Derivation of a Darcy’s Law for a Porous Medium Composed of Two Solid Phases Saturated by a Single- Phase Fluid: A Homogenization Approach

2008 ◽  
Vol 74 (3) ◽  
pp. 349-368 ◽  
Author(s):  
Juan E. Santos ◽  
Dongwoo Sheen
Keyword(s):  
Porous Medium ◽  
Single Phase ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Homogenization Approach ◽  
Solid Phases ◽  
Phase Fluid

Estimating the Coefficient of Inertial Resistance in Fluid Flow Through Porous Media

1974 ◽  
Vol 14 (05) ◽  
pp. 445-450 ◽  
Author(s):  
J. Geertsma
Keyword(s):  
Fluid Flow ◽  
Flow Resistance ◽  
Single Phase ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
High Flow ◽  
Flow Through Porous Media ◽  
Flow Rates ◽  
Flow Through ◽  
Phase Fluid

Abstract The object of this paper is to introduce an empirical, time-honored relationship between inertia coefficient - frequently misnamed "turbulence factor" - permeability, and porosity, based on a combination of experimental data, dimensional analysis, and other physical considerations. The formula can be used effectively for, among other things, the preliminary evaluation of the number of wells in a new gas field and the spacing between them. Introduction It has long been recognized that Darcy's law for single-phase fluid flow through porous media,Equation 1 in which ?=superficial velocity µ=fluid viscosity k=formation permeability p=pressure head, is approximately correct only in a specific flow regime where the velocity ? is low. Single-phase fluid flow in reservoir rocks is often characterized by conditions in favor of this linearized flow law, but important exceptions do occur. They are in particular related to the surroundings of wells producing at high flow rates such as gas wells. For the prediction or analysis of the production behavior of such wells it is necessary to apply a more general nonlinear flow law. The appropriate formula was given in 1901 by Forchheimer1; it readsEquation 2 in which ?=density a=coefficient of viscous flow resistance 1/k ß=coefficient of inertial flow resistance. This equation indicates that in single-phase fluid flow through a porous medium two forces counteract the external force simultaneously - namely, viscous and inertial forces - the latter continuously gaining importance as the velocity ? increases. For low flow rates the viscous term dominates, whereas for high flow rates the inertia term does. The upper limit of practical applicability of Darcy's law can best be specified by some "critical value" orf the dimensionless ratio.Equation 3 which has a close resemblance to the Reynolds number. Observe that ß/a has the dimension of a length. Inertia and Turbulence As the Reynolds number is commonly used as an indicator for either laminar or turbulent flow conditions, the coefficient ß is often referred to as the turbulence coefficient. However, the phenomenon we are interested in has nothing to do with turbulence. The flow regime of concern is usually fully laminar. The observed departure from Darcy's law is the result of convective accelerations and decelerations of the fluid particles on their way through the pore space. Within the flow range normally experienced in oil and gas reservoirs, including the well's surroundings, energy losses caused by actual turbulence can be safely ignored.

Incorporation of Interfacial Areas in Models of Two-Phase Flow

1999 ◽  
Author(s):  
William G. Gray ◽  
Michael A. Celia
Keyword(s):  
Porous Media ◽  
Hydraulic Conductivity ◽  
Single Phase ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Hydraulic Gradient ◽  
Phase Flow ◽  
Single Phase Flow ◽  
Α Phase ◽  
Flow Through

The mathematical study of flow in porous media is typically based on the 1856 empirical result of Henri Darcy. This result, known as Darcy’s law, states that the velocity of a single-phase flow through a porous medium is proportional to the hydraulic gradient. The publication of Darcy’s work has been referred to as “the birth of groundwater hydrology as a quantitative science” (Freeze and Cherry, 1979). Although Darcy’s original equation was found to be valid for slow, steady, one-dimensional, single-phase flow through a homogeneous and isotropic sand, it has been applied in the succeeding 140 years to complex transient flows that involve multiple phases in heterogeneous media. To attain this generality, a modification has been made to the original formula, such that the constant of proportionality between flow and hydraulic gradient is allowed to be a spatially varying function of the system properties. The extended version of Darcy’s law is expressed in the following form: qα=-Kα . Jα (2.1) where qα is the volumetric flow rate per unit area vector of the α-phase fluid, Kα is the hydraulic conductivity tensor of the α-phase and is a function of the viscosity and saturation of the α-phase and of the solid matrix, and Jα is the vector hydraulic gradient that drives the flow. The quantities Jα and Kα account for pressure and gravitational effects as well as the interactions that occur between adjacent phases. Although this generalization is occasionally criticized for its shortcomings, equation (2.1) is considered today to be a fundamental principle in analysis of porous media flows (e.g., McWhorter and Sunada, 1977). If, indeed, Darcy’s experimental result is the birth of quantitative hydrology, a need still remains to build quantitative analysis of porous media flow on a strong theoretical foundation. The problem of unsaturated flow of water has been attacked using experimental and theoretical tools since the early part of this century. Sposito (1986) attributes the beginnings of the study of soil water flow as a subdiscipline of physics to the fundamental work of Buckingham (1907), which uses a saturation-dependent hydraulic conductivity and a capillary potential for the hydraulic gradient.

On steady convection in a porous medium

1972 ◽  
Vol 54 (1) ◽  
pp. 153-161 ◽  
Author(s):  
Enok Palm ◽  
Jan Erik Weber ◽  
Oddmund Kvernvold
Keyword(s):  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Abrupt Change ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Sixth Order ◽  
Steady Convection ◽  
Good Agreement

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.

scholarly journals Décomposition de domaine pour un milieu poreux fracturé : un modèle en 3D avec fractures

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Laila Amir ◽  
Michel Kern ◽  
Jean E. Roberts ◽  
Vincent MARTIN
Keyword(s):  
Porous Medium ◽  
Domain Decomposition ◽  
Single Phase ◽  
Decomposition Methods ◽  
Interface Problem ◽  
Domain Decomposition Methods ◽  
Standard Interface ◽  
International Audience ◽  
Milieu Poreux ◽  
Phase Fluid

International audience In this paper, we are interested in modeling the flow of a single phase fluid in a porous medium with fractures, using domain decomposition methods. In the proposed approach, the fracture is regarded as an active interface, the transmission conditions and the exchanges between the rock and the fracture taking into account the flow in the fracture. The problem to be solved is then a non standard interface problem which takes into account the flow in the fractures.

Darcy's Law for a Heterogeneous Porous Medium

2001 ◽  
Vol 4 (2) ◽  
pp. 14 ◽  
Author(s):  
F. D. Moura Neto ◽  
S. T. Melo
Keyword(s):  
Porous Medium ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Heterogeneous Porous Medium

Gas Separation Using a Membrane

2014 ◽  
Author(s):  
Nawaf Alkhamis ◽  
Ali Anqi ◽  
Dennis E. Oztekin ◽  
Abdulmohsen Alsaiari ◽  
Alparslan Oztekin
Keyword(s):  
Porous Medium ◽  
Natural Gas ◽  
Gas Separation ◽  
Stokes Equations ◽  
Darcy’S Law ◽  
Circular Cross Section ◽  
Darcy's Law ◽  
Mass Separation ◽  
Functional Surface ◽  
Wide Range

Gas-gas separation, to purify natural gas, is simulated using a membrane supported by a porous medium. Removing acidic gasses from the natural gas is gaining attention recently. Computational fluid dynamics simulations are conducted for asymmetric multi-component fluid flows in a channel. The flow system consists of a circular cross-section channel bounded by a porous layer which supports the membrane wall. The Navier-Stokes equations model the flow in the channel, while the flow in the porous medium is modeled by both the Darcy’s law and the extended Darcy’s law. Mass transport equations, including mass diffusion of mixtures of two gasses (CO2 and CH4), are employed to determine the concentration distribution. The membrane will be modeled as a functional surface; where the flux of each component will be determined based on the local partial pressure of each species, composition, and permeability and selectivity of the membrane. The effect of the porous medium on the membrane performance will be determined for a wide range of Reynolds number. The performance of the system will be measured by maximum mass separation with minimal frictional losses.

Numerical modelling of nonlinear effects in laminar flow through a porous medium

1988 ◽  
Vol 190 ◽  
pp. 393-407 ◽  
Author(s):  
O. Coulaud ◽  
P. Morel ◽  
J. P. Caltagirone
Keyword(s):  
Porous Medium ◽  
Pressure Drop ◽  
Nonlinear Term ◽  
Numerical Solutions ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Quadratic Term ◽  
Inertial Effects ◽  
Operator Splitting Methods ◽  
Flow Through

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

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