In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by 
$$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$

 The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

The Forchheimer equation : A theoretical development

Publisher Summary The chapter examines the process of transient heat conduction for a two-phase system in terms of the method of volume averaging. Both one- and two-equation models are developed along with the relevant closure problems that allow determining theoretically the effective transport coefficients. Closure calculations are often carried out using symmetric unit cells. The chapter describes the numerical method used to solve the closure problems and presents selected results for stratified systems and uniform arrays of cylinders. Analytical results are given for stratified systems and a finite-volume numerical method is used for the nodular systems. Numerical results are provided for a two-dimensional array of cylinders embedded in a continuous phase. The chapter attempts to compare results from the two-equation model for transient heat conduction in a two-phase system with results from the one-equation model. Both models are exact in the sense that the effective transport coefficients are determined theoretically and this allows to draw some definite conclusions concerning the assumption of local thermal equilibrium.

One- and Two-Equation Models for Transient Diffusion Processes in Two-Phase Systems

In this paper we develop the averaged form of the Stokes equations in terms of weighting functions. The analysis clearly indicates at what point one must choose a media-specific weighting function in order to achieve spatially smoothed transport equations. The form of the weighting function that produces the cellular average is derived, and some important geometrical theorems are presented.

Transport in ordered and disordered porous media. II: Generalized volume averaging

Transport in ordered and disordered porous media: volume-averaged equations, closure problems, and comparison with experiment

[1] A numerical model based on smoothed particle hydrodynamics (SPH) was used to simulate reactive transport and mineral precipitation in porous and fractured porous media. The stability and numerical accuracy of the SPH-based model was verified by comparing its results with analytical results and finite element numerical solutions. The numerical stability of the model was also verified by performing simulations with different time steps and different number of particles (different resolutions). The model was used to study the effects of the Damkohler and Peclet numbers and pore-scale heterogeneity on reactive transport and the character of mineral precipitation and to estimate effective reaction coefficients and mass transfer coefficients. Depending on the combination of Damkohler and Peclet numbers the precipitation may be uniform throughout the porous domain or concentrated mainly at the boundaries where the solute is injected and along preferential flow paths. The effective reaction rate coefficient and mass transfer coefficient exhibited hysteretic behavior during the precipitation process as a result of changing pore geometry and solute distribution. The changes in porosity and fluid fluxes resulting from mineral precipitation were also investigated. It was found that the reduction in the fluid flux increases with increasing Damkohler number for any particular reduction in the porosity. The simulation results show that the SPH, Lagrangian particle method is an effective tool for studying pore-scale flow and transport. The particle nature of SPH models allows complex physical processes such as diffusion, reaction, and mineral precipitation to be modeled with relative ease.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2005WR004770

A smoothed particle hydrodynamics model for reactive transport and mineral precipitation in porous and fractured porous media

In a previous derivation of Darcy's law, the closure problem was presented in terms of an integro-differential equation for a second-order tensor. In this paper, we show that the closure problem can be transformed to a set of Stokes-like equations and we compare solutions of these equations with experimental data. The computational advantages of the transformed closure problem are considerable.

On the Closure Problem for Darcy's Law

While a pair of N × N matrices can almost always be exactly and simultaneously diagonalized by a generalized eigendecomposition, no exact solution exists in the case of a set with more than two matrices. This problem, termed approximate joint diagonalization (AJD), is instrumental in blind signal processing. When the set of matrices to be jointly diagonalized includes at least N linearly independent matrices, we propose a suboptimal but closed-form solution for AJD in the direct least-squares sense. The corresponding non-iterative algorithm is given the acronym DIEM (DIagonalization using Equivalent Matrices). Extensive numerical simulations show that DIEM is both fast and accurate compared to the state-of-the-art iterative AJD algorithms.

A Direct Algorithm for Nonorthogonal Approximate Joint Diagonalization

—This paper addresses the problem of mobile target detection in multipath scenarios with a passive radar using DVB-T transmitters of opportunity. For such emissions, it has been shown the interest in implementing " mismatched " correlators, reducing both the zero Doppler contribution (ZDC) masking effects and the false alarm rate. A very efficient mismatched reference signal is obtained with the reciprocal filter (or inverse filter) which consist in a modulus frequential equalization of the transmitted signal. We propose here to revisit the reciprocal filter based correlator and to reinterpret it as a so-called Doppler channel detector (CHAD). This new interpretation allows a direct rejection of the ZDC, unifying in one and the same step the main disturbance mitigation and the detector construction. We provide a statistical theoretical study of the performance and a comparison with the matched correlator i.e. the classical cross-ambiguity function (CAF). We demonstrate that CHAD has a random pedestal (a clutter floor level) significantly lower than that of the classical CAF for low Doppler frequency shifts. Numerical experiments on simulated and real data as well validate the mathematical derivations. Index Terms—Passive bistatic radar (PBR), passive covert radar (PCR), passive coherent location (PCL), digital video broadcasting-terrestrial (DVB-T), orthogonal frequency division multiplexing (OFDM), cross-ambiguity function (CAF), mismatched filter, reciprocal filter, clutter.

/pdf/a-unifying-approach-for-disturbance-cancellation-and-target-2t52r85j9f.pdf

A Unifying Approach for Disturbance Cancellation and Target Detection in Passive Radar Using OFDM

Increasing demand of wireless devices contributes to radiofrequency (RF) congestion. Light Fidelity (LiFi) promises to be an interesting alternative by using the visible part of the electromagnetic spectrum instead of the RF part as nearly all existing wireless transmission systems do. A basic LiFi system is composed of one intensity-controlled light-emitting diode (LED) and one receiver device sensitive to very high-frequency (thus invisible to human sight) modulations of the luminous intensity. In most cases, the photoreceptor is a silicon photodiode of PIN (P-type intrinsic N-type) or APD (Avalanche photodiode) conception. Recently, a few studies suggest that photovoltaic (PV) modules could be used to implement outdoor LiFi transmissions, i.e., under direct sunlight exposure. In this paper, we propose to compare the behavior of a PV module and a commercial APD-based photodetector (without any optical lens or colored filter) for experimental LiFi transmissions on both indoor and outdoor conditions. The performance of the two solutions is quantified in terms of various frequency responses like attenuation, signal-to-noise ratio, or bit-error rate. The results show that, while the photodiode exhibits very good performance in indoor conditions, its frequency response is rapidly deteriorating when a sunlight exposure of more than 200 W/m $^2$ is superimposed over the LiFi signal. We demonstrate that a PV module in $V_{oc}$ (open-circuit voltage) condition still operates a LiFi transmission under additional solar illumination.

/pdf/photovoltaic-solar-cells-for-outdoor-lifi-communications-wefygv1w3q.pdf

Photovoltaic Solar Cells for Outdoor LiFi Communications

On etudie le passage d'ecoulements microscopiques a l'echelle du pore, regis par les equations de stokes et de navier-stokes, aux ecoulements macroscopiques dans un milieu poreux, regis par la loi de darcy. Les principaux points d'etudes sont: le passage en revue des theories de prise de moyenne et l'etablissement de l'equivalence de celles-ci et la theorie d'homogeneisation dans le cas de milieux periodiques, la determination numerique par une methode aux differences finies, de tenseurs de permeabilite dans des milieux periodiques anisotropes tridimensionnels, l'etude numerique, par une methode aux elements finis, des non-linearites en regime de navier-stokes dans un treillis de cylindres. La loi macroscopique d'ecoulement obtenue fait intervenir une expression cubique de la vitesse de filtration

Jean Barrere

Papers

On the Closure Problem for Darcy's Law

A Direct Algorithm for Nonorthogonal Approximate Joint Diagonalization

A Unifying Approach for Disturbance Cancellation and Target Detection in Passive Radar Using OFDM

Photovoltaic Solar Cells for Outdoor LiFi Communications

Modélisation des écoulements de Stokes et Navier-Stokes en milieux poreux