Experimental Determination of Hydraulic Conductivity of Pine Samples in the Longitudinal Direction During Convective Drying

In studying the process of drying wood samples, the hydraulic conductivity of pine samples exposed to a convective air flow was measured in the direction perpendicular to the cross-sectional plane of the tree (along the wood grain). The kinetic curves of the drying process were treated with a technique based on the approximate solution of the one-dimensional diffusion equation for hydraulic conductivity of wood with a boundary condition of the third kind. The technique was tested on the basis of the known value of hydraulic conductivity in the direction tangential to annual rings of the tree. It is shown that the hydraulic conductivity in the longitudinal direction is 17 times the hydraulic conductivity in the direction tangential to the annual rings in the cross-sectional plane of the tree. By means of numerical simulation of the process, based on solving the initial-boundary problem for the two-dimensional linear equation of hydraulic conductivity, the effect of anisotropy of hydraulic conductivity coefficients on the dependence of the mean humidity on time and on the local humidity distribution is studied.     

Percolation Cluster Statistics and Conductivity Semi-Variograms

Cluster statistics of percolation theory have been shown to generate expressions for the distribution of hydraulic conductivity values in accord with field studies. Percolation theory yields directly the smallest possible generalized resistance value, R _c, for which a continuous path through an infinite heterogeneous system can avoid all larger resistances. R _c, defines an infinite system hydraulic conductivity. Cluster statistics generate the number of clusters of resistors of a given size with a given R, for which a continuous path through the cluster can avoid resistances larger than R. The probability that a volume of size x ³ falls on a particular cluster gives the probability that volume has a characteristic resistance, R. Determining the semi-variogram of the hydraulic conductivity is now elementary; it is necessary only to determine whether translation h of the center of the volume x ³ removes it from the cluster in question. If the cluster is larger than (x+h)³, then, on the average, the same cluster resistance R will control K. Otherwise, the value of K at x+h will be uncorrelated with its value at x. The condition is then expressed as an integral related to the one, which gives the distribution of K. Then an integral over the derived distribution of K gives the variogram. Results obtained are that the variogram should be similar to either the exponential or Gaussian forms typically in use, if K is a power law function of random variables (as in Poiseuilles Law), or more closely related to the spherical approximation if K is an exponential function of random variables.     

Effective conductivities of thin-interphase composites

A method is presented for approximating the effective conductivity of composite media with thin interphase regions, which is exact to first order in the interphase thickness. The approximations are computationally efficient in the sense the fields need to be computed only in a reference composite in which the interphases have been replaced by perfect interfaces. The results apply whether any two phases of the composite are separated by a single interphase or multiple interphases, whether the conductivities of the composite phases are isotropic or anisotropic, and whether the thickness of an interphase is uniform or varies as a function of position. It is assumed that the conductivities of the interphase materials have intermediate values as opposed to very high or very low conductivities.     

Effective hydraulic conductivities in unsaturated heterogeneous media by Monte Carlo simulation

When modeling flow and transport through unsaturated heterogeneous geological deposits, it may be neither computationally nor technically feasible to account for the actual heterogeneity in the simulations. One would fall short in terms of technical feasibility because there is simply no way that the entire spatial domain could be characterized (e.g., you cannot measure hydraulic conductivity at every location at a site). With respect to computational feasibility, the non-linear nature of the Richards equation (which is used to model the flow process) makes simulation of most sites extremely computationally intensive. The computational roadblock is being dismantled as computer hardware advances, but our inability to precisely characterize geological heterogeneity is expected to remain with us for a very long time. To address this problem, the analyst typically uses average or effective properties to model flow and transport behavior through heterogeneous media. In this paper, a variety of approaches for developing effective unsaturated flow properties are assessed. Computational results have been obtained which give the hydraulic conductivity ratios (K _parallel/K _nomal) for highly nonisotropic layered materials. These results are compared with analytical models. Good agreement was obtained for all soil saturation levels except for extremely dry conditions.This work was performed at Sandia National Laboratories, which is operated for the U.S. Department of Energy under contract number DE-AC04-76DP00789.     

Relation Between Non-Steady Supply Pressure and Flux for a Double Packer Conductivity Meter: An Approximate Analytical Solution

Measurement of the hydraulic coefficients of soil or rock is carried out with the aid of double packer conductivity meters in wells or boreholes. The evaluation of tests is based on the theory of creeping flow through a porous medium. The theory leads to a non-steady cylindrically and axially symmetric heat conduction problem in an infinity domain outside a circular hole with mixed boundary conditions. With the aid of a simple but efficient approximation, which describes the singularity of the boundary value, two of the three integrations in the solution can be evaluated in closed form. The analysis shows that the problem can be reduced to a simple Volterra integral equation, which is easily solved with numerical means. The solution is characterized by one single parameter, namely the aspect ratio of the cylindrical inner cavity.     

Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory

The strong-contrast formulation is used to predict the effective conductivity of a porous material. The distribution, shape and orientation of the two phases are taken into account using two- and three-point probability distribution functions. A new approximation for the three-point probability function appropriate for two-phase media is proposed and discussed. Computed results for the effective conductivity using the strong-contrast formulation are compared to the Voigt and the Hashin-Shtrikman upper-bound estimates. These results show that the predicted effective conductivity is lower than both Voigt and Hashin-Shtrikman bounds. Compared to previous results using the weak-contrast formulation, the strong-contrast formulation seems to provide a better estimate for the effect of the microstructure on the conductivity.     

Hydraulic radius and transport in reconstructed model three-dimensional porous media

Methods for reconstructing three-dimensional porous media from two-dimensional cross sections are evaluated in terms of the transport properties of the reconstructed systems. Two-dimensional slices are selected at random from model three-dimensional microstructures, based on penetrable spheres, and processed to create a reconstructed representation of the original system. Permeability, conductivity, and a critial pore diameter are computed for the original and reconstructed microstructures to assess the validity of the reconstruction technique. A surface curvature algorithm is utilized to further modify the reconstructed systems by matching the hydraulic radius of the reconstructed three-dimensional system to that of the two-dimensional slice. While having only minor effects on conductivity, this modification significantly improves the agreement between permeabilities and critical diameters of the original and reconstructed systems for porosities in the range of 25–40%. For lower porosities, critical pore diameter is unaffected by the curvature modification so that little improvement between original and reconstructed permeabilities is obtained by matching hydraulic radii.     

Using the Sequential Self-calibration Method and Genetic Algorithm Method to Optimally Design Tracer Test to Estimate Conductivity Distribution

A gradient-based inverse method – the sequential self-calibrated method – is combined with a genetic algorithm method to search the optimal design scheme for a tracer test. The sequential self-calibrated method is developed for estimating conductivity distribution in a study domain conditioning on tracer test data. To improve the calculation efficiency, a fast streamline-based approach is used to compute the derivative of concentration with respect to the changes of hydraulic conductivity. Performance of the sequential self-calibrated method has been studied using a synthetic aquifer having a sandwich-like geologic structure where hypothetical tracer tests are conducted. The study results indicate that the locations and number of sampling wells will significantly affect accuracy in the estimates. To maximize estimating accuracy in the sequential self-calibrated method for a fixed number of sampling wells, a genetic algorithm method is applied to search the optimal locations for sampling wells. The results indicate that the optimal sampling well locations depend on the apparent geologic structure and the difference in conductivity values for the various regions. For the sandwich-like structure, when the difference between conductivity values in the two separate regions is large enough, the optimal locations for the sampling wells will be fixed, regardless of conductivity values. The study results also show that based on the optimal sampling-well scheme, estimating accuracy will increase as the number of sampling wells increases, even though the rate of increasing accuracy slows as the number of wells increases.     

Influence of Microbial Growth on Hydraulic Properties of Pore Networks

From laboratory experiments it is known that bacterial biomass is able to influence the hydraulic properties of saturated porous media, an effect called bioclogging. To interpret the observations of these experiments and to predict possible bioclogging effects on the field scale it is necessary to use transport models, which are able to include bioclogging. For these models it is necessary to know the relation between the amount of biomass and the hydraulic conductivity of the porous medium. Usually these relations were determined using bundles of parallel pore channels and do not account for interconnections between the pores in more than one dimension. The present study uses two-dimensional pore network models to study the effects of bioclogging on the pore scale. Numerical simulations were done for two different scenarios of the growth of biomass in the pores. Scenario 1 assumes microbial growth in discrete colonies clogging particular pores completely. Scenario 2 assumes microbial growth as a biofilm growing on the wall of each pore. In both scenarios the hydraulic conductivity was reduced by at least two orders of magnitude, but for the colony scenario much less biomass was needed to get a maximal clogging effect and a better agreement with previously published experimental data could be found. For both scenarios it was shown that heterogeneous pore networks could be clogged with less biomass than more homogeneous ones.     

Accuracy of the Renormalization Method for Computing Effective Conductivities of Heterogeneous Media

The accuracy of the renormalization method for upscaling two-dimensional hydraulic conductivity fields is investigated, using two canonical 2 × 2 blocks: a checkerboard geometry and a geometry in which three of the cells have conductivity K ₁ and the other has conductivity K ₂. The predictions of the renormalization algorithm are compared to the arithmetic, harmonic and geometric means, as well as to theoretical predictions and finite element calculations. For the latter geometry renormalization works well over the entire range of the conductivity ratio K ₂/K ₁, but for the checkerboard geometry the error becomes unbounded as the conductivity ratio grows.     

Electrical Conductivity and Percolation Aspects of Statistically Homogeneous Porous Media

A method of 3-D stochastic reconstruction of porous media based on statistical information extracted from 2-D sections is evaluated with reference to the steady transport of electric current. Model microstructures conforming to measured and simulated pore space autocorrelation functions are generated and the formation factor is systematically determined by random walk simulation as a function of porosity and correlation length. Computed formation factors are found to depend on correlation length only for small values of this parameter. This finding is explained by considering the general percolation behavior of a statistically homogeneous system. For porosities lower than about 0.2, the dependence of formation factor on porosity shows marked deviations from Archie's law. This behavior results from the relatively high pore space percolation threshold (0.09) of the simulated media and suggests a limitation to the applicability of the method to low porosity media. It is additionally demonstrated that the distribution of secondary porosity at a larger scale can be simulated using stochastic methods. Computations of the formation factor are performed for model media with a matrix-vuggy structure as a function of the amount and spatial distribution of vuggy porosity and matrix conductivity. These results are shown to be consistent with limited available experimental data for carbonate rocks.     

A Markov Chain Model for Effective Relative Permeabilities and Capillary Pressure

We present a new method for calculating the effective two-phase parameters of one-dimensional randomly heterogeneous porous media, which avoids the timeconsuming use of simulations on explicit realizations. The procedure is based on the steady state saturation distribution. The idea is to model the local variation of saturation and saturation dependent parameters as Markov chains, in such a way that the effective parameters are given by the asymptotic expectations of the chains. We derive the exact asymptotic moment equations and solve them numerically, based on their second order approximation. The method determines the effective parameters to a high degree of accuracy, even with large variations in rock properties. In particular, the capillary limit and viscous limit effective parameters are recovered exactly. The applicability of the effective parameters in the unsteady state case is studied by comparing the displacement production profiles in heterogeneous media and their homogenized counterpart.     

Macroscopic Conductivity of Vugular Porous Media

A systematic numerical study of the macroscopic electrical conductivity of vugular porous media has been conducted by solving the Laplace equation. The structure of these bimodal media is characterized by the micro and macroporosities and by the micro and macro correlation lengths. The overall correlation function is analyzed. It is shown that mainly depends on the total porosity, and very little on the other parameters. Moreover, similar results are obtained when the microporous medium is considered as a continuum. Finally, these predictions are compared to experimental data for the formation factor and the thermal conductivity of limestones; the agreement is good in the first case, and excellent in the second.     

A micromechanical model for effective conductivity in granular electrode structures

Optimization of composition and microstructure is important to enhance performance of solid oxide fuel cells （SOFC） and lithium-ion batteries （LIB）. For this, the porous electrode structures of both SOFC and LIB are modeled as a binary mixture of electronic and ionic conducting particles to estimate effective transport properties. Particle packings of 10000 spherical, binary sized and randomly positioned particles are created numerically and densified considering the different manufacturing processes in SOFC and LIB： the sintering of SOFC electrodes is approximated geometrically, whereas the calendering process and volume change due to intercalation in LIB are modeled physically by a discrete el- ement approach. A combination of a tracking algorithm and a resistor network approach is developed to predict the con- nectivity and effective conductivity for the various densified structures. For SOFC, a systematic study of the influence of morphology on connectivity and conductivity is performed on a large number of assemblies with different compositions and particle size ratios between 1 and 10. In comparison to percolation theory, an enlarged percolation area is found, es- pecially for large size ratios. It is shown that in contrast to former studies the percolation threshold correlates to varying coordination numbers. The effective conductivity shows not only an increase with volume fraction as expected but also with size ratio. For LIB, a general increase of conductivity during the intercalation process was observed in correlation with increasing contact forces. The positive influence of cal- endering on the percolation threshold and the effective conductivity of carbon black is shown. The anisotropy caused by the calendering process does not influence the carbon black phase.     

Effective permeability of cracked unsaturated porous materials

This study focuses on a theoretical estimation of the effective permeability of unsaturated cracked porous media. The closed-form flow solution around and in a superconductive crack, embedded in an infinite porous matrix under a far-field condition, is recalled first. Then the solution of flow around a completely unsaturated (empty) crack that is considered as an obstruction against the flow is determined. The flow solution for partially saturated crack in special configurations is obtained by superposition of the two basic solutions for superconductive and empty cracks. The contribution of an unsaturated crack, with a given saturation degree, to the effective permeability is estimated by using dilute upscaling scheme. Numerical results obtained by Finite Elements Method, are in good agreement with the theoretical results for weak crack densities but show the additional effect of cracks interaction for higher densities.     

Effective Flux Boundary Conditions for Upscaling Porous Media Equations

We introduce a new algorithm for setting pressure boundary conditions in subgrid simulations of porous media flow. The algorithm approximates the flux in the boundary cell as the flux through a homogeneous inclusion in a homogeneous background, where the permeability of the inclusion is given by the cell permeability and the permeability of the background is given by the ambient effective permeability. With this approximation, the flux in the boundary cell scales with the cell permeability when that permeability is small, and saturates at a constant value when that permeability is large. The flux conditions provide Neumann boundary conditions for the subgrid pressure. We call these boundary conditions effective flux boundary conditions (EFBCs). We give solutions for the flux through ellipsoidal inclusions in two and three dimensions, assuming symmetric tensor permeabilities whose principal axes align with the axes of the ellipse. We then discuss the considerations involved in applying these equations to scale up problems in geological porous media. The key complications are heterogeneity, fluctuations at all length scales, and boundary conditions at finite scales.     

Hydraulic conductivity of rock fractures

The flow of a single-phase fluid through a rough-walled rock fracture is discussed within the context of fluid mechanics. The derivation of the cubic law is given as the solution to the Navier-Stokes equations for flow between smooth, parallel plates - the only fracture geometry that is amenable to exact treatment. The various geometric and kinematic conditions that are necessary in order for the Navier-Stokes equations to be replaced by the more tractable lubrication or Hele-Shaw equations are studied and quantified. In general, this requires a sufficiently low flow rate, and some restrictions on the spatial rate of change of the aperture profile. Various analytical and numerical results are reviewed pertaining to the problem of relating the effective hydraulic aperture to the statistics of the aperture distribution. These studies all lead to the conclusion that the effective hydraulic aperture is less than the mean aperture, by a factor that depends on the ratio of the mean value of the aperture to its standard deviation. The tortuosity effect caused by regions where the rock walls are in contact with each other is studied using the Hele-Shaw equations, leading to a simple correction factor that depends on the area fraction occupied by the contact regions. Finally, the predicted hydraulic apertures are compared to measured values for eight data sets from the literature for which aperture and conductivity data were available on the same fracture. It is found that reasonably accurate predictions of hydraulic conductivity can be made based solely on the first two moments of the aperture distribution function, and the proportion of contact area.     

Percolation approach to the problem of hydraulic conductivity in porous media

While percolation theory has been studied extensively in the field of physics, and the literature devoted to the subject is vast, little use of its results has been made to date in the field of hydrology. In the present study, we carry out Monte Carlo computer simulations on a percolating model representative of a porous medium. The model considers intersecting conducting permeable spheres (or circles, in two dimensions), which are randomly distributed in space. Three cases are considered: (1) All intersections have the same hydraulic conductivity, (2) The individual hydraulic conductivities are drawn from a lognormal distribution, and (3) The hydraulic conductivities are determined by the degree of overlap of the intersecting spheres. It is found that the critical behaviour of the hydraulic conductivity of the system,K, follows a power-law dependence defined byK (N/N _c–1)^x, whereN is the total number of spheres in the domain,N _c is the critical number of spheres for the onset of percolation, andx is an exponent which depends on the dimensionality and the case. All three cases yield a value ofx1.2±0.1 in the two-dimensional system, whilex1.9±0.1 is found in the three-dimensional system for only the first two cases. In the third case,x2.3±0.1. These results are in agreement with the most recent predictions of the theory of percolation in the continuum. We can thus see, that percolation theory provides useful predictions as to the structural parameters which determine hydrological transport processes.     

Effective Thermal Conductivity of Liquid-Saturated Coatings and Their Liquid Vaporisation Behaviour

The effective thermal conductivity is an important element in understanding the thermal response to heating of a paper coating, e.g. during drying in heatset web-offset, and thus it not only affects the drying efficiency but also affects print quality detriments like web fluting. This study examines both the effective thermal conductivity of liquid-saturated ground calcium carbonate coating structures as well as the vaporisation behaviour from these structures. The liquids used for saturation were mineral oil and water in order to resemble ink and fountain solution, respectively, both of which are present in the traditional heatset web-offset process. The effective thermal conductivities of liquid-saturated coating structures are discussed in regard to the corresponding unsaturated systems by using a Lumped Parameter Model. It is shown that the liquid saturation has a dominant effect in determining the effective thermal conductivity. Since this effect is not fully captured by the model, other mechanisms like an apparent pigment–pigment connectivity increase by liquid bridging and the role of liquid in changing the contact resistance during the measurement of thermal conductivity are discussed. In addition, the transformation of three-dimensional structures to an equivalent two-dimensional modelling is evaluated. The vaporisation behaviour of mineral oil and water is studied by a thermogravimetric analysis. By following the changes in maximum evaporation temperatures and evaporation rates, the addition of binder is seen to lead to a reduction in the vaporisation rate of both liquids. Since there is little to no interaction between the liquids and the binder, the confinement caused by the geometry change induced by binder addition is identified as the mechanism resulting in elevated vapour pressure within the structure.