Thermodynamic Analysis of the Darcy Law

1961 ◽  
Vol 28 (2) ◽  
pp. 208-212 ◽  
Author(s):  
R. G. Mokadam
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Gravitational Force ◽  
Equation Of Motion ◽  
Darcy Law ◽  
Navier Stokes ◽  
Proportionality Factor ◽  
Navier Stokes Equation ◽  
Special Case

The Darcy law is used extensively to describe the flow of fluids through porous media. According to this law the fluid flow is linearly dependent upon the pressure gradient and the gravitational force. The proportionality factor is generally known as the permeability of the porous medium. The Darcy law cannot be derived from the Navier-Stokes equation since this equation includes terms which characterize the fluid only. With the help of nonreversible thermodynamics it is possible to develop a general equation of motion of a fluid through a porous body, and obtain the Darcy law as a special case of such an equation.

Conservation of Mass

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Unsaturated Flow ◽  
Flow In Porous Media ◽  
Degree Of Saturation ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Conservation Of Mass ◽  
Special Case

The concept of conservation of mass holds a fundamental role in most problems in fluid physics. For a given problem this concept is cast in the form of an equation of continuity. Such an equation describes a condition—conservation of mass—that must be satisfied in any formal analysis of a problem. Thus an equation of continuity often is one of several complementary equations that are solved simultaneously to arrive at a solution to a flow problem, for example, the flow velocity as a function of coordinate position in a flow field. (Typically these complementary equations, as we will see in later chapters, involve conservation of momentum or energy, or both.) Although we did not explicitly use this idea in analyzing the one-dimensional flow problems at the end of Chapter 3, it turns out that continuity was implicitly satisfied in setting up each problem. We will return to these problems to illustrate this point. We will develop equations of continuity for three general cases: purely fluid flow, saturated single-phase flow in porous media, and unsaturated flow in porous media. The most general of the three equations is that for unsaturated flow, where pores are partially filled with the fluid phase of interest, such that the degree of saturation with respect to that phase is less than one. We will then show that this equation reduces, in the special case in which the degree of saturation equals one, to a simpler form appropriate for saturated single-phase flow. Then, this equation for saturated flow could be reduced further, in the special case in which the porosity equals one, to a form appropriate for purely fluid flow. For pedagogical reasons, however, we shall reverse this order and consider purely fluid flow first. In addition we will consider conservation of a solid or gas dissolved in a liquid, and take this opportunity to introduce Fick’s law for molecular diffusion. For simplicity we will consider only species that do not react chemically with the liquid, nor with the solid phases of a porous medium. Most of the derivations below are based on the idea of a small control volume of specified dimensions embedded within a fluid or porous medium.

scholarly journals SIMULATION OF HYDRO-MECHANICAL PROCESSES IN CENTRIFUGAL PUMPS

2016 ◽  
Vol 2016 (1) ◽  
pp. 100-105
Author(s):  
Ризван Шахбанов ◽  
Rizvan Shakhbanov ◽  
Леонид Савин ◽  
Leonid Savin
Keyword(s):  
Fluid Flow ◽  
Continuity Equation ◽  
Equation Of Motion ◽  
Theoretical Description ◽  
Navier Stokes ◽  
Centrifugal Pumps ◽  
Navier Stokes Equation ◽  
Rotary Pumps ◽  
Rotating Coordinates ◽  
Graphical Presentation

The peculiarities in current and kinematics of hydromechanical processes in centrifugal (rotary) pumps are considered. The theoretical description and graphical presentation of velocity profiles in an impeller are shown. A complex current in an impeller is described with the aid of a continuity equation and Navier-Stokes equation for rotating coordinates. A nonviscous character of fluid flow in the setting of an im-peller is taken into account by means of averaging of the equation of motion for that purpose the equation of a turbulence model is introduced in addition. The scheme of the digitization of a modeling area with the aid of a volumetric endelement grid is presented. As an example a computer model as a part of an impeller is shown.

scholarly journals Coupling Two-Phase Fluid Flow with Two-Phase Darcy Flow in Anisotropic Porous Media

2014 ◽  
Vol 6 ◽  
pp. 871021 ◽  
Author(s):  
Jie Chen ◽  
Shuyu Sun ◽  
Zhangxin Chen
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Domain Decomposition Method ◽  
Darcy Flow ◽  
Navier Stokes ◽  
Free Fluid ◽  
Two Phase ◽  
Anisotropic Porous Medium ◽  
Phase Fluid

This paper reports a numerical study of coupling two-phase fluid flow in a free fluid region with two-phase Darcy flow in a homogeneous and anisotropic porous medium region. The model consists of coupled Cahn-Hilliard and Navier-Stokes equations in the free fluid region and the two-phase Darcy law in the anisotropic porous medium region. A Robin-Robin domain decomposition method is used for the coupled Navier-Stokes and Darcy system with the generalized Beavers-Joseph-Saffman condition on the interface between the free flow and the porous media regions. Obtained results have shown the anisotropic properties effect on the velocity and pressure of the two-phase flow.

Fluid flow over and through a regular bundle of rigid fibres

2016 ◽  
Vol 792 ◽  
pp. 5-35 ◽  
Author(s):  
Giuseppe A. Zampogna ◽  
Alessandro Bottaro
Keyword(s):  
Porous Medium ◽  
Fluid Flow ◽  
Flow Field ◽  
Stokes Equations ◽  
Transversely Isotropic ◽  
Navier Stokes ◽  
Leading Order ◽  
Macroscopic Scale ◽  
Navier Stokes Equations ◽  
Homogenized Model

The interaction between a fluid flow and a transversely isotropic porous medium is described. A homogenized model is used to treat the flow field in the porous region, and different interface conditions, needed to match solutions at the boundary between the pure fluid and the porous regions, are evaluated. Two problems in different flow regimes (laminar and turbulent) are considered to validate the system, which includes inertia in the leading-order equations for the permeability tensor through a Oseen approximation. The components of the permeability, which characterize microscopically the porous medium and determine the flow field at the macroscopic scale, are reasonably well estimated by the theory, both in the laminar and the turbulent case. This is demonstrated by comparing the model’s results to both experimental measurements and direct numerical simulations of the Navier–Stokes equations which resolve the flow also through the pores of the medium.

Numerical Simulation of Supersonic Flow and Particle Motion in Aerodynamic Lenses

2003 ◽  
Author(s):  
Omid Abouali ◽  
Goodarz Ahmadi
Keyword(s):  
Equation Of Motion ◽  
Aerodynamic Performance ◽  
Computational Grid ◽  
Stagnation Pressure ◽  
Sensitivity Analyses ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Flow Stagnation ◽  
Airflow Field ◽  
Particles Trajectories

Airflow and particle motions in aerodynamic lenses are studied. The computational grid is generated with the use of GAMBIT code and FLUENT 5 is used in the analysis. The axisymmetric compressible form of the Navier-Stokes equation is solved and the airflow conditions are evaluated. One-way coupling is assumed in that the air transports the particles, but the effect of dilute particle concentrations on flow field is ignored. The particle equation of motion including drag, lift and Brownian forces is used and the particle trajectories in the aerodynamic a lens are analyzed. In addition, the airflow field and particles motions downstream of the nozzle are also studied. A series of sensitivity analyses on the effect of inlet flow stagnation pressure and backpressure of the nozzle on the aerodynamic performance of the lens is performed. Sample streamlines and particles trajectories in an axisymmetric plane of a combination of three aerodynamic lenses and a nozzle are shown in the figures.

Plastic Flow in Confined Porous Media of Complex Contours

1999 ◽  
Author(s):  
Mario F. Letelier ◽  
César E. Rosas
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Theoretical Study ◽  
Perturbation Parameter ◽  
Resistance Coefficient ◽  
Boundary Perturbation ◽  
Bingham Plastic ◽  
Flow Through ◽  
Central Plug

Abstract A theoretical study of the fully developed fluid flow through a confined porous medium is presented. The fluid is described by the Bingham plastic model for small values of the yield number. The analysis allows for many admissible shapes of the wall contour. The velocity field is computed for several combination of relevant parameters, i.e., the yield number, Darcy resistance coefficient and the boundary perturbation parameter. The wall effect is especially highlighted and the characteristics of the central plug region as well. Plots of isovel curves and velocity profiles are included for a variety of flow and geometry parameters.

Toward a Global Model for FSI in Tube Bundles

2008 ◽  
Author(s):  
Daniel Broc ◽  
Marion Duclercq
Keyword(s):  
Fluid Flow ◽  
Euler Equations ◽  
Dynamic Behaviour ◽  
Navier Stokes ◽  
Inertial Effects ◽  
Navier Stokes Equation ◽  
Dissipative Effects ◽  
Tube Bundles ◽  
Homogenization Methods ◽  
Physical Phenomena

It is well known that a fluid may strongly influence the dynamic behaviour of a structure. Many different physical phenomena may take place, depending on the conditions: fluid at rest, fluid flow, little or high displacements of the structure. Inertial effects can take place, with lower vibration frequencies, dissipative effects also, with damping, instabilities due to the fluid flow (Fluid Induced Vibration). In this last case the structure is excited by the fluid. The paper deals with the vibration of tube bundles in a fluid, under a seismic excitation or an impact. In this case the structure moves under an external excitation, and the movement is influenced by the fluid. The main point in such system is that the geometry is complex, and could lead to very huge sizes for a numerical analysis. Many works has been made in the last years to develop homogenization methods for the dynamic behaviour of tube bundles (/2/ and /3/). The size of the problem is reduced, and it is possible to make numerical simulations on wide tubes bundles with reasonable computer times. These homogenization methods are valid for “little displacements” of the structure (the tubes), in a fluid at rest. The fluid movement is governed by the Euler equations. In this case, only “inertial effects” will take place, with globally lower frequencies. It is well known that dissipative effects due to the fluid may take place, even if the displacements of the tube are no so high, or if the fluid is not still (/4/, /5/, /6/ and /8/). Such effects may be described in the homogenized models by using a Rayleigh damping, but the basic assumption of the model remains the “perfect fluid” hypothesis. It seem necessary, in order to get a best description of the physical phenomena, to build a more general model, based on the general Navier Stokes equation for the fluid. The homogenization of such system will be much more complex than for the Euler equations. The paper doesn’t pretend to give a general solution of the problem, but only points out the most important key points to build such homogenized model for the dynamic behaviour of tubes bundles in a fluid.

Analysis of the Influence of Fluid Flow on the Plasticity of Porous Rock Under an Axially Symmetric Punch

1974 ◽  
Vol 14 (03) ◽  
pp. 271-278 ◽  
Author(s):  
Milos Kojic ◽  
J.B. Cheatham
Keyword(s):  
Plastic Deformation ◽  
Porous Medium ◽  
Porous Media ◽  
Fluid Flow ◽  
Pressure Gradient ◽  
Stress Tensor ◽  
Unit Volume ◽  
Fluid Pressure ◽  
The Body ◽  
Axially Symmetric

Introduction A number of problems occur in the fields of drilling and rock mechanics for which consideration must be given to the interaction of fluid flow and rock deformation. Such problems include those of borehole stability, chip removal from under a drill bit, drilling in the presence of a fluid pressure gradient between the drilling fluid and formation fluid, and drilling by use of hydraulic jets. We have recently developed a general theory of the influence of fluid pressure gradients and gravity on the plasticity of porous media. The solution of the problem considered here serves as an example of the application of that theory. The illustrative problem is to determine the load required on a flat problem is to determine the load required on a flat axially symmetric punch for incipient plasticity of the porous medium under the punch when fluid flows through the bottom face of the punch. The rock is assumed to behave as a Coulomb plastic material under the influence of body forces plastic material under the influence of body forces due to fluid pressure gradients and gravity. Numerical methods that have been used by Cox et al. for analyzing axially symmetric plastic deformation in soils with gravity force are applied to the problem considered here. Involved is an iterative process for determining the slip lines. The fluid flow field ‘used for calculating the fluid pressure gradient is based upon the work by Ham pressure gradient is based upon the work by Ham in his study of the potential distribution ahead of the bit in rotary drilling. The effective stresses in the porous rock and the punch force for incipient plasticity are computed in terms of the fluid plasticity are computed in terms of the fluid pressure and the cohesive strength and internal pressure and the cohesive strength and internal friction of the rock. PLASTICITY OF POROUS MEDIA PLASTICITY OF POROUS MEDIA A recently developed general theory of plasticity of porous media under the influence of fluid flow is summarized in this section. The equation of motion for the porous solid for the case of incipient plastic deformation reduces to the following equilibrium equation:(1) where Ts is the partial stress tensor of the solid; Fs is the body force acting on the solid per unit volume of the solid material; P is the interaction force between the solid and the fluid; and is the porosity, which is defined as the ratio of the pore porosity, which is defined as the ratio of the pore volume to the total volume of the solid-fluid mixture. The partial stress tensor Ts can be considered as the effective stress tensor that is used in sod mechanics. With the acceptance of the effective stress principle defined in Ref. 5, the yield function, f, in the following form is satisfied for plastic deformation of the porous medium. plastic deformation of the porous medium.(2) where EP is the plastic strain tensor and K and the work-hardening parameter. From the equation of motion for the fluid, the interaction force P can be expressed in the form(3) where is the inertial force of the fluid per unit volume of the mixture and F is the body force acting on the fluid per unit volume of fluid. For the case of incipient plastic deformation the solid can be considered static (velocities of the solid particles are zero), and the problem of determining particles are zero), and the problem of determining the fluid flow field is the one usually analyzed in petroleum engineering. petroleum engineering. Consider a flow of be fluid such that the inertial forces of the fluid can be neglected and assume that Darcy's law is applicable. SPEJ P. 271

The Permeability of a Uniformly Vuggy Porous Medium

1973 ◽  
Vol 13 (02) ◽  
pp. 69-74 ◽  
Author(s):  
Graham H. Neale ◽  
Walter K. Nader
Keyword(s):  
Porous Medium ◽  
Flow Field ◽  
Stokes Equation ◽  
Spherical Cavity ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Darcy Equation ◽  
Matrix Permeability ◽  
Flow Vector

Abstract Using the creeping Navier Stokes equation within a spherical cavity and the Darcy equation in the surrounding homogeneous and isotropic porous medium, the flow field in the entire system is evaluated. Applying this result to a representative generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium, an engineering estimation of the interdependence of the matrix permeability km, the vug porosity permeability km, the vug porositytotal volume of vug space 0v = ----------------------------total volume of sample and the system permeability ks of the vuggy porous medium is derived. This interdependence can be expressed by the formula: Introduction The objective of this study is the derivation of an engineering formula that shows the interdependence of matrix permeability, km, vug porosity, 0 v, and system permeability, ks, of a uniformly vuggy porous medium. In the first section, with the above porous medium. In the first section, with the above goal in mind and to satisfy more general interests, we shall study and predict the flow field within a single cavity bounded by a sphere, of radius R, and in the surrounding homogeneous and isotropic porous medium. In the second section, we shall porous medium. In the second section, we shall suggest as a generalizing model of a uniformly vuggy, homogeneous and isotropic porous medium a regular cubic array of monosized spherical cavities. Applying the formula for the pressure field near a single spherical cavity, we shall then develop the sought engineering formula. To describe the creeping flow of the incompressible liquid of viscosity, in the spherical cavity, we shall employ the creeping Navier Stokes equation, .............................(1) The Darcy equation, ,...........................(2) will be used to describe the flow of this liquid in the porous medium of permeability k that fills the space outside the cavity. p designates the liquid pressure referred to datum, denotes the flow pressure referred to datum, denotes the flow vector, and * is used to indicate macroscopically averaged quantities pertaining specifically to a porous medium. porous medium. In hydrodynamics, one generally requests continuity of the pressure, of the flow vector, and of the shear tensor throughout the fundamental domain of the problem - in particular, along the boundary surfaces, which separate subdomains. When applying these principles to this problem, one would impose at the spherical boundary that separates the cavity from the porous medium:continuity of the pressure,continuity of the component of u that is orthogonal to the surface,continuity of the other component of u that is tangential to the surface,continuity of the shear component tangential to the surface. Arguments of this nature have lead to the suggestion of a generalization of the Darcy equation, namely, the Brinkman equation, ...............(3) However, both the necessity and the validity of this generalization have been challenged; indeed, it has been shown that a mathematically consistent solution of our problem may be obtained, using Eqs. 1 and 2 within the respective subdomains, provided one abandons the request for continuity of the shear at the wall of the cavity (compare Boundary Condition d above).** SPEJ P. 69

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