Ln Ks Research Articles

Two‐Dimensional Probabilistic Infiltration Analysis in a Hillslope Using First‐Order Moment Approach

A first-order moment analysis method is introduced to evaluate the pore-water pressure variability within a hillslope due to spatial variability in saturated hydraulic conductivity (Ks ) during rainfall. The influences of the variance of the natural logarithm of Ks (ln Ks ), spatial structure anisotropy of ln Ks , and normalized vertical infiltration flux (q) on the evaluations of the pore-water pressure uncertainty are investigated. Results indicate different responses of pressure head variability in the unsaturated region and the saturated region. In the unsaturated region, a larger variance of ln Ks , a higher spatial structure anisotropy, and a smaller q lead to a larger variability in pressure head, while in the saturated region, the variability in pressure head increases with the increase of variance of ln Ks , the decrease of spatial structure anisotropy, or the increase of q. These variables have great impacts on the range of fluctuation of the phreatic surface within the hillslope. The influences of these three variables on the variance of pressure head within the saturated region are greater than those within the unsaturated region, and the variance of ln Ks has the greatest impact. These results yield useful insight into the effects of heterogeneity on pressure head and uncertainty associated with predicted flow field.

Multi-scale correlations between soil hydraulic properties and associated factors along a Brazilian watershed transect

Soil hydraulic properties are important to the understanding and modeling of hydrological processes at the watershed scale. These properties are presumed to be correlated with associated topographic attributes and other soil factors operating at different intensities and scales. Using multivariate empirical mode decomposition (MEMD), this study examined the multi-scale correlations among two soil hydraulic properties and two topographic attributes and three soil factors along a transect in the Pelotas River Watershed (PRW) situated in Southern Brazil. Soil water content at field capacity (θFC) and saturated hydraulic conductivity (KS) were determined for 100 soil samples taken at 250-m intervals along a 25-km transect that cut through the PRW. The five selected factors were elevation (Ele), slope (Slo), sand content (Sand), bulk density (Bd), and soil organic carbon (SOC). The topographic attributes were derived from a digital elevation model, while those related to soils were determined for the collected samples. The multivariate data series of the two soil hydraulic properties and five associated factors were decomposed into six intrinsic mode functions (IMFs). For θFC, 52.4% of the total variance was separated at IMF1 (scale of 728m) and IMF2 (1,113m), while for ln KS, 35.9% of the variance was separated at IMF1 (728m) and IMF6 (scale of 11,877m). It was found that scale-specific relationships between soil hydraulic properties and the associated factors varied with scale. The associated factors exerted influence on the soil hydraulic properties at their own dominant scale(s). The θFC and ln KS values at each IMF (specific scale) and residue were predicted from the scale-specific associated factors at the same IMF or residue. The total of all the predicted IMFs including the residue was then used to predict the θFC and ln KS at the measurement scale. Sand had the greatest relative importance to the measurement-scale θFC prediction model, while topographic properties were clearly the dominant explanatory factors for the overall ln KS prediction. The overall ln KS predictions using the MEMD outperformed those based on the original data.

Predicting saturated hydraulic conductivity in a sandy grassland using proximally sensed apparent electrical conductivity

Finding a correspondence between soil hydraulic properties, such as saturated hydraulic conductivity (Ks) and apparent electrical conductivity (ECa) as an easily measurable parameter, may be a way forward to estimate the spatial distribution of hydraulic properties at the field scale. In this study, the spatial distributions of Ks, of soil ECa measured by a DUALEM-21S sensor and of soil physical properties were investigated in a sandy grassland. To predict field scale Ks, the statistical relationship between co-located soil Ks, and EMI-ECa was evaluated. Results demonstrated the large spatial variability of all studied properties with Ks being the most variable one (CV=86.21%) followed by ECa (CV≥53.77%). A significant negative correlation was found between ln-transformed Ks and ECa (r=0.83; P≤0.01) at two depths of exploration (0–50 and 0–100cm). This site-specific relation between ln Ks and ECa was used to predict saturated hydraulic conductivity over 0–50cm depth for the whole field. The empirical relation was validated using an independent dataset of measured Ks. The statistical results demonstrate the robustness of this empirical relation with mean estimation error MEE=0.46 (cmh−1), root-mean-square estimation errors RMSEE=0.74 (cmh−1), coefficient of determination r2=0.67 and coefficient of model efficiency Ce=0.64. The relationship was then used to produce a detailed map of Ks for the whole field. The result will allow model predictions of spatially distributed water content in view of irrigation management.

Investigation of solute transport in nonstationary unsaturated flow fields

Abstract. Most theoretical work on analyzing plume spreading at the field scale in a partially saturated heterogeneous formation assumes weak stationarity of velocity field. While this assumption is not applicable to the case of a bounded flow domain, the nonstationarity in fluctuations of unsaturated velocity fields is induced through the presence of the boundary conditions in the flow domain. In this work, we attempt to quantify the large-time macrodispersion in nonstationary unsaturated velocity fields caused by the presence of a fixed head boundary condition. Application of the perturbation-based nonstationary spectral approach leads to an analytical expression for the macrodispersion describing the field-scale dispersive solute flux in terms of the statistical moments of two formation parameters: the Gardener's parameter (α) and the saturated hydraulic conductivity (Ks). The results predicted from the expression indicate that the enhanced unsaturated plume spreading can arise from a larger correlation scale of ln Ks or ln α fields. In addition, the α-parameter takes the role in reducing the field-scale plume spreading.

Solubility of alkanes in a polystyrene matrix

AbstractThe solubilities of seven alkanes in a matrix of uncrosslinked polystyrene were measured at several different temperatures. A gravimetric method reported earlier was used to monitor the sorption of the solvents. The experimental measurements showed unambiguously that the solubility of the alkanes diffusing into a polystyrene matrix reached a constant and reproducible value typical for each system and temperature. These values could be interpreted very well with the Flory–Huggins solubility parameter (χ = a(entropic contribution to x) + b(enthalpic contribution to x)/T[absolute temperature in Kelvin (K)]) or with the van't Hoff equation (ln KS = ΔS/R(gas constant) − ΔH/RT, where KS is the equilibrium constant, ΔS is the entropy change, and ΔH is the enthalpy change). © 2008 Wiley Periodicals, Inc. J Appl Polym Sci, 2008

An investigation on the surface properties of lyocell-based carbon fiber with inverse gas chromatography

The surface thermodynamic properties of two types of cellulose-based carbon fibers (LYCF and RCF from lyocell and rayon precursors, respectively) were studied by inverse gas chromatography (IGC). It was found the molar free energy of adsorption (ΔGmads) of n-alkane probe molecules on the LYCF surface is higher than that of RCF at different temperatures. The interaction of probes on the surfaces of LYCF and RCF becomes stronger with the increasing in the chain length of the n-alkane probes. Simultaneously, the ln Ks is linear to 1/T, where Ks and T are the constant of Henry’s law and temperature, respectively. Similar results were obtained by using Dorris–Gray method (area per methylene unit) and measured probe areas. The results indicate that LYCF and RCF are different in the surface morphology and physical chemistry properties, which suggests that LYCF is a more promising cellulose-based carbon fiber material.

A Comparative Study on Uncertainty Quantification for Flow in Randomly Heterogeneous Media Using Monte Carlo Simulations and Conventional and KL-Based Moment-Equation Approaches

Geological formations are ubiquitously heterogeneous, and the equations that govern flow and transport in such formations can be treated as stochastic partial differential equations. The Monte Carlo method is a straightforward approach for simulating flow in heterogeneous porous media; an alternative based on the moment-equation approach has been developed in the last two decades to reduce the high computational expense required by the Monte Carlo method. However, the computational cost of the moment-equation approach is still high. For example, to solve head covariance up to first order in terms of $\sigma_Y^2$, the variance of log hydraulic conductivity Y = ln Ks, it is required to solve sets of linear algebraic equations with N unknowns for 2N times (N being the number of grid nodes). The cost is even higher if higher-order approximations are needed. Zhang and Lu [J. Comput. Phys., 194 (2004), pp. 773--794] developed a new approach to evaluate high-order moments (fourth order for mean head in terms of $\sigma_Y$, and third order for head variances in terms of $\sigma_Y^2$) of flow quantities based on the combination of Karhunen--Loeve decomposition and perturbation methods. In this study, we systematically investigate the computational efficiency and solution accuracy of three approaches: Monte Carlo simulations, the conventional moment-equation (CME) approach, and the moment-equation approach based on Karhunen--Loeve decomposition (KLME). It is evident that the computational cost for the KLME approach is significantly lower than those required by the Monte Carlo and CME approaches. More importantly, while the computational costs (in terms of the number of times for solving linear algebraic equations with N unknowns) for the CME approach depend on the number of grid nodes, the cost for the KLME approach is independent of the number of grid nodes. This makes it possible to apply the KLME method to solve more realistic large-scale flow problems.

Free energy and kinetics of dissolution of Sokoto rock phosphate and the implications for replenishing phosphorus in the savanna soil of Nigeria

SummaryThe direct application of Sokoto phosphate rock to restore phosphorus in the savanna soil of Nigeria has not been very successful. The dissolution of Sokoto phosphate rock was investigated in three electrolyte solutions – 0.01 m CaCl2, NaCl and KCl – at pH range 3.5–7.0 under laboratory conditions to provide solubility and kinetic data that are required to develop guidelines for direct application in the field. The phosphate rock dissolved in the salt solutions in the order KCl > NaCl > CaCl2. Particle size and ionic strength had no significant effect on the dissolution. The standard free energy of reaction ΔG°R in an acidic solution with no basic cations was −38 kJ mol−1. If Ca2+ ions were in the acidic solution, then ΔG°R increased to 210 kJ mol−1, 170 kJ mol−1 for Na+ ions, and 107 kJ mol−1 for K+ ions in the solution. The theoretical solubility constant (Ks) calculated from the relation ΔG° = −RT ln Ks gave Ks = 106.7 in an acidic solution without basic cations, but decreased to 10−36.8 with Ca2+ ions in solution, 10−29.8 with Na+ ions, and 10−18.8 with K+ ions in solution. At pH ≥ 5.5, the dissolution was more constrained by Ca2+ ions or basic cations in solution than by availability of protons. The kinetics of the dissolution reaction was best described by a power function: Ct = atb, where Ct is the amount of P released from the rock phosphate at time t, and a and b are fitting parameters. An Elovich and a parabolic diffusion expression equally gave satisfactory fits to the dissolution data, suggesting that the rate of dissolution was limited by a combination of film‐ and intra‐particle diffusion. To utilize this rock phosphate as an effective source of P, management practices that increase Ca sinks and the supply of protons to the soil are necessary. In the savanna, increasing the soil's organic matter greatly enhances cation exchange capacity and availability of protons. The practice should provide adequate sinks for Ca2+ and the acidic environment required for the release of P from rock phosphate.

On stochastic modeling of flow in multimodal heterogeneous formations

Most existing stochastic models are developed for unimodal porous media that may be well characterized with only the first two statistical moments. However, the distribution of hydraulic properties, such as hydraulic conductivity, may possess a multiplicity of modes; thus the first two moments may not be adequate to characterize properties of such porous media. In turn, the stochastic models developed for unimodal porous media may not be applicable to flow and transport in a multimodal heterogeneous porous medium. This study investigates under what circumstances the second‐order moment‐based stochastic models are applicable to multimodal heterogeneous porous media. We assume that a porous medium is composed of a number of materials (categories), each of which may have a different mean, variance, and correlation scale. The distribution of materials in the domain is characterized by indicator random variables. We first derive analytical expressions for the mean and covariance of the log saturated hydraulic conductivity (ln Ks) of the multimodal porous medium in terms of categorical proportions, transition probability among categories, and covariances of indicator random variables. We express the covariance in terms of the statistics of materials in the porous medium, which allows us to accurately evaluate the variance and the correlation length of the composite ln Ks field. We then solve the second‐order moment equations for the “equivalent” unimodal field with an exponential covariance with a single correlation scale computed for the composite field. On the other hand, we conduct two sets of Monte Carlo simulations: one with multimodal random fields, and the other with “equivalent” unimodal random fields. Examples for porous media with two materials are given. Numerical experiments show that a bimodal ln Ks field may be well approximated by an equivalent unimodal field when one of the two modes is dominant, under which condition the applicability of the second‐order moment‐based model is subject to the same limitation of relatively small variance as that for unimodal fields. When the bimodal distribution has two more or less equally important modes, although it cannot be adequately represented by an equivalent unimodal distribution, the second‐order moment‐based stochastic model seems to be applicable to systems with larger composite variances than it does for an one‐mode‐dominant distribution.

Conditional moment analysis of steady state unsaturated flow in bounded, randomly heterogeneous soils

We consider steady state unsaturated flow in bounded, randomly heterogeneous soils under the influence of random boundary and source terms. Our aim is to predict pressure heads and fluxes without resorting to Monte Carlo simulation, upscaling, or linearization of the constitutive relationship between unsaturated hydraulic conductivity and pressure head. We represent this relationship through Gardner's exponential model while treating its exponent α as a random constant and saturated hydraulic conductivity Ks as a spatially correlated random field. We linearize the steady state unsaturated flow equations by means of the Kirchhoff transformation and integrate them in probability space to obtain exact integro‐differential equations for the conditional mean and variance‐covariance of transformed pressure head and flux. After approximating these equations recursively to second order in the standard deviation σY of Y = ln Ks, we solve them by finite elements for superimposed mean uniform and divergent flows in the vertical plane, with and without conditioning on measured Y values. Comparison with Monte Carlo solutions demonstrates that whereas our nonlocal solution is nominally restricted to mildly nonuniform media with σY2 ≪ 1, it yields remarkably accurate results for strongly nonuniform media with σY2 at least as large as 2. This accords well with a previous theoretical analysis, which shows that the solution may remain asymptotic for values of σY2 as large as 2.

Compositional constraints on the equation of state and thermal properties of the lower mantle

By extrapolating the lower mantle equation of state (EoS) to P=0, T=290 K, we determine the EoS parameters that are compatible with a mixture of (Mg, Fe)SiO3 perovskite (with a small admixture of Al2O3), (Mg, Fe)O magnesiowüstite and CaSiO3 perovskite in arbitrary proportions and with arbitrary Fe/(Fe+Mg) ratio. The parameters fitted are density, ρ, adiabatic incompressibility, KS, and its pressure derivative, K′S≡(∂KS/∂P)S. The first stage is adiabatic extrapolation to P=0, T=T0, that is, to the foot of the lower mantle adiabat, at which K′0(T0) is allowed to have any value between 3.8 and 4.6, and 1500 K≤T0≤2000 K. It is important to use an equation for which the lower mantle fitting does not prescribe K′0(T0) and this rules out the third-order Birch theory, which gives a seriously wrong value. The further extrapolation to 290 K at P=0 uses thermodynamic relationships with maximum generality, allowing all of the following thermoelastic parameters to be arbitrary functions of temperature: K;ρ; Grüneisen parameter, γ;q=(∂ ln γ/∂ ln V)T, where V is volume; volume coefficient of thermal expansion, α; adiabatic Anderson–Grüneisen parameter, δS=(1/α) (∂ ln KS/∂T)P; and the mixed P, T derivative (∂K′S/∂T)P. The heat capacity at constant volume, CV, is assumed to follow the Debye function, so α is controlled by that. The temperature dependences of the dimensionless parameters γ, q and δS at P=0 are slight. We find γ to be precisely independent of T at constant V. The parameter dK′0/dT increases strongly with T, as well as with the assumed value of K′0(T0), where K′0 is K′S at P=0. The fitting disallows significant parameter ranges. In particular, we find solutions only if K′0(T0)≥4.2 and the 290 K value of K′0 for Mg perovskite is less than 3.8. Conclusions about composition are less secure, partly because of doubt about individual mineral properties. The volume of magnesiowüstite is found to be between 10 and 25 per cent for respective T0 values of 2000 and 1500 K, but the Ca-perovskite volume is no more than 6 per cent and has little influence on the other conclusions. The resulting overall Fe/(Fe+Mg) ratio is 0.12 to 0.15. Although this ratio is higher than expected for a pyrolite composition, the ratio depends critically on the assumed mineral densities; some adjustment of the mineral mix may need to be considered.

Probabilistic Slope Stability Analysis with Stochastic Soil Hydraulic Conductivity

The effects of stochastic hydraulic conductivity on the slope stability of an embankment dam are investigated using a combination of random field simulation, seepage analysis, and slope stability analysis. The hydraulic conductivity distribution is treated as a spatially stationary random field following a lognormal distri- bution. The turning band method is used to generate the spatial variability of the saturated hydraulic conductivity Ks in the domain. Different standard deviations of log hydraulic conductivity are investigated. For each sln Ks value of various realizations of hydraulic conductivity were generated and combined with a numerical s , ln Ks model to simulate water flow in an earth dam with variable Ks. The first-order second-moment reliability index b was employed to characterize the influence of the variability of Ks, and hence, pore-water pressures, on the stability of the downstream slope. A linear relationship between and the standard deviation of the factor sln Ks of safety sF was obtained from the simulation results. A relationship between b and , in which every 0.1 sln Ks increment of results in a decrease of 1.0 in b, is deduced based on the simulation results. Results of a sln Ks Shapiro-Wilk test for goodness of fit indicate that the factor of safety can be assumed to be normally or lognormally distributed when the saturated hydraulic conductivity follows a lognormal distribution and is sln Ks small (#0.5). When is large (>0.5), neither normal nor lognormal distributions provide a reasonable s ln Ks approximation of the factor of safety. Simulation results show that neither standard deviation nor coefficient of variation of the factor of safety is constant when only the variability of hydraulic conductivity is considered. While the results presented are directly applicable only to the particular earth dam geometry and boundary conditions studied, the methodology is general and may be extended to embankments with different boundary conditions. INTRODUCTION AND BACKGROUND Natural soils are highly variable and heterogeneous (Lamb 1966; Peck 1967; Vanmarcke 1977a; DeGroot and Baecher 1993). In practice, methods used for the analysis of slope sta- bility typically are deterministic, with soil properties charac- terized as constants for a given soil layer and each specified layer assumed to be homogeneous. The variability of soil properties presumably is taken into account indirectly for de- sign purposes by selecting a factor of safety, which is chosen based on experience and/or some design criteria (Griffiths and Fenton 1993; Duncan 1996). Many authors have pointed out the limitations of deterministic methods because they do not account explicitly for variability and uncertainty related to soil parameters (Vanmarcke 1977b; Oka and Wu 1990; Soulie et al. 1990; Kulhawy et al. 1991). Methods are needed to quan- tify uncertainties associated with soil properties and to include them in slope stability analysis (Anderson et al. 1982; Whit- tlestone et al. 1995). By incorporating uncertainties quantita- tively, reliability analysis can enhance engineering judgment and facilitate improved decision making (Li and Lumb 1987). Many water-retaining structures in North America are em- bankment dams (Fenton and Griffiths 1996) and the prediction of dam slope stability is of great importance. Hydraulic con- ductivity of the embankment soils is used to calculate all of the hydraulic parameters associated with dam performance, in- cluding seepage rates and amounts, pore-water pressures, and therefore, the state of effective stress. It follows that uncer- tainty associated with the spatial variability of hydraulic con- ductivity leads to uncertainty in estimating pore-water pres-

Probabilistic Slope Stability Analysis with Stochastic Soil Hydraulic Conductivity

The effects of stochastic hydraulic conductivity on the slope stability of an embankment dam are investigated using a combination of random field simulation, seepage analysis, and slope stability analysis. The hydraulic conductivity distribution is treated as a spatially stationary random field following a lognormal distri- bution. The turning band method is used to generate the spatial variability of the saturated hydraulic conductivity Ks in the domain. Different standard deviations of log hydraulic conductivity are investigated. For each sln Ks value of various realizations of hydraulic conductivity were generated and combined with a numerical s , ln Ks model to simulate water flow in an earth dam with variable Ks. The first-order second-moment reliability index b was employed to characterize the influence of the variability of Ks, and hence, pore-water pressures, on the stability of the downstream slope. A linear relationship between and the standard deviation of the factor sln Ks of safety sF was obtained from the simulation results. A relationship between b and , in which every 0.1 sln Ks increment of results in a decrease of 1.0 in b, is deduced based on the simulation results. Results of a sln Ks Shapiro-Wilk test for goodness of fit indicate that the factor of safety can be assumed to be normally or lognormally distributed when the saturated hydraulic conductivity follows a lognormal distribution and is sln Ks small (#0.5). When is large (>0.5), neither normal nor lognormal distributions provide a reasonable s ln Ks approximation of the factor of safety. Simulation results show that neither standard deviation nor coefficient of variation of the factor of safety is constant when only the variability of hydraulic conductivity is considered. While the results presented are directly applicable only to the particular earth dam geometry and boundary conditions studied, the methodology is general and may be extended to embankments with different boundary conditions. INTRODUCTION AND BACKGROUND Natural soils are highly variable and heterogeneous (Lamb 1966; Peck 1967; Vanmarcke 1977a; DeGroot and Baecher 1993). In practice, methods used for the analysis of slope sta- bility typically are deterministic, with soil properties charac- terized as constants for a given soil layer and each specified layer assumed to be homogeneous. The variability of soil properties presumably is taken into account indirectly for de- sign purposes by selecting a factor of safety, which is chosen based on experience and/or some design criteria (Griffiths and Fenton 1993; Duncan 1996). Many authors have pointed out the limitations of deterministic methods because they do not account explicitly for variability and uncertainty related to soil parameters (Vanmarcke 1977b; Oka and Wu 1990; Soulie et al. 1990; Kulhawy et al. 1991). Methods are needed to quan- tify uncertainties associated with soil properties and to include them in slope stability analysis (Anderson et al. 1982; Whit- tlestone et al. 1995). By incorporating uncertainties quantita- tively, reliability analysis can enhance engineering judgment and facilitate improved decision making (Li and Lumb 1987). Many water-retaining structures in North America are em- bankment dams (Fenton and Griffiths 1996) and the prediction of dam slope stability is of great importance. Hydraulic con- ductivity of the embankment soils is used to calculate all of the hydraulic parameters associated with dam performance, in- cluding seepage rates and amounts, pore-water pressures, and therefore, the state of effective stress. It follows that uncer- tainty associated with the spatial variability of hydraulic con- ductivity leads to uncertainty in estimating pore-water pres-

On the spatial variability of parameters of the unsaturated hydraulic conductivity

In situ measurements of discharge‐head (steady state rate of water discharge and water head (Q, H)) pairs were used to estimate the saturated conductivity Ks and the parameter α of the Gardner [1958] model of local unsaturated hydraulic conductivity at N=152 observation points in a field plot (40 m long, 6 m wide, and 2 m deep) by means of the Geulph permeameter method. Twenty‐two of these (Q, H) pairs yielded negative Ks and were excluded from the subsequent analyses. Estimates of ln Ks and ln α from N=130 observation points were used to estimate parameters of the spatial covariance and the drift functions of the two soil properties, using the restricted maximum likelihood estimation procedure and the weighted least squares procedure, respectively. For both soil properties a deterministic drift was identified, suggesting that in the site under investigation, approximately 59% and 37% of the total variability of ƒ=ln Ks and a=ln α, respectively, may stem from large‐scale variations. Results of the analyses of the uncorrelated residuals of ln Ks and ln α suggest that both follow an approximately normal distribution and are moderately cross‐correlated (ρƒa=0.68). The fitted covariance models of both soil properties exhibited slight statistical anisotropy in the horizontal plane and a significant anisotropy in the vertical planes. Components of the range of ln Ks were 2.5 m and 0.6 m in the horizontal and the vertical directions, respectively, while those of ln α were 1.4 m and 0.4 m, respectively. Estimates of the correlation scales of both properties in the horizontal plane were associated with relatively large uncertainty, which probably stemmed from the less than optimum selection of the number and locations of the observation points in this plane. The results of the model validation test, however, indicated that the fitted covariance and drift models were theoretically consistent with the observations of both soil properties.

An efficient method for simulating steady unsaturated flow in random porous media: Using an analytical perturbation solution as initial guess to a numerical model

Numerical simulation of flow through multidimensional heterogeneous soils under unsaturated conditions is a computationally intensive task. The governing unsaturated flow equation is nonlinear. The degree of nonlinearity depends on the unsaturated hydraulic properties of the soil and the degree of heterogeneity. In this paper, we present a highly efficient method to simulate unsaturated steady state flow through random porous media. Our method utilizes the analytical (approximate) solution derived by a perturbation‐spectral method as an initial guess solution for a numerical model to simulate two‐dimensional vertical infiltration problems. It is found that this approach, which we call “ASIGNing,” reduces the required CPU time by one to two orders of magnitude. ASIGNing is demonstrated to operate successfully under a wide variety of boundary conditions which may substantially deviate from those imposed on the initial guess solution, A large range of mean and variances in the independent variables ln Ks and α or alternatively In α, has been tested and it is shown that the method works well for variances of the unsaturated hydraulic conductivity σln2K′ ≤ 5 and average 〈α〉 ≤ 0.1 [cm−1].

Stochastic analysis of unsaturated flow: One‐dimensional Monte Carlo simulations and comparisons with spectral perturbation analysis and field observations

A numerical experiment was designed to study the stochastic behavior of one‐dimensional transient unsaturated flow in a Monte Carlo setting. Soil hydraulic properties, log‐saturated hydraulic conductivity ln Ks, pore size distribution parameter α, and the specific water capacity C are assumed to be statistically homogeneous random fields described by exponential correlation functions with identical correlation lengths. Fifty realizations of each soil hydraulic property, with statistical properties obtained from a field experiment, are generated by using a nearest‐neighbor model. Numerical solutions of the one‐dimensional flow equation are used repeatedly to obtain realizations of soil water pressure head ψ and flux q corresponding to the realizations of In Ks, α, and C. The dependence of ψ and q on the statistical properties of ln Ks, α, and C and the magnitude of the hydraulic head gradient G prevailing at the lower boundary are investigated. In addition, results of Monte Carlo analysis and spectral perturbation analyses are compared with field observations. The greatest variability of ψ and q are observed when soil hydraulic properties are uncorrelated and have large variances and integral scales and when G at the lower boundary is unity. The Monte Carlo and spectral perturbation analyses tend to agree reasonably well for the flow domains in which ergodicity of local soil hydraulic properties is assured. Results of the Monte Carlo and spectral perturbation analyses are also supported by field observations.

Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 1. Statistically Isotropic Media

Steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques. The hydraulic conductivity K is related to the capillary pressure head ψ by K = Ks exp (−αψ), where Ks is the saturated conductivity, and α is a soil parameter. A general formulation is presented for the case with Ks and α represented as statistically homogeneous spatial random fields. In part 1, solutions are developed assuming α is constant and representing Ks variability by one‐dimensional and three‐dimensional isotropic random fields. Results are obtained for head variances and covariance functions, effective hydraulic conductivities, variances of the unsaturated hydraulic conductivity, flux variances, and variance of pressure gradient. When the parameter α is relatively large, corresponding to coarse textured soils, the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one‐dimensional vertical flow. The effective conductivity is dependent on the correlation scale of ln Ks and the mean hydraulic gradient.

Stochastic Analysis of Unsaturated Flow in Heterogeneous Soils: 2. Statistically Anisotropic Media With Variable α

Steady unsaturated flow in heterogeneous soil with an arbitrarily oriented mean hydraulic gradient is analyzed using spectral solutions of the stochastic perturbation equation which describes the capillary pressure head ψ. The unsaturated hydraulic conductivity is related to ψ by K = Ks exp (−αψ), where Ks is the saturated hydraulic conductivity and α is a soil parameter, and both Ks and α are treated as three‐dimensional statistically homogeneous, anisotropic random fields. Analytical results are obtained for the capillary pressure head variance and the effective (mean) unsaturated hydraulic conductivity. The head variance depends upon the degree of anisotropy of the ln Ks covariance; when α is random, the head variance increases with mean capillary pressure head. The effective hydraulic conductivity for arbitrary orientation of the mean hydraulic gradient J is shown to have tensorial properties, but its components depend on the magnitude and direction of J and the orientation of the stratification in the soil. When α is random, the degree of anisotropy of the effective conductivity depends strongly on mean capillary pressure.

Heat resistance ofCitrobacter freundii in media with various water activities

The heat resistance ofCitrobacter freundii NCTC 9750 between 45–65°C in media with various water activities has been determined. At a water activity of nearly 1.00, the Arrhenius plot of the death rate shows a sharp breakpoint at 56.5°C, suggesting the existence of at least two different thermal inactivation processes causing lethality of the bacterial cell. The activation energy below 56.5°C is 0.4186 MJ/mol (100 000 cal/mol), above 56.5°C it is 0.1863 MJ/mol (44 500 cal/mol). After addition of sucrose (1.8 mol/l) or NaCl (0.77 mol/l) to the heating medium, such a breakpoint is not observed. The activation energy for these processes are, for sucrose; 0.2097 MJ/mol, for NaCl; 0.3641 MJ/mol. However, at an NaCl concentration of 1.54 mol/l there is a breakpoint at 53.3°C. The influence of the sucrose concentration on the heat resistance can be described by the formula: ln kS=ln kO−a [sucrose]. Such a simple correlation does not exist for the influence of NaCl or glycerol. The heat inactivation of whole cells ofC. freundii was also measured with a differential scanning calorimeter. The first irreversible conformation change took place at 323 K, the main conformation change at 343 K.