Carol Braester

Effective and relative permeabilities of anisotropie porous media

A model composed of a three-dimensional orthogonal network of capillary tubes was used to simulate the flow behavior in an unsaturated anisotropic soil. The anisotropy in the networks permeability was introduced by randomly selecting the radii in the three mutually orthogonal directions of the network tubes from three different lognormal probability distributions, one for each direction. These three directions were assumed to be the principal directions of anisotropy. The sample was gradually drained, with only tubes smaller than a certain diameter remaining full at each degree of saturation. Computer experiments were conducted to determine the networks effective permeability as a function of saturation. The main conclusion was that the relationship between saturation and effective permeability depends on direction. Consequently the concept of relative permeability used in unsaturated flow should be limited to isotropic media and not extended to anisotropic ones.

Determination of formation permeability by double-packer tests

Abstract The double-packer test is a classical field method for in situ determination of formation permeability. The formulae used in this calculation are based on the assumption that the formation is continuous, homogeneous and isotropic, and that the borehole is cased (impervious) except for the region of injection between the packers. These ideal conditions are rarely realized in practice, and the present paper deals with the error which thus in the calculated permeability is included, for an inhomogeneous, anisotropic or discontinuous formation, as well as for an uncased (pervious) borehole. The investigation takes the form of a numerical simulation of double-packer tests in hypothetical formations of a priori known permeability, with the rate of flow and the water overpressure at quasi steady state measured as in a real packer test. This information is subsequently used as input for calculating the permeability in the idealized model, and the calculated permeabilities are compared with the actual ones. It has been shown that for homogeneous and isotropic formations, Dagans, Hvorslevs and Moyes formulae yield similar results, cased and uncased boreholes show practically the same permeability value, and a low permeability around the borehole accounts for a significant distortion of the representative value for the formation as a whole. For an anisotropic formation with horizontal vertical principal axes of anisotropy, the calculated results more or less coincide with the horizontal value.

Denitrification in laboratory sand columns: Carbon regime, gas accumulation and hydraulic properties

Microbiological denitrification in a sandy matrix was studied by means of laboratory sand columns operated at continuous and pulse feed regimes. Gas production resulting from the biological activity played a major role in modifying the hydraulic properties of the column, leading to decreases in hydraulic conductivity and porosity, higher water velocities through the column, higher dispersion and anomalies in the head difference to flow rates ratios. All of these effects were more pronounced when formate, the carbon source used, was supplied continuously: microbial activity and gas production were concentrated at the top of the column, leading to almost complete clogging. When the formate was supplied in pulses, activity and gas production were dispersed, leading to relative uniformity in the physical parameters measured and a homogeneous appearance of the column. The results suggest that in a futurein situ aquifer denitrification plant, pulse application of the carbon source is prefereable to a continuous supply regime.

The effect of a weak heterogeneity of a porous medium on natural convection

The results of an investigation on the effect of a weak heterogeneity of a porous medium on natural convection are presented. A medium heterogeneity is represented by spatial variations of the permeability and of the effective thermal conductivity. As a general rule the existence of horizontal thermal gradients in heterogeneous porous media provides a sufficient condition for the occurrence of natural convection. The implications of this condition are investigated for horizontal layers or rectangular domains subject to isothermal top and bottom boundary conditions. Results lead to a restriction on the classes of thermal conductivity functions which allow a motionless solution. Analytical solutions for rectangular weak heterogeneous porous domains heated from below, consistent with a basic motionless solution, are obtained by applying the weak nonlinear theory. The amplitude of the convection is obtained from an ordinary non-homogeneous differential equation, with a forcing term representative of the medium heterogeneity with respect to the effective thermal conductivity. A smooth transition through the critical Rayleigh number is obtained, thus removing a bifurcation which usually appears in homogeneous domains with perfect boundaries, at the critical value of the Rayleigh number. Within a certain range of slightly supercritical Rayleigh numbers, a symmetric thermal conductivity function is shown to reinforce a symmetrical flow while antisymmetric functions favour an antisymmetric flow. Except for the higher-order solutions, the weak heterogeneity with respect to permeability plays a relatively passive role and does not affect the solutions at the leading order. In contrast, the weak heterogeneity with respect to the effective thermal conductivity does have a significant effect on the resulting flow pattern.

Dispersion in Sub-Representative Elementary Volume Fracture Networks: Percolation Theory and Random Walk Approaches

A law for dispersion in fracture networks below the representative elementary volume (REV) is established by analyzing random walks in two-dimensional fracture networks in conjunction with percolation theory. Irregular fracture networks near the percolation threshold were obtained by removing some of the fractures of a regular orthogonal network, consisting of fractures of equal length and different apertures, drawn randomly from a lognormal distribution. The random walk was directed by an exact solution of flow through the network, and Monte Carlo simulations were performed to track particles through the fracture system. The percolation theory analysis indicates a proportionality between the mean square displacement and time raised to the power 1.27, in excellent agreement with the simulations in the fracture networks, which indicate a proportionality with time raised to the power 1.3.

On the Flow of Two Immscible Fluids in Fractured Porous Media

Publisher Summary The chapter discusses on the flow of two immiscible fluids in fractured porous media. There may be three orders of inhomogeneity in a porous medium. The first is because of inhomogeneity in the granular structure of the rock; it results from the pore- or grain-size distribution, the variability in grainshapes, and so on. The second because of the presence of macrojoints, small tectonic fissures, vugs, and solution channels which form a network of interconnected channels. The third occurs because of the Inhomogeneity that results from the presence of distinct layers or other forms of zoning of different media (for example, the different permeability). The chapter deals with those cases in which inhomogeneity of the second type is present and is distributed more or less uniformly throughout the porous medium domain. The chapter discusses those cases where the system of fractures produces “porous blocks” completely surrounded by fractures. Such a medium is well represented by a rock containing joints at a more or less uniform spacing. The chapter presents to state the equations and analyze the medium parameters which describe immiscible displacement in a fractured porous medium.

On the movement of an LNAPL lens on the water table

The penetration of light nonaqueous phase liquids (LNAPLs) in quantities that lead to an accumulation in the form of a lens above the water table is considered. First, the three-phase vertical gravity-capillary equilibrium of water, NAPL, and air above the water table is specified. The hypothesis of ‘vertical equilibrium phase distribution’ is used to derive averaged asymptotic equations describing NAPL flow as a thin lens floating above the water table. Some problems of unsteady NAPL lens movement and the development of a NAPL mound, spreading along an inclined or horizontal phreatic surface are discussed and the analytical solutions are obtained.

Solute transport in fracture channel and parallel plate models

Many studies of flow and solute transport in fractured rocks are based on a conceptual model whereby flow occurs between parallel plates which approximate the fracture walls. Recently, it has been observed that this representation is not consistent with actual fractures. Real fractures are actually composed of a complex system of void spaces of varying aperture, and contact areas which are closed to flow. In such fractures, flow takes place through a network of channels and dead-end regions. The present investigation analyzes and compares the behavior of solute breakthrough curves in these channels and parallel plate representations. Numerical experinents show that breakthrough curves obtained with channel models are characterized by a long tail and jumps in the solute concentration, consistent with those observed experimentally. It is concluded that the considered channel models provide a sound explanation for the behavior of real breakthrough curves, which cannot be reproduced by parallel plate models.

The effect of perfectly conducting side walls on natural convection in porous media

Abstract An investigation of natural convection in a porous medium heated from below or above, and bounded by perfectly conducting side walls, shows that a motionless solution is impossible, except for a particular side wall temperature variation. Hence, convection occurs regardless of the value of Rayleigh number and regardless of whether the fluid is heated from below or from above. Numerical solutions for identical uniform temperatures imposed on both side walls (no temperature difference between the side walls) show that when heating from below, a subcritical flow results mainly near the side walls, which amplifies and extends over the entire domain under supercritical conditions.

Pressure transient response of stochastically heterogeneous fractured reservoirs

A stochastic model for flow through inhomogeneous fractured reservoirs of double porosity, based on Barenblattet al.s continuum approach, is presented. The fractured formation is conceptualized as an interconnected fracture network surrounding porous blocks, and amenable to the continuum approach. The block permeability is negligible in comparison to that of the fractures, and therefore the reservoir permeability is represented by the permeability of the fracture network. The fractured reservoir inhomogeneity is attributed to the fracture network, while the blocks are considered homogeneous. The mathematical model is represented by a coupled system of partial differential random equations, and a general solution for the average and for the correlation moments of the fracture pressure are obtained by the Neumann expansion (or Adomian decomposition). The solution for pressure is represented by an infinite series and an approximate solution for radial flow, is obtained by retaining the first two terms of the series. The purpose of this investigation is to get an insight on the pressure behavior in inhomogeneous fractured reservoirs and not to obtain type curves for determination of reservoir properties, which owing to the nonuniqueness of the solution, is impossible. For the present analysis we assumed an ideal reservoir with cylindrical symmetric inhomogeneity around the well. Fractured rock reservoirs being practically inhomogeneous, it is of interest to compare the pressure behavior of such reservoirs, with Warren and Rootss solution for (ideal) homogeneous reservoirs, used as a routine for determining the fractured reservoir characteristic parametersλ andΩ, using the results of well tests. The comparison of the results show that inhomogeneous and homogeneous reservoirs exhibit a similar pressure behavior. While the behavior is identical, the same drawdown or a build-up pressure curve may be fitted by different characteristic dimensionless parametersλ andΩ, when attributed to an inhomogeneous or a homogeneous reservoir. It is concluded that the ambiguity in determining the fractured reservoirλ andΩ, makes questionable the usefulness of determination of these parameters. Computations were also carried out to determine the correlation between the fracture pressure at the well and the fracture pressure at different points.