this work is the in vitro cartilage cell culture on a fibrous biodegradable polymer scaffold placed in a bioreactor. A seeding process using a liquid containing cells (chondrocytes) initiates the culture and an imposed continuous flow through the scaffold allows both the transport of nutrients necessary for cell-growth and of metabolic waste products. This work will attempt to contribute to the study of the hydrodynamics and transport through the fibrous scaffold at different stages of growth, both having a key role in the process of cell growth and on the final quality of the cultured cartilage. The hydrodynamics in the scaffold and in particular the relationship between macroscopic experimentally accessible properties such as the permeability and the porosity have first been studied. For this purpose, the formalism of volume averaging is employed and the associated closure problem is solved numerically with an artificial compressibility algorithm on the basis of a finite volume scheme on a Marker and Cell type of grid. Fibrous media with different microscopic structures are studied. Through a theoretical study, assuming local mass equilibrium, a macroscopic one-equation model describing the reactive transport (advection/diffusion/reaction) of the two species in a three-phase system composed of the cell-phase, a fluid phase and a solid phase is proposed. The volume averaging method is used to develop macroscopic transport equations and associated closure problems. Resolution of the latter over a unit cell representative of a pseudo-periodic medium allows the determination of effective macroscopic properties without any adjustable parameters. The dimensionless form of the closure problems involving advective, diffusive and reactive terms are numerically solved for any 3D geometrical configuration using a finite volume formulation using appropriate schemes. The velocity field input to the model is obtained by the resolution of the Navier-Stokes problem using a modified QUICK scheme and an Artificial Compressibility algorithm. The numerical tool is then validated by comparing its results to those presented in the literature for 2-D unit cells and under-classes of our model (namely, diffusion, diffusion/reaction and diffusion/advection problems). The complete problem involving convection, diffusion and reaction in the three phase system is then studied for different parameters. More precisely, the influence of a cell Peclet number and the solid and cell volume fractions on the dispersion tensor has been studied.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10985/97532007-01-01T00:00:00ZAHMADI-SENICHAULT, AzitaLASSEUX, DidierLETELLIER, SamuelFlow and transport in fibrous media are encountered in a wide variety of domains ranging from biotechnology to filtration in chemical engineering. The context of this work is the in vitro cartilage cell culture on a fibrous biodegradable polymer scaffold placed in a bioreactor. A seeding process using a liquid containing cells (chondrocytes) initiates the culture and an imposed continuous flow through the scaffold allows both the transport of nutrients necessary for cell-growth and of metabolic waste products. This work will attempt to contribute to the study of the hydrodynamics and transport through the fibrous scaffold at different stages of growth, both having a key role in the process of cell growth and on the final quality of the cultured cartilage. The hydrodynamics in the scaffold and in particular the relationship between macroscopic experimentally accessible properties such as the permeability and the porosity have first been studied. For this purpose, the formalism of volume averaging is employed and the associated closure problem is solved numerically with an artificial compressibility algorithm on the basis of a finite volume scheme on a Marker and Cell type of grid. Fibrous media with different microscopic structures are studied. Through a theoretical study, assuming local mass equilibrium, a macroscopic one-equation model describing the reactive transport (advection/diffusion/reaction) of the two species in a three-phase system composed of the cell-phase, a fluid phase and a solid phase is proposed. The volume averaging method is used to develop macroscopic transport equations and associated closure problems. Resolution of the latter over a unit cell representative of a pseudo-periodic medium allows the determination of effective macroscopic properties without any adjustable parameters. The dimensionless form of the closure problems involving advective, diffusive and reactive terms are numerically solved for any 3D geometrical configuration using a finite volume formulation using appropriate schemes. The velocity field input to the model is obtained by the resolution of the Navier-Stokes problem using a modified QUICK scheme and an Artificial Compressibility algorithm. The numerical tool is then validated by comparing its results to those presented in the literature for 2-D unit cells and under-classes of our model (namely, diffusion, diffusion/reaction and diffusion/advection problems). The complete problem involving convection, diffusion and reaction in the three phase system is then studied for different parameters. More precisely, the influence of a cell Peclet number and the solid and cell volume fractions on the dispersion tensor has been studied.Two-phase inertial flow in homogeneous porous media: A theoretical derivation of a macroscopic model
http://hdl.handle.net/10985/9745
Two-phase inertial flow in homogeneous porous media: A theoretical derivation of a macroscopic model
LASSEUX, Didier; AHMADI-SENICHAULT, Azita; ABBASIAN ARANI, Ali Akbar
The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ , the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient ∇ pα α to the filtration or Darcy velocity vα in a coupled nonlinear form explicitly given by : (equations) In these equations, Fαα and Fακ are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors K∗αα and K∗ακ and the permeability and viscous drag tensors Kα and Kακ are intrinsic and are those defined the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10985/97452008-01-01T00:00:00ZLASSEUX, DidierAHMADI-SENICHAULT, AzitaABBASIAN ARANI, Ali AkbarThe purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ , the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient ∇ pα α to the filtration or Darcy velocity vα in a coupled nonlinear form explicitly given by : (equations) In these equations, Fαα and Fακ are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors K∗αα and K∗ακ and the permeability and viscous drag tensors Kα and Kακ are intrinsic and are those defined the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided.On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media
http://hdl.handle.net/10985/9726
On the stationary macroscopic inertial effects for one phase flow in ordered and disordered porous media
LASSEUX, Didier; ABBASIAN ARANI, Ali Akbar; AHMADI-SENICHAULT, Azita
We report on the controversial dependence of the inertial correction to Darcy’s law upon the filtration velocity (or Reynolds number) for one-phase Newtonian incompressible flow in model porous media. Our analysis is performed on the basis of an upscaled form of the Navier-Stokes equation requiring the solution of both the micro-scale flow and the associated closure problem. It is carried out with a special focus on the different regimes of inertia (weak and strong inertia) and the crossover between these regimes versus flow orientation and structural parameters, namely porosity and disorder. For ordered structures, it is shown that (i) the tensor involved in the expression of the correction is generally not symmetric, despite the isotropic feature of the permeability tensor. This is in accordance with the fact that the extra force due to inertia exerted on the structure is not pure drag in the general case; (ii) the Forchheimer type of correction (which strictly depends on the square of the filtration velocity) is an approximation that does not hold at all for particular orientations of the pressure gradient with respect to the axes of the structure; and (iii) the weak inertia regime always exists as predicted by theoretical developments. When structural disorder is introduced, this work shows that (i) the quadratic dependence of the correction upon the filtration velocity is very robust over a wide range of the Reynolds number in the strong inertia regime; (ii) the Reynolds number interval corresponding to weak inertia, that is always present, is strongly reduced in comparison to ordered structures. In conjunction with its relatively small magnitude, it explains why this weak inertia regime is most of the time overlooked during experiments on natural media. In all cases, the Forchheimer correction implies that the permeability is different from the intrinsic one.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/10985/97262011-01-01T00:00:00ZLASSEUX, DidierABBASIAN ARANI, Ali AkbarAHMADI-SENICHAULT, AzitaWe report on the controversial dependence of the inertial correction to Darcy’s law upon the filtration velocity (or Reynolds number) for one-phase Newtonian incompressible flow in model porous media. Our analysis is performed on the basis of an upscaled form of the Navier-Stokes equation requiring the solution of both the micro-scale flow and the associated closure problem. It is carried out with a special focus on the different regimes of inertia (weak and strong inertia) and the crossover between these regimes versus flow orientation and structural parameters, namely porosity and disorder. For ordered structures, it is shown that (i) the tensor involved in the expression of the correction is generally not symmetric, despite the isotropic feature of the permeability tensor. This is in accordance with the fact that the extra force due to inertia exerted on the structure is not pure drag in the general case; (ii) the Forchheimer type of correction (which strictly depends on the square of the filtration velocity) is an approximation that does not hold at all for particular orientations of the pressure gradient with respect to the axes of the structure; and (iii) the weak inertia regime always exists as predicted by theoretical developments. When structural disorder is introduced, this work shows that (i) the quadratic dependence of the correction upon the filtration velocity is very robust over a wide range of the Reynolds number in the strong inertia regime; (ii) the Reynolds number interval corresponding to weak inertia, that is always present, is strongly reduced in comparison to ordered structures. In conjunction with its relatively small magnitude, it explains why this weak inertia regime is most of the time overlooked during experiments on natural media. In all cases, the Forchheimer correction implies that the permeability is different from the intrinsic one.