Probabilistic sensitivity and modeling of two-dimensional transport in porous media

A reliability approach is used to develop a probabilistic model of two-dimensional non-reactive and reactive contaminant transport in porous media. The reliability approach provides two important quantitative results: an estimate of the probability that contaminant concentration is exceeded at some location and time, and measures of the sensitivity of the probabilistic outcome to likely changes in the uncertain variables. The method requires that each uncertain variable be assigned at least a mean and variance; in this work we also incorporate and investigate the influence of marginal probability distributions. Uncertain variables includex andy components of average groundwater flow velocity,x andy components of dispersivity, diffusion coefficient, distribution coefficient, porosity and bulk density. The objective is to examine the relative importance of each uncertain variable, the marginal distribution assigned to each variable, and possible correlation between the variables. Results utilizing a two-dimensional analytical solution indicate that the probabilistic outcome is generally very sensitive to likely changes in the uncertain flow velocity. Uncertainty associated with dispersivity and diffusion coefficient is often not a significant issue with respect to the probabilistic analysis; therefore, dispersivity and diffusion coefficient can often be treated for practical analysis as deterministic constants. The probabilistic outcome is sensitive to the uncertainty of the reaction terms for early times in the flow event. At later times, when source contaminants are released at constant rate throughout the study period, the probabilistic outcome may not be sensitive to changes in the reaction terms. These results, although limited at present by assumptions and conceptual restrictions inherent to the closed-form analytical solution, provide insight into the critical issues to consider in a probabilistic analysis of contaminant transport. Such information concerning the most important uncertain parameters can be used to guide field and laboratory investigations.     

Determination of two‐dimensional laboratory‐scale dispersivities

A tracer test was conducted in a laboratory chamber representing a two‐dimensional aquifer to investigate the longitudinal dispersivity (α_L) and the ratio (α_T/α_L) of transverse to longitudinal dispersivity of sandy aquifer materials. Dispersive parameters were obtained by matching the observed chloride plumes at 9 hours and 16 hours after tracer injection with those simulated by a flow and transport model. The best match was found for α_L = 0·2 ? 0·25 cm and α_T/α_L = 0·2. The ratio of α_T/α_L = 0·2 was within the range of laboratory values reported in the literature. Sensitivity analysis revealed that the tracer plume concentration and shape were more sensitive to variations in longitudinal dispersivity than to the ratio of transverse to longitudinal dispersivity. This result contrasted with findings of others, showing that the dispersivity ratio greatly affects contaminant plume shape. However, our experimental boundary conditions restricted expansion of the plume normal to the direction of flow and thus affected the parameter estimation. Copyright © 2004 John Wiley & Sons, Ltd.     

Form-drag instability in a barotropic and a baroclinic atmosphere

We discuss the form-drag instability for a quasi-geostrophic channel flow. We first study the characteristics of this instability in a barotropic flow, considering in detail the influence of the meridional scale and discussing which structure of the perturbation zonal flow must be chosen in order to describe properly this instability.We then consider a continuous quasi-geostrophic channel model in which the topography enters only through the bottom boundary condition, and we discuss how in this case the effects of the form-drag are felt by the mean zonal flow through the ageostrophic mean meridional circulation. Because the meridional structure of the perturbation zonal flow cannot simply be extended from the barotropic to the continuous case, we show how to modify it properly.We then study the baroclinic model in the particular case of constant (in the vertical) basic-state zonal flow and show how this case closely resembles the barotropic, demonstrating the barotropic nature of the form-drag instability.Symbols _t is the partial derivative with respect tot. - _x is the partial derivative with respect tox. - _y is the partial derivative with respect toy. - represents the geostrophic stream function. - u is the eastward component of the geostrophic wind. - v is the northward component of the geostrophic wind. - u _a is the eastward component of the ageostrophic wind. - v _a is the northward component of the ageostrophic wind. - w is the vertical component of the wind. - f is the Coriolis parameter=2 sin f _o+y. - f _o is the Coriolis parameter evaluated at mid-latitude. - N is the Brunt-Vaisala frequency. - [A] is the zonal (x) average ofA at constantp andy. - <A> is the horizontal (x andy) average ofA at constantp     

Is fractal dispersion subdiffusive or superdiffusive? A theoretical investigation

A probabilistic approach is used to simulate particle tracking along a fractal path. The particle tracking is modelled as the sum of elementary steps with independent random variables. An exponential distribution is obtained for each elementary step and a Gamma distribution or probability density function is then deduced. The relationship between fractal dispersivity and the elementary step is given. It is theoretically demonstrated that the fractal dispersion is subdiffusive and that the linear dispersion coefficient and the linear dispersivity decrease with the mean travel distance. A review of some fractal models showing an increase of the linear dispersivity is given and an explanation that shows why these models may be not correct is proposed. It is shown that the results presented are in agreement with other studies relating to the application of the fractional calculus to diffusion transport. Lastly, a relation between the fractal dimension and the order of the fractional Langevin equation is proposed. Copyright © 2007 John Wiley & Sons, Ltd.     

Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer

We perform global sensitivity analysis (GSA) through polynomial chaos expansion (PCE) on a contaminant transport model for the assessment of radionuclide concentration at a given control location in a heterogeneous aquifer, following a release from a near surface repository of radioactive waste. The aquifer hydraulic conductivity is modeled as a stationary stochastic process in space. We examine the uncertainty in the first two (ensemble) moments of the peak concentration, as a consequence of incomplete knowledge of (a) the parameters characterizing the variogram of hydraulic conductivity, (b) the partition coefficient associated with the migrating radionuclide, and (c) dispersivity parameters at the scale of interest. These quantities are treated as random variables and a variance-based GSA is performed in a numerical Monte Carlo framework. This entails solving groundwater flow and transport processes within an ensemble of hydraulic conductivity realizations generated upon sampling the space of the considered random variables. The Sobol indices are adopted as sensitivity measures to provide an estimate of the role of uncertain parameters on the (ensemble) target moments. Calculation of the indices is performed by employing PCE as a surrogate model of the migration process to reduce the computational burden. We show that the proposed methodology (a) allows identifying the influence of uncertain parameters on key statistical moments of the peak concentration (b) enables extending the number of Monte Carlo iterations to attain convergence of the (ensemble) target moments, and (c) leads to considerable saving of computational time while keeping acceptable accuracy.     

Upper air mean dry-bulb and dew-point temperature profiles over Gauhati airport

Summary The best fit curves for upper air mean dry-bulb and dew-point temperatures over Gauhati airport (26°05N, 91°43E, 49 metres a.m.s.l.), for the month of April, have been calculated with the equation,x=A+By+Cy ²,y being the log value in mb of the isobaric level under consideration andx, the mean dry-bulb or dew-point temperature as the case may be, at the isobaric level under consideration. The values of constantsA, B andC for morning dry-bulb and dew-point curves come to be –29.54559, –93.65766 and +37.35048 and –118.84791, –31.15503 and +25.63585 respectively and values of these constants for evening curves come to be –35.86214, –94.15694 and +38.61870 and –127.55970, –29.97192 and +26.36538 respectively. These best fit curves help in finding out mean desired temperatures at any isobaric level in forecasting of thunderstorms and hailstorms, at a station, by dry-bulb and dew-point temperature anomaly technique propsed earlier by the authors.     

INDUCTIVE SOUNDING OF A STRATIFIED EARTH WITH TRANSITION LAYER RESTING ON DIPPING ANISOTROPIC BEDS *

The generalised three layer boundary value problem with a transition layer sand-witched between an isotropic overburden and dipping anisotropic substratum is discussed assuming that plane electro-magnetic waves are incident normally over the air-earth inter-face. The tangential electric (E_y) and magnetic (H_x) fields and the expression for surface-impedance (E_y/H_x) have been evaluated at the earth's surface. Through numerical analysis it is shown that changes in the values of the parameters m (coefficient of anisotropy), h (thickness of the transition layer), α (angle of inclination of the dipping beds), and b (conductivity ratio between substratum and upper layer) modify the amplitude and phase-variation curves (with skindepth) significantly.     

Towards improved post‐processing of hydrologic forecast ensembles

Forecast ensembles of hydrological and hydrometeorologial variables are prone to various uncertainties arising from climatology, model structure and parameters, and initial conditions at the forecast date. Post‐processing methods are usually applied to adjust the mean and variance of the ensemble without any knowledge about the uncertainty sources. This study initially addresses the drawbacks of a commonly used statistical technique, quantile mapping (QM), in bias correction of hydrologic forecasts. Then, an auxiliary variable, the failure index (γ), is proposed to estimate the ineffectiveness of the post‐processing method based on the agreement of adjusted forecasts with corresponding observations during an analysis period prior to the forecast date. An alternative post‐processor based on copula functions is then introduced such that marginal distributions of observations and model simulations are combined to create a multivariate joint distribution. A set of 2500 hypothetical forecast ensembles with parametric marginal distributions of simulated and observed variables are post‐processed with both QM and the proposed multivariate post‐processor. Deterministic forecast skills show that the proposed copula‐based post‐processing is more effective than the QM method in improving the forecasts. It is found that the performance of QM is highly correlated with the failure index, unlike the multivariate post‐processor. In probabilistic metrics, the proposed multivariate post‐processor generally outperforms QM. Further evaluation of techniques is conducted for river flow forecast of Sprague River basin in southern Oregon. Results show that the multivariate post‐processor performs better than the QM technique; it reduces the ensemble spread and is a more reliable approach for improving the forecast. Copyright © 2012 John Wiley & Sons, Ltd.     

Analytical solutions to the transient,unsaturated transport of water and contaminants through horizontal porous media

We present a range of analytical solutions to the combined transient water and solute transport for horizontal flow. We adopt the concept of a scale and time dependent dispersivity used for contaminant transport in aquifers and apply it to transient, unsaturated horizontal flow to develop similarity solutions for both constant solute concentration and solute flux boundary conditions. Through the use of a specific form of the water profile as used by Brutsaert [Water Resour Res 1968:4;785], the solute profiles can be reduced to a simple quadrature. We also derive a solution for the instantaneous injection of water and solute into a horizontal media for an arbitrary dispersivity. It is found that the solute concentration remains constant in both space and time as the water redistributes, suggesting that the solute does not disperse relative to the water.     

Analytical power series solutions to the two-dimensional advection–dispersion equation with distance-dependent dispersivities

As is frequently cited, dispersivity increases with solute travel distance in the subsurface. This behaviour has been attributed to the inherent spatial variation of the pore water velocity in geological porous media. Analytically solving the advection–dispersion equation with distance-dependent dispersivity is extremely difficult because the governing equation coefficients are dependent upon the distance variable. This study presents an analytical technique to solve a two-dimensional (2D) advection–dispersion equation with linear distance-dependent longitudinal and transverse dispersivities for describing solute transport in a uniform flow field. The analytical approach is developed by applying the extended power series method coupled with the Laplace and finite Fourier cosine transforms. The developed solution is then compared to the corresponding numerical solution to assess its accuracy and robustness. The results demonstrate that the breakthrough curves at different spatial locations obtained from the power series solution show good agreement with those obtained from the numerical solution. However, owing to the limited numerical operation for large values of the power series functions, the developed analytical solution can only be numerically evaluated when the values of longitudinal dispersivity/distance ratio e_L exceed 0·075. Moreover, breakthrough curves obtained from the distance-dependent solution are compared with those from the constant dispersivity solution to investigate the relationship between the transport parameters. Our numerical experiments demonstrate that a previously derived relationship is invalid for large e_L values. The analytical power series solution derived in this study is efficient and can be a useful tool for future studies in the field of 2D and distance-dependent dispersive transport. Copyright © 2008 John Wiley & Sons, Ltd.     

Notes on the variation of magnetization within basalt lava flows and dikes

Summary The magnetic properties of basaltic rocks are dominated by the contained primary Fe–Ti oxides. At solidus temperature (1000°C) the composition of these primary oxides is restricted to titanomagnetite (Fe_3-xTi_xO₄) and hemoilmenites (Fe_2-yTi_yO₃). The examination of 269 chemical analyses of the primary Fe–Ti oxides in basalts (in sensu lato) gives an average ofx=0.61 (T _c=168°C) for the titanomagnetites andy=0.89 (T _c=–121°C) for the hemoilmenites. If distinction is made between tholeiites, alkali basalts and andesites, a clear difference for thex-values is observed: the average for tholeiitesx=0.64 (T _c=144°C), for alkali basaltsx=0.52 (T _c=253°C), for andesitesx=0.38 (T _c=341°C).Environment of crystallization and cooling rate are major interrelated factors influencing subsequent changes in the mineralogy of the primary Fe–Ti oxides and resulting magnetic properties. This has been tested by studying the variation of magnetization and some of its parameters in three different basalt rock units: a dike, 180 cm, and two lava flows, 3 m and 33 m thick, respectively. Grain size and oxidation state of the titanomagnetites control the variation of magnetization in these basalt units.     

Modelling effects of spatial variability of saturated hydraulic conductivity on autocorrelated overland flow data: linear mixed model approach

The mixed linear model approach was introduced and applied in studying the effects of spatial variation of the saturated hydraulic conductivity (K _s ) on the variation of the overland flow. Analysis was carried out with 2,000 rainfall-runoff events, all generated through transformation of real, observed rainfall events and different spatially variable K _s fields in a small (12 ha) agricultural catchment. The parameters accounting for the variation in the generation method were the coefficient of variation (cv) and correlation length (L _x L _y ) of K _s both having two levels of values obtained from field measurements of other studies. The analysis showed that the combinations with both parameters having the smaller or bigger value during flow peaks only caused different response in the overland flow. However, the parameters were statistically significant only at the 10% level. Most of the flow variation was explained by the event dynamics. The mixed models were able to model the structure of the data efficiently with less restrictive assumptions than for example the analysis of variance, hence producing more reliable results. The method was able to take into account autocorrelation of the test series, correlation between the factors and unequal variances. The usefulness of the method was supported by the fact that the conclusions drawn by it were confirmed by simple, conventional methods of a previous study, added with statistical criteria and confidence levels for each calculation moment. The findings of the study can be utilized in practise for example when designing the field sampling experiments.     

An extended theory of equi-risk line for a general release rule

The expression of equi-risk line derived by the authors represents the relationship between discharge capacityy ₀ ^u and storage capacityz ₀ ^u to keep flood frequency under a certain risk level represented by the return periodT, i.e.,z ₀/z ₀ ^u ={(y ₀ ^u –y ₀ ^u )/y ₀ ^u } ^S , wherey ₀ ^u andz ₀ ^u areT-year probability peak discharge and total volume of a hydrograph. The shape parametersS is evaluated in this paper for various release rules of the storage facilities and correlations of durations and peaks of hydrographs. The expression forS is: , whereS ₀ andS are the values ofS forp=0 and , andp is the exponent of a general storage-release relation,q=az' ^p, wherea is the storage constant, andz' andq are the volume of stored water and the corresponding release. The values ofS ₀ andS are expressed in terms of the correlation coefficient of durations and peaks of inflow hydrographs.     

Offset dependent amplitudes: Effects of wavefront and reflector curvature in homogeneous and layered media

We analyze the effect of reflector curvature on the angular dependence of reflection amplitude using ray theory. Defining the curvature effect, CE, as the ratio of reflected amplitude from a curved boundary to that from a flat boundary at the same depth, we obtain 1/CE ²=(1+Z/a ^xcos²)(1+Z/a ^y), where is the angle of incidence,Z the depth of the boundary, anda ^x anda ^y are the principal radii of curvature of the reflector in the plane, of incidence and in the perpendicular plane, respectively. At =0 this reduces to the formula given byHilterman (1975). The angular dependence ofCE involves onlya ^x, which appears to shorten at wider angles. This causes an augmentation of the dimming effect of an anticlinal geometry at far offsets. For synclinal structures, the amplitude increases with offset when |Z/a^x|<1 and decreases with offset when |Z/a^x|<1.In addition we examine the effects of wavefront curvature and of a layered overburden in modulating the curvature effect. We find that there is a significant difference in the curvature effect between plane waves and spherical waves impacting on a curved boundary. In addition results are given showing simple examples of the effect of layered overburden in distorting the curvature effect of a horizontal, but curved, reflector and, more interestingly, of a dipping, but planar, bed. These results strongly suggest that we still have but a rudimentary knowledge, in general, of the combined effects of wavefront curvature, reflector curvature and overburden layering in modulating and/or producing amplitude with offset behavior observed on seismic records.     

Convergent Radial Tracing of Viral and Solute Transport in Gneiss Saprolite

Deeply weathered crystalline rock aquifer systems comprising unconsolidated saprolite and underlying fractured bedrock (saprock) underlie 40% of sub-Saharan Africa. The vulnerability of this aquifer system to contamination, particularly in rapidly urbanizing areas, remains poorly understood. In order to assess solute and viral transport in saprolite derived from Precambrian gneiss, forced-gradient tracer experiments using chloride and Escherichia coli phage ΦX174 were conducted in southeastern Uganda. The bacteriophage tracer was largely unrecovered; adsorption to the weathered crystalline rock matrix is inferred and enabled by the low pH (5.7) of site ground water and the bacteriophage's relatively high isoelectric point (pI = 6.6). Detection of the applied ΦX174 phage in the pumping well discharge at early times during the experiment traces showed, however, that average ground water flow velocities exceed that of the inert solute tracer, chloride. This latter finding is consistent with observations in other hydrogeological environments where statistically extreme sets of microscopic flow velocities are considered to transport low numbers of fecal pathogens and their proxies along a selected range of linked ground water pathways. Application of a radial advection-dispersion model with an exponentially decaying source term to the recovered chloride tracer estimates a dispersivity (α) of 0.8 ± 0.1 m over a distance of 4.15 m. Specific yield (S_y) is estimated to be 0.02 from volume balance calculations based on tracer experiments. As single-site observations, our estimates of saprolite S_y and α are tentative but provide a starting point for assessing the vulnerability of saprolite aquifers in sub-Saharan Africa to contamination and estimating quantitatively the impact of climate and abstraction on ground water storage.     

The α-effect in 2D fast dynamos

We look at the large-scale dynamo properties of spatially periodic, time dependent, helical 2D flows of the form u(x, t)?=?(? _y ?ψ?(x, y, t), ?? _x ?ψ?(x, y, t), ?ψ (x, y, t). These flows act as kinematic fast dynamos and are able to generate a mean magnetic field uniform and constant in the xy-plane but whose direction varies periodically along z with wavenumber k. Using Mean Field Electrodynamics, the generation mechanism can be understood in terms of a k-dependent α-effect, which depends on the magnetic Reynolds number, R _m . We calculate this effect for different motions and investigate how its limit as k?→?0 depends on R _m and on the properties of the flows such as their spatial structure or correlation time. This work generalises earlier studies based on 2D steady flows to motions with time dependence.     

Quasi-Saturated Layer: Implications for Estimating Recharge and Groundwater Modeling

This study presents an extension of the concept of “quasi-saturation” to a quasi-saturated layer, defined as the uppermost dynamic portion of the saturated zone subject to water table fluctuations. Entrapped air here may cause substantial reductions in the hydraulic conductivity (K) and fillable pore water. Air entrapment is caused by a rising water table, usually as a result of groundwater recharge. The most significant effects of entrapped air are recharge overestimation based on methods that use specific yield (S_y), such as the water table fluctuation method (WTF), and reductions in K values. These effects impact estimation of fluid flow velocities and contaminant migration rates in groundwater. In order to quantify actual groundwater recharge rates and the effects of entrapped air, numerical simulations with the FEFLOW (Version 7.0) groundwater flow model were carried out using a quasi-saturated layer for a pilot area in Rio Claro, Brazil. The calculated recharge rate represented 16% of the average precipitation over an 8-year period, approximately half of estimates using the WTF method. Air entrapment amounted to a fillable porosity of 0.07, significant lower that the value of 0.17 obtained experimentally for S_y. Numerical results showed that the entrapped air volume in the quasi-saturated layer can be very significant (0.58 of the air fraction) and hence can significantly affect estimates of groundwater recharge and groundwater flow rates near the water table.     

Diffusion models of slope development

This paper represents a systematic investigation of slope evolution diffusion models and has the following sections: (1) The model of slope development with linear coefficient k = k₀x; (2) The model of slope development with quadratic coefficient in x; (3) Slope development model with vertical lowering of base level (downcutting); (4) Slope development model with the base level a horizontal variable (undercutting); (5) Steady-state regime of undercut slopes; (6) Model of a pediment and scree slope formation. The comparison is made between mathematical and classical methods of slope evolution analysis.     

Assessing environmental risks through fuzzy parameterized probabilistic analysis

A fuzzy parameterized probabilistic analysis (FPPA) method was developed in this study to assess risks associated with environmental pollution-control problems. FPPA integrated environmental transport modeling, fuzzy transformation, probabilistic risk assessment, fuzzy risk quantification into a general risk assessment framework, and was capable of handling uncertainties expressed as fuzzy-parameterized stochastic distributions. The proposed method was applied to two environmental pollution problems, with one being about the point-source pollution in a river system with uncertain water quality parameters and the other being concerned with groundwater contaminant plume from waste landfill site with poorly known contaminant physical properties. The study results indicated that the complex uncertain features had significant impacts on modeling and risk-assessment outputs; the degree of impacts of modeling parameters were highly dependent on the level of imprecision of these parameters. The results also implied that FPPA was capable of addressing vagueness or imprecision associated with probabilistic risk evaluation, and help generate risk outputs that could be elucidated under different possibilistic levels. The proposed method could be used by environmental managers to evaluate trade-offs involving risks and costs, as well as identify management solutions that sufficiently hedge against dual uncertainties.     

An effective scale-dependent dispersivity deduced from a purely convective flow field

Abstract

In the case of straight flow but with hydraulic conductivity varying in a transverse direction, the distribution of hydraulic conductivity has been determined for which the breakthrough curve due to convection only will have the same analytical form as the onedimensional convection/dispersion equation solution at the outlet end of a porous medium. That distribution is found exactly and it is very similar to the lognormal distribution. This result is significant since field evidence indicates that the logarithm of hydraulic conductivity is normally distributed. For the case considered, a simple relation between dispersivity and the coefficient of variation of hydraulic conductivity is found. One can thus determine very simply dispersivity in terms of the parameters of the distribution of hydraulic conductivity. This is particularly useful to estimate dispersivity in various cells of finite difference or finite element models when the distribution of hydraulic conductivity is not stationary, i.e. varies in space.