Flow in porous media I: A theoretical derivation of Darcy's law

Abstract. Current theories for water flow in porous media are valid for scales much smaller than those at which problem of public interest manifest themselves. This provides a drive for upscaled flow equations with their associated upscaled parameters. Upscaling is often achieved through volume averaging, but the solution to the resulting closure problem imposes severe restrictions to the flow conditions that limit the practical applicability. Here, the derivation of a closed expression of the effective hydraulic conductivity is forfeited to circumvent the closure problem. Thus, more limited but practical results can be derived. At the Representative Elementary Volume scale and larger scales, the gravitational potential and fluid pressure are treated as additive potentials. The necessary requirement that the superposition be maintained across scales is combined with conservation of energy during volume integration to establish consistent upscaling equations for the various heads. The power of these upscaling equations is demonstrated by the derivation of upscaled water content-matric head relationships and the resolution of an apparent paradox reported in the literature that is shown to have arisen from a violation of the superposition principle. Applying the upscaling procedure to Darcy's Law leads to the general definition of an upscaled hydraulic conductivity. By examining this definition in detail for porous media with different degrees of heterogeneity, a series of criteria is derived that must be satisfied for Darcy's Law to remain valid at a larger scale.

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Study on the Generalized Darcy's Law for Bingham and Herschel-Bulkley Fluids

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.433-435.1933 ◽

2013 ◽

Vol 433-435 ◽

pp. 1933-1936

Author(s):

Jing Wen Cui ◽

Zhi Shang Liu ◽

Yu Chen Zhang

Keyword(s):

Porous Media ◽

Polymer Solution ◽

Heavy Oil ◽

Darcy’S Law ◽

Darcy's Law ◽

Flow In Porous Media ◽

Type Model ◽

Drilling Fluids ◽

Flow Models ◽

Type Flow

Extra-heavy oil, polymer solution and some drilling fluids are typical non-Newtonian Herschel-Bulkley fluids, which behave as sheer-thinning with yield stress. In this paper, the Generalized Darcy's law for Herschel-Bulkley fluids flow in porous media was formulated, by the same way formulating the Generalized Darcy's Law for Bingham fluids. Then, the applications of the two type flow models were compared; Bingham type model was still widely applied due to its conciseness and relatively satisfied accuracy. In addition, the Generalized Darcys Law was revised to describe thixotropic non-Newtonian fluids conceptually.

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An Investigation of the Flow Regime for Hele-Shaw Flow

Society of Petroleum Engineers Journal ◽

10.2118/2140-pa ◽

1969 ◽

Vol 9 (04) ◽

pp. 434-442

Author(s):

R.C. Smith ◽

R.A. Greenkorn

Keyword(s):

Reynolds Number ◽

Porous Media ◽

Darcy’S Law ◽

Darcy's Law ◽

Creeping Flow ◽

Flow In Porous Media ◽

Two Dimensional ◽

Ideal Flow ◽

Plate Spacing ◽

Perturbation Pressure

Abstract Hele-Shaw cells are used to model creeping flow through porous media (where Darcy's law is valid). The effects of inertia on flow about obstructions in a Hele-Shaw cell can be calculated by a perturbation method if one can determine a solution to Laplace's equation. Results of a computer solution for flow about circular, square and elliptical obstructions are presented These results show that for a modified presented These results show that for a modified Reynolds number of less than 1, the inertia terms are small; and for values of less than 3, the average streamline predicts the ideal flow. Therefore, the analogy might be used for studying flow in porous media up to a modified Reynolds number of at least 3. Introduction The nature of fluid flow in porous media is of interest in the fields of soil mechanics, ground water flow, petroleum production, filtration and flow, in packed beds. Because it is very difficult to study the phenomenological behavior of flow in porous media, homologs and analogs are used to study flow characteristics. A Hele-Shaw model, made of two closely spaced plates - usually glass - is often used as an analogy to two-dimensional flow in porous media. Hele-Shaw showed experimentally that the streamline configuration for creeping flow around an obstacle located between two closely spaced parallel plates is the same as for two-dimensional parallel plates is the same as for two-dimensional ideal flow about the same obstacle. Stokes verified these observations mathematically. The usual equation of motion for flow in porous media is Darcy's law. The form of the mathematical statement of Darcy's law is identical, within a multiplicative constant, to the expression for the average velocity over the place gap in the plane of a Hele-Shaw model. These models may be used to describe flow in both homogeneous and heterogeneous porous media. In the mathematical proof of the Hele-Shaw analogy it is assumed that the convective terms in the Navier-Stokes equations are negligible and that the equations of motion degenerate to Laplace's equation, with pressure the dependent variable. Whenever a Hele-Shaw model is used as an analogy to flow in porous media, the validity of this assumption is in question. Riegels showed that if convection is not neglected, the velocity distribution around a cylindrical obstruction in the flow field depends on a Reynolds number, the plate spacing, and a dimension characteristic of the obstacle. Riegels solution, a perturbation solution, uses the boundary condition that the flow rate into the obstacle averaged over the plate gap at any point on the obstacle is zero. The method requires that a solution to Poisson's equation for the perturbation pressure be found. Riegels evaluated this solution pressure be found. Riegels evaluated this solution for the case of the cylindrical obstruction. His method may be simplified by eliminating the need for solving Poisson's equation for the perturbation pressure. Instead, an analytic expression for the pressure. Instead, an analytic expression for the perturbation pressure gradient is obtained (valid perturbation pressure gradient is obtained (valid for arbitrary shapes) and used to eliminate pressure from the equations for the perturbation velocities. The results show, for symmetrical shapes, that if N'Re less than 1, the convective acceleration terms are small, and that the average velocities represent ideal flow up m at least N'Re 3, where: ..........................................(1) L is a characteristic dimension of the obstacle perpendicular to flow, b is the plate spacing, mu is perpendicular to flow, b is the plate spacing, mu is viscosity, va is velocity of approach and p is density. SPEJ P. 434

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