Contaminant transport in fractured media: analytical solution and sensitivity study considering pulse,Dirac delta and sinusoid input sources

The analytical solution of one‐dimensional transport for a single species radioactive nuclide, considering the decay term in a single fracture for pulse, Dirac delta, and single sinusoid input sources, has been studied using the Laplace transform method. The dimensionless concentration of the radioactive nuclide in the fracture appears to be a function of space, elapsed time, dispersivity, retardation factor, half‐life of the nuclide, and release time. By comparing different values of groundwater velocity, retardation factor, dispersivity, and release time, the results show that the c/c₀ ratio agrees with the nature of the physical and chemical characteristics of the nuclide in fracture transportation. The dimensionless concentration peak value from a small retardation factor is found to be more sensitive, within a time frame ranging from 10 years to a few hundreds years, than the case with a larger retardation factor for H‐3. Except for a small variation in the peak value, the result is almost the same for pulse and sinusoid inputs when considering the H‐3 nuclide. Analytical solutions during the preliminary screening phase are suitable for performance assessment on radioactive waste disposal sites under a one‐dimensional single fracture condition. Copyright © 2002 John Wiley & Sons, Ltd.     

Uncertainty propagation of hydrodispersive transfer in an aquifer: an illustration of one-dimensional contaminant transport with slug injection

It is evident that the hydrodynamic dispersion coefficient and linear flow velocity dominate solute transport in aquifers. Both of them play important roles characterizing contaminant transport. However, by definition, the parameter of contaminant transport cannot be measured directly. For most problems of contaminant transport, a conceptual model for solute transport generally is established to fit the breakthrough curve obtained from field testing, and then suitable curve matching or the inverse solution of a theoretical model is used to determine the parameter. This study presents a one-dimensional solute transport problem for slug injection. Differential analysis is used to analyze uncertainty propagation, which is described by the variance and mean. The uncertainties of linear velocity and hydrodynamic dispersion coefficient are, respectively, characterized by the second-power and fourth-power of the length scale multiplied by a lumped relationship of variance and covariance of system parameters, i.e. the Peclet number and arrival time of maximum concentration. To validate the applicability for evaluating variance propagation in one-dimensional solute transport, two cases using field data are presented to demonstrate how parametric uncertainty can be caught depending on the manner of sampling.     

Determination of both exposure time and denudation rate from an in situ-produced 10Be depth profile: A mathematical proof of uniqueness. Model sensitivity and applications to natural cases

Measurements of radioactive in situ-produced cosmogenic nuclide concentrations in surficial material exposed to cosmic rays allow either determining the long-term denudation rate assuming that the surface studied has reached steady-state (where production and losses by denudation and radioactive decay are in equilibrium) (infinite exposure time), or dating the initiation of exposure to cosmic rays, assuming that the denudation and post-depositional processes are negligible. Criteria for determining whether a surface is eroding or undergoing burial as well as quantitative information on denudation or burial rates may be obtained from cosmogenic nuclide depth profiles. With the refinement of the physical parameters involved in the production of in situ-produced cosmogenic nuclides, a unique well-constrained depth profile now permits determination of both the exposure time and the denudation rate affecting a surface. In this paper, we first mathematically demonstrate that the exponential decrease of the in situ-produced ¹⁰Be concentrations observed along a depth profile constrains a unique exposure time and denudation rate when considering both neutrons and muons. In the second part, an improved chi-square inversion model is described and tested in the third part with actual measured profiles.     

Upscaling of a dual-permeability Monte Carlo simulation model for contaminant transport in fractured networks by genetic algorithm parameter identification

The transport of radionuclides in fractured media plays a fundamental role in determining the level of risk offered by a radioactive waste repository in terms of expected doses. Discrete fracture networks methods can provide detailed solutions to the problem of modeling the contaminant transport in fractured media. However, within the framework of the performance assessment (PA) of radioactive waste repositories, the computational efforts required are not compatible with the repeated calculations that need to be performed for the probabilistic uncertainty and sensitivity analyses of PA. In this paper, we present a novel upscaling approach, which consists in computing the detailed numerical fractured flow and transport solutions on a small scale and use the results to derive the equivalent continuum parameters of a lean, one-dimensional dual-permeability, Monte Carlo simulation model by means of a genetic algorithm search. The proposed upscaling procedure is illustrated with reference to a realistic case study of $ {}^{239}{\text{Pu}} $ migration taken from literature.     

Multiple decision-making criteria in the transport and burial of hazardous and radioactive wastes

We evaluate the application of various statistical measures for the identification of optimal financial strategies in environmental projects that may be burdened by the consequences of low-probability, high-cost events. Our particular application lies in the area of transport and burial of hazardous and radioactive wastes but our approach applies to a wide range of problems where the utility structure is of the form of gains minus losses, and where limited and/or catastrophic failures may be encountered. We utilize four statistical measures, the expected value, variance, volatility and cumulative probability to compare the outcomes of limited and catastrophic spills. The maximum expected monetary value which is frequently used in the environmental and gas and oil industries as the sole criterion for the selection of optimum actions is seen to lead to erroneous decisions and fails to unambiguously differentiate the economic consequences of limited and catastrophic failures in a project. We demonstrate that unwarranted inclusion of catastrophic scenarios into the decision-making analysis can substantially alter the perspective of a project and guide a corporation away from an investment that could be profitable even under a limited liability case. We conclude by providing a decision-making procedure for cases where the probabilities associated with future events and/or the monetary returns are characterized not by sharp estimates but rather are represented by a range of values.     

Eulerian spatial moments for solute transport in three-dimensional heterogeneous,dual-permeability media

A Eulerian analytical method is developed for nonreactive solute transport in heterogeneous, dual-permeability media where the hydraulic conductivities in fracture and matrix domains are both assumed to be stochastic processes. The analytical solution for the mean concentration is given explicitly in Fourier and Laplace transforms. Instead of using the fast fourier transform method to numerically invert the solution to real space (Hu et al., 2002), we apply the general relationship between spatial moments and concentration (Naff, 1990; Hu et al., 1997) to obtain the analytical solutions for the spatial moments up to the second for a pulse input of the solute. Owing to its accuracy and efficiency, the analytical method can be used to check the semi-analytical and Monte Carlo numerical methods before they are applied to more complicated studies. The analytical method can be also used during screening studies to identify the most significant transport parameters for further analysis. In this study, the analytical results have been compared with those obtained from the semi-analytical method (Hu et al., 2002) and the comparison shows that the semi-analytical method is robust. It is clearly shown from the analytical solution that the three factors, local dispersion, conductivity variation in each domain and velocity convection flow difference in the two domains, play different roles on the solute plume spreading in longitudinal and transverse directions. The calculation results also indicate that when the log-conductivity variance in matrix is 10 times less than its counterpart in fractures, it will hardly influence the solute transport, whether the conductivity field is matrix is treated as a homogeneous or random field.     

Assessing the validity of a lower-dimensional representation of fractures for numerical and analytical investigations

Due to their high aspect ratio fractures are often conceptualized as lower-dimensional structures embedded into the surrounding host matrix. This simplification is typically made within the context of numerical simulation, for the inverse estimation of the matrix-diffusion coefficient from break-through curves or for the derivation of analytical solutions describing flow and transport in a fracture–matrix system. It is generally justified by the so called Lauwerier assumption stating that the transversal dispersion inside the fracture is infinitely fast therefore hampering the formation of gradients across the width of the fracture. In this study we want to verify the applicability of such lower-dimensional modeling. To that end we investigate the occurrence of fracture-scale gradients in a simplified fracture–matrix model by virtue of analytical as well as numerical investigations. The relevant processes modeled are advection, dispersion, matrix diffusion and linear decay. In addition, we also investigate the impact on the inverse estimation of matrix-diffusion coefficients through analytical solutions, which assume a lower-dimensional fracture. Results show that a lower-dimensional modeling of fractures will only lead to errors for early periods of the time-dependent solution. Such errors may however, extent to the steady state if fast radioactive decay is considered. The estimation of the matrix-diffusion coefficient too is affected by the assumption of a lower-dimensional fracture. We see errors as big as 20% for the estimation procedure, the value of which depends on the ratio of the matrix-diffusion vs. the transversal dispersion coefficient. Our analysis suggest that a lower-dimensional representation of fractures is justified for many typical conditions and that special attention must only be paid in a confined number of cases.     

泥岩裂缝性储层地震勘探方法初探

针对胜利油田罗家地区以构造裂缝为主、与断层关系密切的特点,本文采用地震资料相干分析和方差分析技术,力求精细而准确地确定该区目的层的断裂系统;提出了地震吸收系数分析技术,期望为确定目的层段的裂缝发育分布提供重要的证据;利用全方位地震信息进行AVA、VVA方法的研究,可望定量地得到目的层段的裂缝方位和密度分布情况;针对AVA和VVA技术的不足,采用了不同方位角阻抗的变化来检测裂缝发育情况的IPVA方法.实例分析表明,该套泥岩裂缝储层勘探方法取得了良好的效果,在泥岩裂缝油气藏定量检测方面获得了重要进展.     

Complex multiple cosmogenic nuclide concentration and histories in the arid Rio Lluta catchment,northern Chile

Terrestrial cosmogenic nuclide (TCN) concentrations measured in river sediments can be used to estimate catchment‐wide denudation rates. By investigating multiple TCN the steadiness of sediment generation, transport and depositional processes can be tested. Measurements of ¹⁰Be, ²¹Ne and ²⁶Al from the hyper‐ to semi‐arid Rio Lluta catchment, northern Chile, yield average single denudation rates ranging from 12 to 75 m Myr^–1 throughout the catchment. Paired nuclide analysis reveals complex exposure histories for most of the samples and thus the single nuclide estimates do not exclusively represent catchment‐wide denudation rates. The lower range of single nuclide denudation rates (12–17 m Myr^–1), established with the noble gas ²¹Ne, is in accordance with palaeodenudation rates derived from ²¹Ne/¹⁰Be and ²⁶Al/¹⁰Be ratio analysis. Since this denudation rate range is measured throughout the system, it is suggested that a headwater signal is transported downstream but modulated by a complex admixture of sediment that has been stored and buried at proximal hillslope or terrace deposits, which are released during high discharge events. That is best evidenced by the stable nuclide ²¹Ne, which preserves the nuclide concentration even during storage intervals. The catchment‐wide single ²¹Ne denudation rates and the palaeodenuation rates contrast with previous TCN‐derived erosion rates from bedrock exposures at hillslope interfluves by being at least one order of magnitude higher, especially in the lower river course. These results support earlier studies that identified a coupling of erosional processes in the Western Cordillera contrasting with decoupled processes in the Western Escarpment and in the Coastal Cordillera. Copyright © 2008 John Wiley & Sons, Ltd.     

Cosmogenic 21Ne analysis of individual detrital grains: Opportunities and limitations

We use a numerical model describing cosmogenic nuclide acquisition in sediment moving through the upper Gaub River catchment to evaluate the extent to which aspects of source area geomorphology and geomorphological processes can be inferred from frequency distributions of cosmogenic ²¹Ne (²¹Ne_c) concentrations in individual detrital grains. The numerical model predicts the pathways of sediment grains from their source to the outlet of the catchment and calculates the total ²¹Ne_c concentration that each grain acquires along its pathway. The model fully accounts for variations in nuclide production due to changes in latitude, altitude and topographic shielding and allows for spatially variable erosion and sediment transport rates. Model results show that the form of the frequency distribution of ²¹Ne_c concentrations in exported sediment is sensitive to the range and spatial distribution of processes operating in the sediment's source areas and that this distribution can be used to infer the range and spatial distribution of erosion rates that characterise the catchment. The results also show that lithology can affect the form of the ²¹Ne_c concentration distribution indirectly by exerting control on the spatial pattern of denudation in a catchment. Model results further indicate that the form of the distribution of ²¹Ne_c concentrations in the exported sediment can also be affected by the acquisition of ²¹Ne_c after detachment from bedrock, in the diffusive (hillslope) and/or advective (fluvial) domains. However, for such post‐detachment nuclide acquisition to be important, this effect needs to at least equal the nuclide acquisition prior to detachment from bedrock. Copyright © 2009 John Wiley and Sons, Ltd.     

Contaminant transport in one‐dimensional single fractured media: semi‐analytical solution for three‐member decay chain with pulse and Heaviside input sources

A semi‐analytical solution of the one‐dimensional transport for considering a three‐member decay chain in a single fracture with pulse and Heaviside input sources has been studied using the Laplace transform and its numerical inversion. The results reveal that breakthrough curves of dimensionless concentration for the decay chain of Np‐237, U‐233, and Th‐229 in the fracture can be well demonstrated in the temporal and spatial domains. The conditions with and without retardation effects are also compared. During the preliminary screening phase the solutions are suitable for performance assessment on radioactive waste disposal sites under a one‐dimensional single fracture condition. Copyright © 2007 John Wiley & Sons, Ltd.     

UNCERTAINTY AND SENSITIVITY ANALYSES OF SEDIMENT TRANSPORT FORMULAS

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Probabilistic risk and uncertainty analysis for bioremediation of four chlorinated ethenes in groundwater

Groundwater contamination risk assessment for health-threatening compounds should benefit from a stochastic environmental risk assessment which considers the effects of biological, chemical, human behavioral, and physiological processes that involve elements of biotic and abiotic aquifer uncertainty, and human population variability. This paper couples a complex model of chemical degradation and transformation with movement in an aquifer undergoing bioremediation to generate a health risk analysis for different population cohorts in the community. A two-stage Monte Carlo simulation has separate stages for population variability and aquifer uncertainty yielding a computationally efficient and conceptually attractive algorithm. A hypothetical example illustrates how risk variance analysis can be conducted to determine the distribution of risk, and the relative impact of uncertainty and variability in different sets of parameters upon the variation of risk values for adults, adolescents, and children. The groundwater example considers a community water supply contaminated with chlorinated ethenes. Biodegradation pathways are enhanced by addition of butyrate. The results showed that the contribution of uncertainty to the risk variance is comparable to that of variability. Among the uncertain parameters considered, transmissivity accounted for the major part of the output variance. Children were the most susceptible and vulnerable population cohort.     

Incorporation of conceptual and parametric uncertainty into radionuclide flux estimates from a fractured granite rock mass

Detailed numerical flow and radionuclide simulations are used to predict the flux of radionuclides from three underground nuclear tests located in the Climax granite stock on the Nevada Test Site. The numerical modeling approach consists of both a regional-scale and local-scale flow model. The regional-scale model incorporates conceptual model uncertainty through the inclusion of five models of hydrostratigraphy and five models describing recharge processes for a total of 25 hydrostratigraphic–recharge combinations. Uncertainty from each of the 25 models is propagated to the local-scale model through constant head boundary conditions that transfer hydraulic gradients and flow patterns from each of the model alternatives in the vicinity of the Climax stock, a fluid flux calibration target, and model weights that describe the plausibility of each conceptual model. The local-scale model utilizes an upscaled discrete fracture network methodology where fluid flow and radionuclides are restricted to an interconnected network of fracture zones mapped onto a continuum grid. Standard Monte Carlo techniques are used to generate 200 random fracture zone networks for each of the 25 conceptual models for a total of 5,000 local-scale flow and transport realizations. Parameters of the fracture zone networks are based on statistical analysis of site-specific fracture data, with the exclusion of fracture density, which was calibrated to match the amount of fluid flux simulated through the Climax stock by the regional-scale models. Radionuclide transport is simulated according to a random walk particle method that tracks particle trajectories through the fracture continuum flow fields according to advection, dispersion and diffusional mass exchange between fractures and matrix. The breakthrough of a conservative radionuclide with a long half-life is used to evaluate the influence of conceptual and parametric uncertainty on radionuclide mass flux estimates. The fluid flux calibration target was found to correlate with fracture density, and particle breakthroughs were generally found to increase with increases in fracture density. Boundary conditions extrapolated from the regional-scale model exerted a secondary influence on radionuclide breakthrough for models with equal fracture density. The incorporation of weights into radionuclide flux estimates resulted in both noise about the original (unweighted) mass flux curves and decreases in the variance and expected value of radionuclide mass flux.     

USING SPECTRAL ANALYSIS TO DETECT SENSOR NOISE AND CORRECT TURBULENCE INTENSITY AND SHEAR STRESS ESTIMATES FROM EMCM FLOW RECORDS

Electromagnetic current meters (EMCMs) are frequently used to gather turbulent velocity records in rivers and estuaries. Experience has shown that, on occasion, the output of these sensors can be affected by contamination from various noise sources. These noises may be limited to narrow bands of frequencies and thus fail to produce conspicuous increases in observed signal variance. Such ‘narrow-band’ noises can be difficult to identify from simple inspection of signal traces or variance levels, yet degrade estimates of turbulence statistics, in particular covariances (used to calculate Reynolds shear stress). This paper demonstrates the usefulness of spectral analysis to detect and characterize narrow-band noise components in turbulent flow records. Statistical principles underlying the use of spectral analysis for noise detection are briefly reviewed. Examples of u and v velocity spectra and cospectra are then presented from actual EMCM velocity records from flume and field deployments that were found to be contaminated by such noises. The sensitivity of the shear stress estimates to even minor noise levels is demonstrated. The use of spectral analysis to correct variance (turbulence intensity) and covariance (shear stress) estimates obtained from records contaminated by narrow-band noise is also illustrated.     

Gas tracer transport in a heterogeneous fracture in two-phase flow conditions. Experimental and modeling results

Large amounts of gas can result from anaerobic corrosion of metals and from chemical and biological degradation of organic substances in underground repositories for radioactive waste. Gas generation may lead to the formation of a buoyant gas phase bubble (i.e. zone with increased gas saturation surrounded by water) and to the migration of radioactive gaseous species. In this situation, gaseous species migration is controlled by (1) advection, dispersion and diffusion within the gas bubble, and (2) dissolution in the water surrounding the gas bubble and diffusion of the dissolved species away from the interface. A number of gas tracer tests were performed in the framework of the GAs Migration (GAM) project to study the role played by dissolution/diffusion phenomena in gas transport. Tracers were selected to display a large range of solubility and diffusion coefficients, which should have led to significant chromatographic separation in the breakthrough curves (BTCs) of the tracers. However, measured BTCs displayed much smaller chromatographic separation than expected. These curves were interpreted using (1) a numerical model of multiphase flow and tracer transport in the fracture plane and diffusion into the immobile water, and (2) a simple two box model. Results showed that dissolution/diffusion into immobile water regions played a small role, and tailing appears to have been largely controlled by diffusion into dead gas volumes, such as boreholes.     

Shear dispersion in a fracture with porous walls

We present an analytical expression for the shear dispersion during solute transport in a coupled fracture–matrix system. The dispersion coefficient is obtained in a fracture with porous walls by taking into account an accurate boundary condition at the interface between the matrix and fracture, and the results were compared with those in a non-coupled system. The analysis presented identifies three regimes: diffusion-dominated, transition, and advection-dominated. The results showed that it is important to consider the exchange of solute between the fracture and matrix in development of the shear dispersion coefficient for the transition and advection-dominated regimes. The new dispersion coefficient is obtained by imposing the continuity of concentrations and mass fluxes along the porous walls. The resulting equivalent transport equation revealed that the effective velocity in a fracture increases while the dispersion coefficient decreases due to mass transfer between the matrix and fracture. A larger effective advection term leads to greater storage of mass in the matrix as compared with the classical double-porosity model with a non-coupled dispersion coefficient. The findings of this study can be used for modeling of tracer tests as well as fate, transport, and remediation of groundwater contaminants in fractured rocks.     

Gas tracer transport through a heterogeneous fracture zone under two phase flow conditions: Model development and parameter sensitivity

Large amounts of gas can result from anaerobic corrosion of metals and from chemical and biological degradation of organic substances in underground repositories for radioactive waste. Gas generation may lead to the formation of a gas phase bubble and to the migration of radioactive gaseous species. Transport occurs in, at least, in two forms: (1) gas bubble, migration is controlled by advection, dispersion and diffusion in the gas phase, and (2) within water pockets, the dissolved species migrate mainly by diffusion. We consider a two-dimensional system representing an isolated heterogeneous fractured zone. A dipole gas flow field is generated and gas tracers are injected. The delay in the breakthrough curves is studied. A simple method is used to solve the gas species transport equations in multiphase conditions. This method is based on a formal analogy between the equations of gas transport in a two phase system and the equations of solute tracer transport in water saturated systems. We perform a sensitivity analysis to quantify the relevance of the various transport mechanisms. We find that gas tracer migration is very sensitive to gas tracer solubility, which affects gas tracer transport of both mobile and immobile zones, and shows high sensitivity to diffusion in the gas phase, to heterogeneity and to gas pressure, but the largest sensitivity was observed with respect to injection borehole properties, i.e. borehole volume and water filled fraction.     

Three-dimensional numerical analysis of magma transport through a pre-existing fracture in the crust

Magmas are transported through pre-existing fractures in many repeatedly erupting volcanoes. The study of this special process of magma transport is fundamentally important to understand the mechanisms and conditions of volcanic eruptions. In this paper, we numerically simulate the magma propagation process through a pre-existing vertical fracture in the crust by using the combined finite difference method (FDM), finite element method (FEM) and discontinuous deformation analysis (DDA) approach. FDM is used to analyze magma flow in the pre-existing fracture, FEM is used to calculate the opening of the fracture during magma intrusion, and DDA is used to deal with the contact of the closed fracture surfaces. Both two-dimensional (2D) and three-dimensional (3D) examples are presented. Parametric studies are carried out to investigate the influence of various physical and geometric parameters on the magma transport in the pre-existing fracture. We have considered magma chamber depth ranging from 7 km to 10 km under the crust surface, magma viscosity ranging from 2 × 10⁻² to 2 × 10⁻⁷ MPa s, and the density difference between the magma and host rock ranging from 300 to 700 kg/m³. The numerical results indicate that (1) the fluid pressure p varies gradually along the depth, (2) the shape of the magma body during propagation is like a torch bar and its width ranges from 2 m to 4 m approximately in the 3D case and 10 m to 50 m in the 2D case for the same physical parameters used, (3) the crust surface around the pre-existing fracture begins to increase on both sides of the fracture, forms a trough between them, then gradually uplifts during the transport of the magma, and finally takes the shape of a crater when the magma reaches the surface. We have also examined the influence of physical and geometric parameters on the minimum overpressure for magma transport in the 3D case. The numerical results show that our numerical technique presented in this paper is an effective tool for simulating magma transport process through pre-existing fractures in the crust.     

Concentration rebound following in situ chemical oxidation in fractured clay

A two-dimensional, transient-flow, and transport numerical model was developed to simulate in situ chemical oxidation (ISCO) of trichloroethylene and tetrachloroethylene by potassium permanganate in fractured clay. This computer model incorporates dense, nonaqueous phase liquid dissolution, reactive aquifer material, multispecies matrix diffusion, and kinetic formulations for the oxidation reactions. A sensitivity analysis for two types of parameters, hydrogeological and engineering, including matrix porosity, matrix organic carbon, fracture aperture, potassium permanganate dosage, and hydraulic gradient, was conducted. Remediation metrics investigated were the relative rebound concentrations arising from back diffusion and percent mass destroyed. No well-defined correlation was found between the magnitude of rebound concentrations during postremedy monitoring and the amount of contaminant mass destroyed during the application. Results indicate that all investigated parameters affect ISCO remediation in some form. Results indicate that when advective transport through the fracture is dominant relative to diffusive transport into the clay matrix (large System Peclet Number), permanganate is more likely to be flushed out of the system and treatment is not optimal. If the System Peclet Number is too small, indicating that diffusion into the matrix is dominant relative to advection through the fracture, permanganate does not traverse the entire fracture, leading to postremediation concentration rebound. Optimal application of ISCO requires balancing advective transport through the fracture with diffusive transport into the clay matrix.