Dispersion in doublet-type flows through highly anisotropic porous formations

Steady doublet-type flow takes place in a porous formation, where the log-transform $Y = \ln K$ of the spatially variable hydraulic conductivity $K$ is regarded as a stationary random field of two-point autocorrelation $\rho _Y$ . A passive solute is injected at the source in the porous formation and we aim to quantify the resulting dispersion process between the two lines by means of spatial moments. The latter depend on the distance $\ell$ between the lines, the variance $\sigma ^2_Y$ of $Y$ and the (anisotropy) ratio $\lambda$ between the vertical and the horizontal integral scales of $Y$ . A simple (analytical) solution to this difficult problem is obtained by adopting a few simplifying assumptions: (i) a perturbative solution, which regards $\sigma ^2_Y$ as a small parameter, of the velocity field is sought; (ii) pore-scale dispersion is neglected; and (iii) we deal with a highly anisotropic formation ( $\lambda \lesssim 0.1$ ). We focus on the longitudinal spatial moment, as it is of most importance for the dispersion mechanism. A general expression is derived in terms of a single quadrature, which can be straightforwardly carried out once the shape of $\rho _Y$ is specified. Results permit one to grasp the main features of the dispersion processes as well as to assess the difference with similar mechanisms observed in other non-uniform flows. In particular, the dispersion in a doublet-type flow is observed to be larger than that generated by a single line. This effect is explained by noting that the advective velocity in a doublet, unlike that in source/line flows, is rapidly increasing in the far field owing to the presence there of the singularity. From the standpoint of the applications, it is shown that the solution pertaining to $\lambda \to 0$ (stratified formation) provides an upper bound for the dispersion mechanism. Such a bound can be used as a conservative limit when, in a remediation procedure, one has to select the strength as well as the distance $\ell$ of the doublet. Finally, the present study lends itself as a valuable tool for aquifer tests and to validate more involved numerical codes accounting for complex boundary conditions.

Truncation Effect on Estimation of Transport Parameters for Slug-injection Tracer Tests

For slug-injection tracer tests, tracer concentrations below the detection limit of the measurement instrument can cause truncation of the observed data. This study investigated the truncation effect on the estimation error of parameters based on analytical solutions and the results of a laboratory-scale experiment. Spatial moment analysis was performed to estimate the measured total mass and transport parameters, including the pore velocity and the longitudinal and transverse dispersivities. Increasing the travel distance and detection limit caused the measured mass and dispersivities to be underestimated regardless of the dimensionality because hydrodynamic dispersion occurs with increasing travel distance, which smoothens the concentration front. The one- and two-dimensional cases showed that the truncation effect on the measured mass and longitudinal dispersivity depended on dimensionality. In contrast, the pore velocity showed no such dependence; the center of mass did not change as the unmeasured portion due to truncation was increased because the plume, which exhibited a Gaussian distribution, was truncated symmetrically. In the experiment, the measured mass and dispersivities likewise depended on the travel distance and detection limit, but there were large differences in the detection limit at which the dimensionless parameter reached a value of zero between the experimental results and analytical solution. This is because the initial plume in the experiment was of a finite size. Thus, experimental design factors such as the scale, device, and dimensionality should be considered to minimize the estimation error of transport parameters, excluding the pore velocity.

Generalized Empirical Likelihood-Based Kernel Estimation of Spatially Similar Densities

This study concerns the estimation of spatially similar densities, each with a small number of observations. To achieve flexibility and improved efficiency, we propose kernel-based estimators that are refined by generalized empirical likelihood probability weights associated with spatial moment conditions. We construct spatial moments based on spline basis functions that facilitate desirable local customization. Monte Carlo simulations demonstrate the good performance of the proposed method. To illustruate its usefulness, we apply this method to the estimation of crop yield distributions that are known to be spatically similar.

Population dynamics with spatial structure and an Allee effect

Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

Population dynamics with spatial structure and an Allee effect

Allee effects describe populations in which long-term survival is only possible if the population density is above some threshold level. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to population extinction, whereas initial densities above the threshold eventually asymptote to some positive carrying capacity density. Mean field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as short-range competition and dispersal. The influence of such non mean-field effects has not been studied in the presence of an Allee effect. To address this we develop an individual-based model (IBM) that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework accurately recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure that mean-field models neglect. For example, we show that there are cases where the mean-field model predicts extinction but the population actually survives and vice versa. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

Effects of Rainfall Spatial Distribution on the Relationship between Rainfall Spatiotemporal Resolution and Runoff Prediction Accuracy

We studied how rainfall spatial distribution affects the relationship between rainfall spatiotemporal resolution and runoff prediction accuracy under real field conditions. We gathered radar rainfall and discharge data for three rainfall events. These rainfall-runoff events were then reproduced using a kinematic wave model. Modeling accuracy was estimated quantitatively using the Nash–Sutcliffe model efficiency coefficient and peak discharge ratio. Normalized root-mean-square error ( nRMSE ), skewness ( S k ), and second scaled spatial moment of catchment rainfall ( δ 2 ) were employed to quantify rainfall spatial distribution characteristics. By relating the accuracy of modeling results to the rainfall spatial characteristics using various rainfall spatiotemporal resolutions, we found that the modeling results converged to a value as the nRMSE , | S k | and | 1 − δ 2 | decreased. That is, rainfall spatial distributions affect the relationship between lower limit of rainfall spatiotemporal resolution for runoff models and runoff prediction accuracy.