Showing papers in "Transport in Porous Media in 2001"

Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests

Abstract: The macrodispersion experiments (MADE) at the Columbus Air Force Base in Missis-sippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Levy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (α), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order α = 1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamin-ation event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.

Role of Molecular Diffusion in Contaminant Migration and Recovery in an Alluvial Aquifer System

Abstract: Highly-resolved simulations and flow and transport in an alluvial system at the Lawrence Livermore National Laboratory (LLNL) site explore the role of diffusion in the migration and recovery of a conservative solute. Heterogeneity is resolved to the hydrofacies scale with a discretization of 10.0, 5.0 and 0.5m in the strike, dip and vertical directions of the alluvial-fan system. Transport simulations rely on recently developed random-walk techniques that accurately account for local dispersion processes at interfaces between materials with contrasting hydraulic and transport properties. Solute migration and recovery by pump and treat are shown to be highly sensitive to magnitude of effective diffusion coefficient. Further, transport appears significantly more sensitive to the diffusion coefficient than to local-scale dispersion processes represented by a dispersivity coefficient. Predicted hold back of solute mass near source locations during ambient migration and pump-and-treat remediation is consistent with observations at LLNL, and reminiscent of observations at the MADE site of Columbus Air Force Base, Mississippi. Results confirm the important role of diffusion in low-conductivity materials and, consequently, its impact on efficacy of pump-and-treat and other remedial technologies. In a typical alluvial system on a decadal time scale this process is, in part, fundamentally nonreversible because the average thickness of low-K hydrofacies is considerably greater than the mean-square length of penetration of the solute plume.

Salt water intrusion in a three-dimensional groundwater system in The Netherlands: a numerical study

Abstract: Salt water intrusion is investigated in a coastal groundwater system in the northern part of the province Noord-Holland, The Netherlands. Density dependent groundwater flow is modeled in three-dimensions with MOCDENS3D. This computer code is a version of MOC3D (Konikow et al., 1996) that has been adapted to simulate transient density-driven groundwater flow. Results from the model suggests that in this Dutch hydrogeologic system a severe and irreversible salinisation is already occurring. Within a few tens to hundreds of years, the salinity of the shallow aquifer is estimated to increase substantially. This salinisation process is a result of human activities such as the reclamation of the low-lying areas during the past centuries. Without changing the present boundary conditions, seepage into the low-lying areas will decrease slightly because of predicted increases in groundwater salinity. However, the rate in salt load through the Holocene aquitard into the low-lying areas will increase significantly due to an increase in salinity in the shallow aquifer. In addition, a relative sea level rise of 0.5 m per century will intensify the salinisation process, causing an enormous increase in salt load in all low-lying areas in this part of The Netherlands.

A Theoretical Model of Hysteresis and Dynamic Effects in the Capillary Relation for Two-phase Flow in Porous Media

Abstract: It is well known that the relationship between capillary pressure and saturation, in two- phase flow problems demonstrates memory effects and, in particular, hysteresis. Explicit represent- ation of full hysteresis with a myriad of scanning curves in models of multiphase flow has been a difficult problem. A second complication relates to the fact that P c -S relationships, determined under static conditions, are not necessarily valid in dynamics. There exist P c -S relationships which take into account dynamic effects. But the combination of hysteretic and dynamic effects in the capillary relationship has not been considered yet. In this paper, we have developed new models of capillary hysteresis which also include dynamic effects. In doing so, thermodynamic considerations are employed to ensure the admissibility of the new relationships. The simplest model is constructed around main imbibition and drainage curves and assumes that all scanning curves are vertical lines. The dynamic effect is taken into account by introducing a damping coefficient in P c -S equation. A second-order model of hysteresis with inclined scanning curves is also developed. The simplest version of proposed models is applied to two-phase incompressible flow and an example problem is solved.

Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media

Abstract: A smoothed particle hydrodynamics (SPH) model is presented for the study of diffusion in spatially periodic porous media. The method of SPH is formulated to solve the convection–diffusion equation for tracer diffusion under steady state and transient conditions. Solutions obtained using SPH are compared with other available solutions and the model is used to calculate diffusion coefficients of spatially periodic porous media for the steady state diffusion problem. Diffusion coefficients are then used to calculate nondimensional diffusivities of the media. The effects of media properties on the values of nondimensional diffusivity are also presented.

Modeling and Analysis of Seawater Intrusion in the Coastal Aquifer of Eastern Cap-Bon, Tunisia

Abstract: A numerical model that treats density-dependent variably saturated flow and miscible salt transport is used to investigate the occurrence of seawater intrusion in the ‘Korba’ aquifer of the eastern coast of Cap-Bon in northern Tunisia. We examine the interplay between pumping regimes and recharge scenarios and its effect on the saline water distribution. More localized simulations are used to examine, in vertical cross sections, the effects of well location and soil type and the role of the vadose zone in possible remediation actions. The exploratory simulations suggest interesting interactions between the unsaturated zone and the saltwater–freshwater interface with possible implications for groundwater exploitation from shallow unconfined coastal aquifers, involving in one case feedback between seawater intrusion and the high pressure head gradients around the pumping-induced drawdown cone and in another case threshold-like interface displacement for tight soils such as clays. The data processing steps undertaken in this GIS and modeling study are described in some detail, and a critical assessment is given of the data availability and of the requirements for successful monitoring and modeling of seawater intrusion risks in heavily exploited coastal aquifers such as those found in the semi-arid regions of the Mediterranean basin. It is shown how, with the aid of GIS, reasonably reliable information can be assembled from maps, surveys, and other sources of geospatial and hydrogeological data, an approach that is necessary in the many regions of the world with acute water resource problems but with limited means for undertaking systematic data acquisition and environmental monitoring actions. Nonetheless the need for more concerted monitoring of relevant parameters and processes and of closer coordination between monitoring and modeling is stressed. An idea of the extent of over-exploitation of the Korba aquifer is obtained by examining the pumping and rainfall/infiltration data, and the simulation results support groundwater pumping as the mechanism for and seawater intrusion as the origin of the salt contamination observed in the soils and subsurface waters of the Korba plain.

Derivation of the Forchheimer Law via Homogenization

Abstract: In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcy's law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.

Application of the Carman–Kozeny Correlation to a High‐Porosity and Anisotropic Consolidated Medium: The Compressed Expanded Natural Graphite

Abstract: The Carman–Kozeny correlation is applied to a medium which is consolidated, highly porous and anisotropic: the expanded then compressed natural graphite. The effective textural properties (i.e. the mean pore diameter, porosity and tortuosity) have been measured by a mercury porosimeter and a heterogeneous diffusion cell. The texture and the permeability (according to the Darcy's law) measured for the two main directions of these orthotropic porous media change over a very wide range depending on their apparent mass densities. Experimental data show that only a part of the total porosity participates in the gas flow in steady state conditions.

The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media

Abstract: A physical picture of contaminant transport in highly heterogeneous porous media is presented. In any specific formation the associated governing transport equation is valid at any time and space scale. Furthermore, the advective and dispersive contributions are inextricably combined. The ensemble average of the basic transport equation is equivalent to a continuous time random walk (CTRW). The connection between the CTRW transport equation, in a limiting case and the familiar advection-dispersion equation (ADE) is derived. The CTRW theory is applied to the results of laboratory experiments, field observations, and simulations of random fracture networks. All of these results manifest dominant non-Gaussian features in the transport, over different scales, which are accounted for quantitatively by the theory. The key parameter β controlling the entire shape of the contaminant plume evolution and breakthrough curves is advanced as a more useful characterization of the transport than the dispersion tensor, which is based on moments of the plume. The role of probabilistic approaches, such as CTRW, is appraised in the context of the interplay of spatial scales and levels of uncertainty. We then discuss a hybrid approach, which uses knowledge of non-stationary aspects of a field site on a larger spatial scale (trends) with a probabilistic treatment of unresolved structure on a smaller scale (residues).

Mitigation of Seawater Intrusion by Pumping Brackish Water

Abstract: This paper presents a technique to restore the balance between freshwater and saline water in coastal aquifers in order to mitigate seawater intrusion problems. Brackish water can be pumped from the dispersion zone and then used to develop green lands in the coastal areas or to irrigate certain types of crops. A two-dimensional finite element model (2D-FED), has been employed to verify this technique. The model is based on the dispersion zone approach with a variable density flow. Simulations were preformed in the vertical view and equiconcentration and equipotential lines were plotted for different locations of brackish water pumping. In all of the tested runs the width of dispersion zone has reduced significantly due to brackish water pumping. The quality of the pumped water differs according to the location of pumping. A study case on the Madras aquifer in India is presented.

A Highly Conductive Porous Medium for Solid–Gas Reactions: Effect of the Dispersed Phase on the Thermal Tortuosity

Abstract: The heat transfer in a highly conductive material constituted by a graphite matrix in which a granular phase is dispersed is studied. The effective thermal conductivity of this anisotropic porous composite medium used in solid–gas reactors can vary largely with the component fractions. The effect of the dispersed grains on the deformable structure of the matrix is considered. A model developed on the basis of thermal tortuosity by analogy with mass transfer is adequately correlated with experimental results.

Radial Flow in a Bounded Randomly Heterogeneous Aquifer

Abstract: Flow to wells in nonuniform geologic formations is of central interest to hydrogeologists and petroleum engineers. There are, however, very few mathematical analyses of such flow. We present analytical expressions for leading statistical moments of vertically averaged hydraulic head and flux under steady-state flow to a well that pumps water from a bounded, randomly heterogeneous aquifer. Like in the widely used Thiem equation, we prescribe a constant pumping rate deterministically at the well and a constant head at a circular outer boundary of radius L. We model the natural logarithm Y = lnT of aquifer transmissivity T as a statistically homogeneous random field with a Gaussian spatial correlation function. Our solution is based on exact nonlocal moment equations for multidimensional steady state flow in bounded, randomly heterogeneous porous media. Perturbation of these nonlocal equations leads to a system of local recursive moment equations that we solve analytically to second order in the standard deviation of Y. In contrast to most stochastic analyses of flow, which require that log transmissivity be multivariate Gaussian, our solution is free of any distributional assumptions. It yields expected values of head and flux, and the variance–covariance of these quantities, as functions of distance from the well. It also yields an apparent transmissivity, T a, defined as the negative ratio between expected flux and head gradient at any radial distance. The solution is supported by numerical Monte Carlo simulations, which demonstrate that it is applicable to strongly heterogeneous aquifers, characterized by large values of log transmissivity variance. The two-dimensional nature of our solution renders it useful for relatively thin aquifers in which vertical heterogeneity tends to be of minor concern relative to that in the horizontal plane. It also applies to thicker aquifers when information about their vertical heterogeneity is lacking, as is commonly the case when measurements of head and flow rate are done in wells that penetrate much of the aquifer thickness. Potential uses include the analysis of pumping tests and tracer test conducted in such wells, the statistical delineation of their respective capture zones, and the analysis of contaminant transport toward fully penetrating wells.

Estimation of Macrodispersion by Different Approximation Methods for Flow and Transport in Randomly Heterogeneous Media

Abstract: We present two- and three-dimensional calculations for the longitudinal and transverse macrodispersion coefficient for conservative solutes derived by particle tracking in a velocity field which is based on the linearized flow equation. The simulations were performed upto 5000 correlation lengths in order to reach the asymptotic regime. We used a simulation method which does not need any grid and therefore allows simulations of very large transport times and distances. Our findings are compared with results obtained from linearized transport, from Corrsin's Conjecture and from renormalization group methods. All calculations are performed with and without local dispersion. The variance of the logarithm of the hydraulic conductivity field was chosen to be one to investigate realistic model cases. While in two dimensions the linear transport approximation seems to be very good even for this high variance of the logarithmic hydraulic conductivity, in three dimensions renormalization group results are closer to the numerical calculations. Here Dagan's theory and the theory of Gelhar and Axness underestimate the transverse macrodispersion by far. Corrsin's Conjecture always overestimates the transverse dispersion. Local dispersion does not significantly influence the asymptotic behavior of the various approximations examined for two-dimensional and three-dimensional calculations.

General Anisotropic Effective Medium Theory for the Effective Permeability of Heterogeneous Reservoirs

Abstract: One of the techniques to calculate the effective property of a heterogeneous medium is the effective medium theory. The present paper presents a general mathematical formulation for the effective medium approximation using a self-consistent choice of the effective permeability, to apply it to the case of a general anisotropic 2D medium and to the case of a 3D isotropic medium with randomly oriented ellipsoidal inclusions. The 2D results are compared with analytical results and with a homogenization technique with good result. The 3D correlations are used to derive percolation thresholds in two-phase systems with a large permeability contrast, which are compared to numerical results from the literature, also with good results.

Laminar flow through irregularly-shaped pores in sedimentary rocks

Abstract: Steady-state, laminar flow of an incompressible fluid through prismatic tubes of irregular but constant cross-section is investigated. Several approximations for the hydraulic conductance (Saint-Venant, Aissen, hydraulic radius), some of which were originally proposed for the mathematically analogous problem of torsion of a prismatic elastic bar, are examined and tested for regular geometric shapes for which analytical solutions exist. For such shapes, the Saint-Venant and Aissen approximations are typically within 15% of the exact conductance, whereas the hydraulic radius approximation may be in error by as much as 50%. Conformal mapping and the boundary element method are then used to study the hydraulic conductance of sandstone pores from SEM images of Berea and Massilon sandstone. For these irregular shapes, the hydraulic radius approximation is much more accurate than either the Saint-Venant or Aissen approximation. Moreover, the errors in the hydraulic radius approximation may be of either sign, and thereby partially cancel out when large numbers of pores are considered, whereas the other two methods tend always to overestimate the hydraulic conductance of rock pores.

Sand Erosion in Axial Flow Conditions

Abstract: A mathematical model of sand erosion in axial flow conditions is presented. The basic mass balance equations and sand erosion constitutive equation were given in Vardoulakis et al. (1996). As opposed to reference Vardoulakis et al. (1996), we consider here the extreme case where convection is null and hydrodynamic dispersion dominates. In addition, Brinkman's extension of Darcy's law is adopted to account for a smooth transition between channel flow and Darcian flow. The set of governing PDE's is presented in dimensionless form and is solved numerically. In concordance with the basic constitutive equation for erosion kinetics, the analysis shows that erosion progresses in time as a ‘front’ of high transport concentration. This result is justified by the highly non-linear character of the erosion source term which dominates in the diffusion-like governing equation.

Three-Phase Capillary Pressure and Relative Permeability Relationships in Mixed-Wet Systems

Abstract: A simple process-based model of three-phase displacement cycles for both spreading and non-spreading oils in a mixed-wet capillary bundle model is presented. All possible pore filling sequences are determined analytically and it is found that the number of pore occupancies that are permitted on physical grounds is actually quite restricted. For typical non-spreading gas/oil/water systems, only two important cases need to be considered to see all types of allowed qualitative behaviour for non-spreading oils. These two cases correspond to whether water or gas is the ‘intermediate-wetting’ phase in oil-wet pores as determined by the corresponding contact angles, that is, cos θogw > 0 or cos θogw < 0, respectively. Analysis of the derived pore occupancies leads to the establishment of a number of relationships showing the phase dependencies of three-phase capillary pressures and relative permeabilities in mixed-wet systems. It is shown that different relationships hold in different regions of the ternary diagram and the morphology of these regions is discussed in terms of various rock/fluid properties. Up to three distinct phase-dependency regions may appear for a non-spreading oil and this reduces to two for a spreading oil. In each region, we find that only one phase may be specified as being the ‘intermediate-wetting’ phase and it is only the relative permeability of this phase and the capillary pressure between the two remaining phases that depend upon more than one saturation. Given the simplicity of the model, a remarkable variety of behaviour is predicted. Moreover, the emergent three-phase saturation-dependency regions developed in this paper should prove useful in: (a) guiding improved empirical approaches of how two-phase data should be combined to obtain the corresponding three-phase capillary pressures and relative permeabilities; and (b) determining particular displacement sequences that require additional investigation using a more complete process-based 3D pore-scale network model.

Effects of Clay Dispersion on Aquifer Storage and Recovery in Coastal Aquifers

Abstract: Cyclic injection, storage, and withdrawal of freshwater in brackish aquifers is a form of aquifer storage and recovery (ASR) that can beneficially supplement water supplies in coastal areas. A 1970s field experiment in Norfolk, Virginia, showed that clay dispersion in the unconsolidated sedimentary aquifer occurred because of cation exchange on clay minerals as freshwater displaced brackish formation water. Migration of interstitial clay particles clogged pores, reduced permeability, and decreased recovery efficiency, but a calcium preflush was found to reduce clay dispersion and lead to a higher recovery efficiency. Column experiments were performed in this study to quantify the relations between permeability changes and clay mineralogy, clay content, and initial water salinity. The results of these experiments indicate that dispersion of montmorillonite clay is a primary contributor to formation damage. The reduction in permeability by clay dispersion may be expressed as a linear function of chloride content. Incorporating these simple functions into a radial, cross-sectional, variable-density, ground-water flow and transport model yielded a satisfactory simulation of the Norfolk field test – and represented an improvement over the model that ignored changes in permeability. This type of model offers a useful planning and design tool for ASR operations in coastal clastic aquifer systems.

A Direct Determination of the Transient Exchange Term of Fractured Media Using a Continuous Time Random Walk Method

Abstract: In two recent papers, Nœtinger and Estebenet, 2000; Nœtinger et al., submitted, we set-up a method allowing to compute both the transient and steady-state exchange terms between the matrix and fractured regions of a naturally fractured porous medium using continuous time random walk methods (CTRW). The goal of the present paper is to show that a new version of the CTRW algorithm provides a direct determination of the so called transient exchange function f(t) (or its Laplace transform f(s)) widely used in well test interpretation. It is shown that this function is directly linked with the probability density of the first escape time in the fractured region of a Brownian particle launched initially in the matrix region. This new interpretation allows relating directly the exchange coefficient α∞ with the mean escape time of brownian particles in the matrix. From a practical point of view, these new results allow to derive a simpler version of the CTRW method. In addition, we obtain a considerable speed up of the CTRW method for up-scaling fractured reservoirs.

Numerical investigations of the steady state relative permeability of a simplified porous medium

Abstract: The purpose of this paper is to investigate, by flow simulations in a uniform pore-space geometry, how the co and countercurrent steady state relative permeabilities depend on the following parameters: phase saturation, wettability, driving force and viscosity ratio The main results are as follows: (i) with few exceptions, relative permeabilities are convex functions of saturation; (ii) the cocurrent relative permeabilities increase while the countercurrent ones decrease with the driving force; (iii) with one exception, phase 2 relative permeabilities decrease and phase 1 relative permeabilities increase with the viscosity ratio M = μ1/μ2; (iv) the countercurrent relative permeabilities are always less than the cocurrent ones, the difference being partly due to the opposing effect of the viscous coupling, and partly to different levels of capillary forces; (v) the pore-level saturation distribution, and hence the size of the viscous coupling, can be very different between the cocurrent and the countercurrent cases so that it is in general incorrect to estimate the full mobility tensor from cocurrent and countercurrent steady state experiments, as suggested by Bentsen and Manai (1993)

A Numerical Study of the Distribution of Water in Partially Saturated Porous Rock

Abstract: The distribution of water and air phases in small blocks of porous sandstone is examined by using a simulated annealing technique that finds the minimum interfacial energy distributions at different saturations. Simulations are based on existing sandstone microstructures that were determined by X-ray microtomography. At low saturations, some of the water is distributed in films along the walls of larger pore spaces, and connects to pendular structures in the crevices and smaller pores. As the amount of water in the pores increases the water films become thicker and pores fill from the pendular structures. The distribution of water voxels in the pore space is examined by calculating interfacial areas, by classifying water voxels as to whether they lie within films or clusters, and by determining the size and distribution of these film clusters. An exponential relationship is found between the fraction of water voxels in the films and the degree of saturation. In addition, the dependency of small-sample electrical conductivity on saturation is examined by using a random walk method.

Modeling Biogeochemical Processes in Leachate-Contaminated Soils: A Review

Abstract: During subsurface transport, reactive solutes are subject to a variety of hydrological, physical and biochemical processes. The major hydrological and physical processes include advection, diffusion and hydrodynamic dispersion, and key biochemical processes are aqueous complexation, precipitation/dissolution, adsorption/desorption, microbial reactions, and redox transformations. The addition of strongly reduced landfill leachate to an aquifer may lead to the development of different redox environments depending on factors such as the redox capacities and reactivities of the reduced and oxidised compounds in the leachate and the aquifer. The prevailing redox environment is key to understanding the fate of pollutants in the aquifer. The local hydrogeologic conditions such as hydraulic conductivity, ion exchange capacity, and buffering capacity of the soil are also important in assessing the potential for groundwater pollution. Attenuating processes such as bacterial growth and metal precipitation, which alter soil characteristics, must be considered to correctly assess environmental impact. A multicomponent reactive solute transport model coupled to kinetic biodegradation and precipitation/dissolution model, and geochemical equilibrium model can be used to assess the impact of contaminants leaking from landfills on groundwater quality. The fluid flow model can also be coupled to the transport model to simulate the clogging of soils using a relationship between permeability and change in soil porosity. This paper discusses the different biogeochemical processes occurring in leachate-contaminated soils and the modeling of the transport and fate of organic and inorganic contaminants under such conditions.

Steady-State Source Flow in Heterogeneous Porous Media

Abstract: The flow of fluids in heterogeneous porous media is modelled by regarding the hydraulic conductivity as a stationary random space function. The flow variables, the pressure head and velocity field are random functions as well and we are interested primarily in calculating their mean values. The latter had been intensively studied in the past for flows uniform in the average. It has been shown that the average Darcy's law, which relates the mean pressure head gradient to the mean velocity, is given by a local linear relationship. As a result, the mean head and velocity satisfy the local flow equations in a fictitious homogeneous medium of effective conductivity. However, recent analysis has shown that for nonuniform flows the effective Darcy's law is determined by a nonlocal relationship of a convolution type. Hence, the average flow equations for the mean head are expressed as a linear integro-differential operator. Due to the linearity of the problem, it is useful to derive the mean head distribution for a flow by a source of unit discharge. This distribution represents a fundamental solution of the average flow equations and is called the mean Green function G d (x). The mean head G d(x) is derived here at first order in the logconductivity variance for an arbitrary correlation function ρ(x) and for any dimensionality d of the flow. It is obtained as a product of the solution G d (0)(x) for source flow in unbounded domain of the mean conductivity K A and the correction Ψ d (x) which depends on the medium heterogeneous structure. The correction Ψ d is evaluated for a few cases of interest. Simple one-quadrature expressions of Ψ d are derived for isotropic two- and three-dimensional media. The quadratures can be calculated analytically after specifying ρ (x) and closed form expressions are derived for exponential and Gaussian correlations. The flow toward a source in a three-dimensional heterogeneous medium of axisymmetric anisotropy is studied in detail by deriving Ψ3 as function of the distance from the source x and of the azimuthal angle θ. Its dependence on x, on the particular ρ(x) and on the anisotropy ratio is illustrated in the plane of isotropy (θ=0) and along the anisotropy axis (θ = π/2). The head factor k * is defined as a ratio of the head in the homogeneous medium to the mean head, k *=G d (0)/G d=Ψ d −1. It is shown that for isotropic conductivity and for any dimensionality of the flow the medium behaves as a one-dimensional and as an effective one close and far from the source, respectively, that is, lim x→0 k *(x) = K H/K A and lim x→∞ k *(x) = K efu/K A, where K A and K H are the arithmetic and harmonic conductivity means and K efu is the effective conductivity for uniform flow. For axisymmetric heterogeneity the far-distance limit depends on the direction. Thus, in the coordinate system of ρ(x) principal directions the limit values of k * are obtained as $$\begin{gathered} \tilde H_{{\text{cont}}}^ * (\mathcal{A},\mathcal{A}) = H(K_{\text{h}}^{{\text{efu}}} ) \subset _{{\text{diff}}}^ * (\mathcal{A},\mathcal{A}) = \mathcal{V}^ * ,{\text{ where }}\mathcal{A}\mathcal{U}\mathcal{R}q(\mathfrak{g})\mathfrak{k}\mathcal{U}_q (gl(2N + 1))\mathcal{R}\mathcal{B}(N) \subset \mathcal{U}_q \mathcal{F} \hfill \\ (K_{\text{h}}^{{\text{efu}}} K_{\text{v}}^{{\text{efu}}} )^{1/2} /K_{\text{A}} {\text{ for }}\theta = 0{\text{ and }}K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{for }}\theta = \pi /2 \hfill \\ \end{gathered} $$ . These values differ from the corresponding components $$K_{\text{h}}^{{\text{efu}}} /K_{\text{A}} {\text{ and }}K_{\text{v}}^{{\text{efu}}} /K_{\text{A}} $$ of the effective conductivities tensor for uniform flow for θ = 0 and π/2, respectively. The results of the study are applied to solving the problem of the dipole well flow. The dependence of the mean head drop between the injection and production chambers on the anisotropy of the conductivity and the distance between the chambers is analyzed.

Application of a Soil Water Hysteresis Model to Simple Water Retention Curves

Abstract: The Parlange hysteresis model is reformulated as a pair of recurrence relations to provide relationships between wetting and drying phases to any order. The model is applied to the classical van Genuchten model for soil water retention used as the main wetting curve. The nonphysical behaviour of these retention curves is related to the existence of a point of inflection in the van Genuchten model when it is used for the wetting boundary. Where the van Genuchten form is used as the main drying curve, the Parlange hysteresis model provides an ordinary differential equation describing the main wetting curve. A number of simple analytical solutions, relating to particular values of the parameters of the van Genuchten model, then provide forms for the main wetting curve. In general, a numerical integration is required to generate the main wetting curve, for general values of the parameters of the van Genuchten model. The recurrence relations for the hysteresis cycling are still applicable, even when the main wetting curve is only known numerically. The new main wetting curves do not have inflection points and there is no nonphysical behaviour. The model is then applied to the experimental data of Viaene et al. (1994)

Dispersion in Heterogeneous Porous Media: One-Equation Non-equilibrium Model

Abstract: In this paper, the method of large-scale averaging is used to develop two different one-equation models describing dispersion in heterogeneous porous media. The first model represents the case of large-scale mass equilibrium, while the second represents the asymptotic behavior of a two-equation model obtained by large-scale averaging. It is shown that a one-equation, non-equilibrium model can be developed even when the intrinsic large-scale averaged concentrations for each region are not equal. The solution of this non-equilibrium model is equivalent to the asymptotic behavior of the two-equation model.

The Utilization of Electrokinetics in Geotechnical and Environmental Engineering

Abstract: Electrokinetics can be utilized to solve many problems in geotechnical and environmental engineering. The processes which occur in association with electrokinetics are complicated and difficult to control. The success of the application depends on certain conditions which are controlled by many parameters. It is therefore important to understand these processes so that methods and procedures can be optimized. In this paper theoretical considerations, results of tests and examples of in situ applications are presented and discussed. The presented examples show ways of controlling and improving the efficiency of electrokinetic processes.

Permeability of self-affine fractures

Abstract: The permeability of self-affine fractures with various roughness exponents H is investigated by direct three-dimensional numerical simulations. A scaling behavior with an exponent H is exhibited in the self-affine scale range. Permeability can be related to the fractional open area and to the percolation probability by simple models.

Hydraulic Behavior and Contaminant Transport in Multiple Porosity Media

Abstract: The hydraulic behavior and contaminant transport of aquifers containing distinct families of fractures are investigated by using the multiple porosity continuum model. We consider that the conditions are such that a horizontal 2D flow takes place. By writing the continuity of mass (including exchange terms between the various families of fractures) and Darcy's law for each family of fractures, macroscopic equations for both confined and unconfined flow are obtained. A classification procedure and geometrical idealization of the individual fractures for each family is proposed which enables the calculation of the exchange coefficients. Equations for the description of the contaminant transport in the field scale for both confined and unconfined aquifer are developed. It turns out that the adequate formulation of the macroscopic equations and their sink-source term depends on whether the aquifer investigated is confined or unconfined, and also on the value of an nondimensional parameter describing the transfer process at the microscopic scale (connection Peclet number). Numerical investigation of representative problems offers some insight into the behavior of double and triple porosity aquifers.

Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium

Abstract: The classical mass balance equation is usually used to model the transfer of humidity in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hy- draulic conductivity are defined as functions of geometry and local characteristics of the porous medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second medium (the mortar), on the global behavior of the system is investigated. Numerical results for the Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of the model in relation to the effective hydrodynamic parameters of the porous structure is presented.

Coupled diffusion-dissolution around a fracture channel : The solute congestion phenomenon

Abstract: The paper studies the coupled diffusion-dissolution process in reactive porous media, separated by a fracture channel. An aggressive solute, corresponding for e.g., to a complete demineralization that dissolves the solid skeleton of the surrounding porous material, is prescribed at the inlet of the fracture. By means of asymptotic dimensional analysis it is shown that for large times the diffusion length in the fracture develops with the quadratic root of time. In comparison with the 1D-Stefan Problem, in which the dissolution front evolves with the square root of time, this indicates that the overall solute evacuation through the structure slows down in time. This phenomenon is referred to as a diffusive solute congestion in the fracture. This asymptotic behavior is confirmed by means of model-based simulation, and the relevant material parameters, related to only the chemical equilibrium condition, are identified. The obtained results suggest that the presence of a small crack does not significantly increase the propagation of the dissolution front in the porous bulk, and hence the overall chemical degradation of the porous material. The same applies to other diffusion induced demineralization, mineralization, sorption and melting processes, provided that the convective transport of the solute in the crack is small in comparison with the solute diffusion. The result is relevant for several problems in durability mechanics: calcium leaching of concrete in nuclear waste containment, mineralization and demineralization in bone remodeling, chloride penetration, etc.