Time-dependent Dispersivity Research Articles

Hydraulic dispersion of diurnal reactive constituents in an open channel eutrophic flow

Summary Closely related to the solar photocycle, plankton growth in eutrophic waters displays a diurnal variation because of photosynthesis and respiration. Presented in this paper is an analytical study of the diurnal variation of mean concentration of plankton and nutrient in an open channel eutrophic flow initiated by an instantaneous emission. The evolution of the concentration is shown driven by the combination of hydraulic dispersion and diurnal reaction between plankton and nutrient. The analytical solution for longitudinal distribution of concentration is rigorously derived and illustrated, based on the time dependent hydraulic dispersivity. Numerical results are presented and characterized by the reaction rate, yield factor and period for the diurnal reaction and the P e clet number of the flow. For typical applications such as ecological risk assessment and environmental impact assessment, the upper and lower limits of critical length and duration of five typical pollutant concentrations are concretely illustrated for given concentration criterions. Remarkable diurnal variations are revealed up to around one third in the critical length and duration for plankton, and about ten percent for nutrient.

Special Issue on Toxics and Pathogens in the Environment

Special Issue on Toxics and Pathogens in the Environment

Concentration Profiles and Spatial Moments for Reactive Transport through Porous Media

AbstractWe use an advection-dispersion-reaction equation, which accounts for both physical and chemical nonequilibrium, to study the effect of heterogeneity of the porous media on a spatial concentration profile. Distance-dependent and time-dependent dispersivity is used to account for the heterogeneity. A sensitivity analysis has been performed to illustrate the effect of various factors on breakthrough curves for reactive transport through homogeneous porous media. An exponential function for the dispersion coefficient is used to simulate experimental data for spatial moments of nonreactive and reactive chemicals. An implicit finite-difference method has been used for the numerical solution of the governing equations. The distance-dependent dispersion coefficient resulted in higher peak value of concentration than the equivalent time-dependent dispersion coefficient. With an increase in time, the spatial concentration profile is steeper for the distance-dependent dispersion coefficient. Experimental dat...

Time-Dependent Dispersivity of Linearly Sorbing Solutes in a Single Fracture with Matrix Diffusion

Field studies show that the variance of travel distance often increases nonlinearly with time elapsed after release of solute tracers. The nonlinear relationship between variance of travel distance and time is attributed to the heterogeneity of the porous media. To describe the transport in such a heterogeneous system, a time-dependent dispersivity is necessary. Though more attention has been devoted toward the study of non-Fickian dispersion at early time, there are no known studies that explicitly describe the dispersivity behavior in a fracture–matrix-coupled system. The observation from numerical results suggests that dispersivity has a time-dependent behavior and it reaches asymptotic values after a long time. The preasymptotic behavior of a solute front in fracture is characterized by increasing effective dispersivity with time. The role of fracture and matrix transport parameters on this behavior is analyzed for linearly sorbing solutes. Approximate expression is provided for the time-dependent dispersivity of the solute front in a single fracture with matrix diffusion and the expression for the time required to attain the asymptotic behavior is also obtained. A comparison of the front dispersivity behavior between parallel multiple fractures with a constant aperture width model and smooth parallel multiple fractures with a varying aperture width model is done.

Effect of sorption intensities on dispersivity and macro-dispersion coefficient in a single fracture with matrix diffusion

Matrix diffusion and sorption are among the key processes impacting the efficiency of natural attenuation in the subsurface. While these processes have been studied extensively in fractured media, limited information exists on the sorption nonlinearity. To address this shortfall, a numerical model has been developed that couples matrix diffusion and nonlinear sorption at the scale of a single fracture using the dual-porosity concept. The study is limited to a constant continuous-solute-source boundary condition. The influence of sorption intensities on dispersivity and macro-dispersion coefficient is investigated using a method of spatial moments. Results suggest that mixing of solutes is significantly lowered by nonlinear sorptive behavior, with respect to the mixing caused by matrix diffusion for linearly sorbing solutes. Also, the magnitude of time dependent dispersivity during the pre-asymptotic regime is lower for nonlinearly sorbing solutes with respect to the linearly sorbing solutes. Reduced mixing is also observed for nonlinearly sorbing solutes under combined mechanisms of matrix diffusion and decay.

Specifying Scale‐dependent Dispersivity in Numerical Solutions of the Convection–Dispersion Equation

Many studies have incorporated various scale‐dependent dispersivity functions into the convection–dispersion equation (CDE). For a given function, there are different ways of specifying the dispersivity values in the discretized subdomains of a numerical solution. These ways would depend on how the function's time or space variable is built into the numerical scheme and on how the different ways may be interrelated. No study has addressed how these different ways may affect resulting breakthrough curves (BTCs) or their applicability to solute transport problems. In this study, five ways were specified and designated as local time, average time, apparent time, local distance, or apparent distance dependent dispersivity in the numerical scheme. The main objective was to demonstrate relationships among these ways by assuming that dispersion in the scale‐dependent CDE was quasi‐Fickian and implicitly based on an estimation of the first moment of the BTC at the given location. How these ways affected calculated BTCs was numerically tested using generalized linear and power scale–dependent dispersivity functions for a one‐dimensional problem with initial solute distribution represented as a Dirac delta function. For either type of function, differences were obvious in BTCs obtained for solute transport conditions with small apparent Peclet numbers using the alternative ways of specifying scale‐dependent dispersivity in the numerical scheme. The differences decreased with increasing apparent Peclet number. The numerical test results also showed that the derived relationships were not applicable for numerically calculating the spatial solute distribution at a given time. These test results were expected to be generally applicable because any other initial‐value solute transport problem can be solved as a superposition of this test problem.

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.

A CONCEPTUAL FRACTAL MODEL FOR DESCRIBING TIME-DEPENDENT DISPERSIVITY 1

Field studies show that the variance of travel distance often increases nonlinearly with time elapsed after release of solute tracers. The nonlinear relationship between variance of travel distance and time is attributed to the heterogeneity of the porous media. To describe the transport in such a heterogeneous system, a time-dependent dispersivity is necessary. In this paper, the basis for the formulation of the Wheatcraft-Tyler model is discussed and its inadequacy and inconsistencies are presented. As a consequence of the above, we developed a new model, which is similar in appearance to that of Wheatcraft and Tyler. The proposed model indicates that the variance of travel distance for a tracer may grow with time raised to the power of D, where D is close to the fractal dimension of the fractal stream tube and ranges from 1.0 to 2.0. When D equals 2.0, the dispersivity will increase linearly with time, which is consistent with the Mercado model. The applicability of our model was evaluated with three field scale transport experiments: the Cape Cod, the Borden, and the Columbus sites. The obtained D values for all three sites were significantly greater than 1.0. Comparison between our model and the fractional-order advection-dispersion equation reveals that the D value in our model is related to the fractional order.

A stream tube model for miscible flow

Large-scale dispersion in heterogeneous porous media is studied by using a simple model based on stochastic calculation of convective flow in a bundle of stream tubes. The advantage of this approach is that there is a local relationship between velocity and permeability in the 1-dimensional space of the stream tubes. Dispersion is due to the variation in stream tube cross-section, related to the permeability field. First, the arrival times of the tracer in the stream tubes are related to the stochastic properties of the permeability field (variance and covariance). Then, transport equations are derived from the moments of the arrival times. The results agree with more complicated studies. For a permeability field with long-range correlation, the transport equation is not unique. It depends on the assumptions involving moments higher than two. Assuming a Gaussian shape for the tracer flux leads to equations similar to the ones obtained in previous studies of time-dependent dispersivity. Without this approximation, the equation is non-local (integrodifferential) and leads to a memory effect. In the last part of this paper, the general results are illustrated with several correlation functions for the permeability field: purely random, exponential and power law covariance, and perfectly layered media.

A stream tube model for miscible flow

Large-scale dispersion in heterogeneous porous media is studied by using a simple model based on stochastic calculation of convective flow in a bundle of stream tubes. The advantage of this approach is that there is a local relationship between velocity and permeability in the 1-dimensional space of the stream tubes. Dispersion is due to the variation in stream tube cross-section, related to the permeability field. First, the arrival times of the tracer in the stream tubes are related to the stochastic properties of the permeability field (variance and covariance). Then, transport equations are derived from the moments of the arrival times. The results agree with more complicated studies. For a permeability field with long-range correlation, the transport equation is not unique. It depends on the assumptions involving moments higher than two. Assuming a Gaussian shape for the tracer flux leads to equations similar to the ones obtained in previous studies of time-dependent dispersivity. Without this approximation, the equation is non-local (integrodifferential) and leads to a memory effect. In the last part of this paper, the general results are illustrated with several correlation functions for the permeability field: purely random, exponential and power law covariance, and perfectly layered media.