Flow through isotropic granular porous media

A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation.     

Mathematical modelling of flow through consolidated isotropic porous media

A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.     

Macroscopic modelling of transport phenomena in porous media. 1: The continuum approach

This is the first of two papers presenting a systematic development of a continuum model of a porous medium and of transport processes occurring in it. The concept of a Representative Elementary Volume (REV) as opposed to any arbitrary volume of averaging quantities at the micro-scale, is quantified. A universal criterion for selecting the size of an REV as a function of measurable characteristics of a porous medium and selected tolerance levels of estimation errors, is developed. The rules of spatial averaging are extended by including the effects of both the configuration of the solid matrix and of interphase transfer phenomena within an REV.     

On the spatial-temporal averaging method for modeling transport in porous media

The spatial-temporal averaging procedure is considered with a nonhomogeneous distribution of elementary domains in the spatial-temporal space and the probabilistic interpretation of the ST-averaging is also given. Several averaging theorems and corollaries about the averages of spatial and temporal derivatives are presented and rigorously proved which allow elementary domain to vary in space and time. The macroscopic transport equation in the most general condition and the simplified macroscopic equation under the special form of distributions are developed which may be reduced to the classical macroscopic transport equation as the spatial-temporal average degenerates into the volume average.     

A scale-dependent equation for solute transport in porous media

The governing equation describing solute transport in porous media is reformulated using standard volume averaging techniques. The alternative formulation is based on a modified definition of the deviation, which allows for variation of macroscopic velocity across the REV. The new equation contains additional scale-dependent terms which are functions of the size of the averaging volume (REV). This result indicates that the scale-dependent nature of the dispersion phenomenon is inherent even at the scale of the REV.     

Magnetic Resonance Imaging of Hydrodynamic Dispersion in a Saturated Porous Medium

By using nuclear magnetic resonance imaging (NMRI) we have been able to analyse dispersion at the microscopic scale during steady-state flow through water-saturated glass beads. The flow rate through the porous medium was chosen high enough in order to neglect the influence of molecular diffusion on dispersion. Velocity statistics were measured, by NMRI, within slices of increasing thickness perpendicular to the direction of flow. It took more than two bead diameters before a representative elementary volume (REV) for the mean velocity was reached. This was in a region in the middle of the column that was not influenced by the boundary conditions. There the velocity variance decreased exponentially as a function of the slice thickness, due we consider to the formation of an interconnecting streamline network. The exponential decrease in the velocity variance reflects the transition from a local pattern of stochastic–convective flow to a convective–dispersion regime at the scale of the REV. We found that the point-like preferential influx and efflux boundary condition increased velocity variances and thus enhanced longitudinal hydrodynamic dispersion. Using the transverse correlation length of longitudinal velocity variance, we derived a mean transverse dispersivity that agreed well with Saffmans (1959) model. So we have been able to provide for the first time a direct observation verification of a part of Saffmans (1959) conjectures. By NMRI we observed this value to be independent of the observation scale of the slice thickness.     

Free Flow at the Interface of Porous Surfaces: A Generalization of the Taylor Brush Configuration

A solution to the problem of shallow laminar water flow above a porous surface is essential when modeling phenomena such as erosion, resuspension, and mass transfer between the porous media and the flow above it. Previous studies proposed theoretical, experimental, and numerical insight with no single general solution to the problem. Many studies have used the Brinkman equation, while others showed that it does not represent the actual interface flow conditions. In this paper we show that the interface macroscopic velocity can be accurately modeled by introducing a modification to the Brinkman equation. A moving average approach was proved to be successful when choosing the correct representative elementary volume and comparing the macroscopic solution with the average microscopic flow. As the size of the representative elementary volume was found to be equal to the product of the square root of the permeability and an exponential function of the porosity, a general solution is now available for any brush configuration. Given the properties of the porous media (porosity and permeability), the flow height and its driving force, a complete macroscopic solution of the interface flow is obtained.     

The use of a data envelopment analysis efficient rule to produce a steady and high cooperation frequency in a spatial evolvement game

We use the data envelopment analysis (DEA) method to estimate the relative efficiency of the current strategy for each player in a game by taking the spatial distribution of strategies as input and the total payoff as output. Based on the optimal value of the DEA model, we present a DEA efficient rule to update the strategy in evolutionary games. Simulations of the prisoner’s dilemma game (PDG) [4] and the snowdrift game (SG) on two-dimensional regular lattices of four, six, and eight neighbors with periodic boundary conditions, are carried out; the results show the emergence of high and stable cooperator frequency. The heuristic analysis of the DEA efficient rule are discussed in detail. Our work may be helpful in exploring the promotion of cooperator behavior.     

STRENGTH PROPERTIES OF JOINTED ROCK MASSES BASED ON THE HOMOGENIZATION METHOD

This paper aims to determine the strength properties of jointed rock masses by means of the homogenization method.To reflect the microstructure of jointed rock masses,a representative element volume (R...     

Effects of the Correlation Length on the Hydraulic Parameters of a Fracture Network

We consider the influences of correlation length and aperture variability on the REV, the equivalent permeability of a fracture network, and the uncertainty in the equivalent permeability using a two-dimensional orthogonal bond percolation model. The percolation threshold, correlation length, effective conductivity, and coefficient of variation of the effective conductivity are investigated over statistically representative multiple realizations with Monte Carlo simulations in 2D fracture networks that have log-normally distributed individual fracture permeabilities. We show that although the aperture variability is large, the REV and the correlation length are similar near the percolation threshold. In contrast, when the fracture density is much larger than the percolation threshold they diverge as the aperture variability increases. We characterize the effects of correlation length and aperture variability on effective conductivity with a simple function. From the coefficient of variation analysis, the correlation length can be a criterion for evaluating which conceptual model is appropriate for describing the flow system for a given fracture network when aperture variability is sufficiently small. However, discrete fracture network models are recommended for flow simulation models because of the difficulty of REV estimation and the uncertainty in equivalent hydraulic parameters when aperture variability is large.