Generalized Analytical Solutions of The Advection-Dispersion Equation with Variable Flow and Transport Coefficients

Demand has increased for analytical solutions to determine the velocities and dispersion coefficients that describe solute transport with spatial, temporal, or spatiotemporal variations encountered in the field. However, few analytical solutions have considered spatially, temporally, or spatiotemporally dependent dispersion coefficients and velocities. The proposed solutions consider eight cases of dispersion coefficients and velocities: both spatially dependent, both spatiotemporally dependent, both temporally dependent, spatiotemporally dependent dispersion coefficient with spatially dependent velocity, temporally dependent dispersion coefficient with constant velocity, both constant, spatially dependent dispersion coefficient with spatiotemporally dependent velocity, and constant dispersion coefficient with temporally dependent velocity. The spatial dependence is linear, while the temporal dependence may be exponential, asymptotical, or sinusoidal. An advection–dispersion equation with these variable coefficients was reduced to a non-homogeneous diffusion equation using the pertinent coordinate transform method. Then, solutions were obtained in an infinite medium using Green’s function. The proposed analytical solutions were validated against existing analytical solutions or against numerical solutions when analytical solutions were unavailable. In this study, we showed that the proposed analytical solutions could be applied for various spatiotemporal patterns of both velocity and the dispersion coefficient, shedding light on feasibility of the proposed solution under highly transient flow in heterogeneous porous medium.

ACHROMATIZATION OF THE VOLUME HOLOGRAPHIC OPTICAL ELEMENS (part 2)

The problem of achromatization of a single volume holographic element is solved in this work by choosing the dispersion coefficients of the material at the stages of recording and use. The condition for achromatization of a single volume holographic element was obtained in this work, namely, the difference in the dispersion coefficients of this element at the stages of use and recording should be inversely proportional to the recording wavelength.

Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows

In this study, the performance of two routing procedures were evaluated to estimate the two-dimensional dispersion coefficients. The two-dimensional Stream-Tube Routing Procedure (2D ST-RP) has been widely used to obtain the dispersion coefficients from measured concentration-time curves under the frozen cloud assumption. Meanwhile, the Spatial Routing Procedure (2D S-RP) employs the spatial distributions of concentration to estimate the dispersion coefficients. The performance of the two routing methods were evaluated in aspect of the validity of the frozen cloud assumption and the applicability in the non-Fickian mixing. From the estimation results of dispersion coefficients, the results by the 2D ST-RP included errors due to skewed concentration-time curves which were created by violating the frozen cloud assumption. On the other hand, the 2D S-RP provides accurate dispersion coefficients in the same condition. The estimated results of dispersion coefficients in the meandering channel show that both the 2D ST-RP and the 2D S-RP contained errors due to the non-Fickian mixing properties of the test case. Even with the discrepancies, the 2D S-RP presented more appropriate spatial variabilities along the meander cycle than the results by the 2D ST-RP.

Dispersion Coefficients For Puff at Different Stability Conditions

Slade gave 3 sets of dispersion coefficients for puff i.e, for Unstable, Neutral & Very Stable conditions for 100m and 4000m spread along wind direction. These dispersion coefficients were adopted for stabilities B,D&F respectively. The dispersion coefficients for the remaining 3 stabilities i.e, A,C&E were arrived at by comparing the puff dispersion with the corresponding plumes Pasquil-Gifford had given curves for the different plume spread during A to F stabilities in the same way dispersion curves for puff are plotted for stabilities A to F.

Upscaling of pore-scale mixing and reaction through effective dispersion coefficients

<p>We utilize effective dispersion coefficients to capture the evolution of the mixing interface between two initially segregated species due to the coupled effect of pore-scale heterogeneity and molecular diffusion. These effective dispersion coefficients are defined as the average spatial variance of the solute plume that evolves from a pointlike injection (the transport Green function). We numerically investigate the effective longitudinal dispersion coefficients in two porous media of different structure heterogeneity  and through different Péclet number regimes for each medium. We find that, as distance traveled increases (or time spent), the solute experiences the pore-scale velocity field heterogeneity due to advection and transverse diffusion, resulting in an evolution of the dispersion coefficients. They evolve from the value of molecular diffusion at early time, then undergo an advection dominated regime, to finally reach the value of hydrodynamic dispersion at late times. This means that, at times smaller than the characteristic diffusion time, the effective dispersion coefficients can be notably smaller than the hydrodynamic dispersion coefficient. Therefore, mismatches between pore-scale reaction data from experiment or simulations and Darcy scale predictions based on temporally constant hydrodynamic dispersion can be explained through these differences. We use the effective dispersion coefficients to approximate the transport Green function and to quantify the incomplete mixing occurring at the pore-scale. We evaluate the evolution of two initially segregated species via this methodology. The approach correctly predicts the amount of chemical reaction occuring in reactive bimolecular particle tracking simulations. These results shed light on the upscaling of pore-scale incomplete mixing and demonstrates that the effective dispersion is an accurate measure for the width of the mixing interface between two reactants. </p>

Impact of flow conditions on pore-scale solute mixing: experiments in heterogeneous 2D porous media

<p>Solute mixing mediated by flow in porous media plays a significant role in controlling reaction rates in subsurface environments. In many practical cases, incomplete mixing—inhomogeneous solute concentrations—occurs at the pore-scale, limiting local and thus upscaled reaction rates, and renders their prediction based on effective dispersion coefficients derived from dispersion models (or by assuming Taylor-Aris dispersion) inaccurate. We perform solute transport experiments in transparent, quasi-two-dimensional, soil analog models to investigate the relationships between pore-scale solute dispersion and mixing under different flow conditions. We use Fluorescein as a conservative tracer and record its fluorescence intensity in monochrome images at fixed time intervals. We convert the fluorescence intensity to solute concentration fields based on a calibration curve obtained with various homogeneous solute concentrations and subsequently compute concentration gradients. Our images provide evidence for incomplete mixing at the pore-scale and show strong gradients transverse to the overall flow direction. We fit the mean longitudinal concentration profile to an analytical solution of the advection-dispersion equation and compute the effective longitudinal dispersion coefficient. Based on the lamellar mixing theory, we also infer an effective diffusion coefficient relevant to the mean concentration gradient’s dynamics. By comparing these two diffusion/dispersion coefficients in saturated flow conditions, we show that while their values are similar at low Péclet, their scaling behaviors as a function of Péclet are different. Hence, as pointed out by several previous studies, modeling reactive transport processes requires accounting for a mixing behavior driven by a diffusive process that cannot entirely be described by the solute dispersion coefficient. We extend this work by varying the saturation degree in the experiments and our samples' structural heterogeneity to investigate how flow desaturation and porous medium structure impact solute mixing.</p>