Particle Travel Times Research Articles

Dispersion and connectivity in flow through fractured networks

Abstract A stochastic modeling technique in a 2‐D discrete fracture network with two orthogonal sets of fractures has been developed to observe mass transport dispersion, and to investigate the scaling law related to the mean square displacement of particle travel time, with various percolation probabilities. A law with a fractal dimension for dispersion in the discrete fracture network is estimated by analyzing the random walk with percolation theory, and by using the particle tracking method. Emphasis is placed on understanding how fracture connectivity influences dispersion in flows through fractured networks. Simulation results show that the distribution of masses is skewed or multimodal due to the low interconnection of fractures, and has an approximately Gaussian distribution at the high level of interconnection. Dispersion in fractured media is affected by the connectivity of fractures. Based on percolation theory analysis, numerical results show that there exists a threshold for fracture interconn...

Uncertainty and Sensitivity Analyses of Groundwater Travel Time in a Two-Dimensional Variably-Saturated Fractured Geologic Medium

ABSTRACTThis paper presents a method for sensitivity and uncertainty analyses of a hypothetical nuclear waste repository located in a layered and fractured unconfined aquifer. Groundwater travel time (GWTlT) has been selected as the performance measure. The repository is located in the unsaturated zone, and the source of aquifer recharge is due solely to steady infiltration impinging uniformly over the surface area that is to be modeled. The equivalent porous media concept is adopted to model the fractured zone in the flow field. The evaluation of pathlines and travel time of water particles in the flow domain is performed based on a Lagrangian concept. The Bubnov-Galerkin finite-element method is employed to solve the primary flow problem (non-linear), the equation of motion, and the adjoint sensitivity equations. The matrix equations are solved with a Gaussian elimination technique using sparse matrix solvers. The sensitivity measure corresponds to the first derivative of the performance measure (GWTT) with respect to the parameters of the system. The uncertainty in the computed GWTT is quantified by using the first-order second-moment (FOSM) approach, a probabilistic method that relies on the mean and variance of the system parameters and the sensitivity of the performance measure with respect to these parameters. A test case corresponding to a layered and fractured, unconfined aquifer is then presented to illustrate the various features of the method.

An exact solution of solute transport by one-dimensional random velocity fields

The problem of one-dimensional transport of passive solute by a random steady velocity field is investigated. This problem is representative of solute movement in porous media, for example, in vertical flow through a horizontally stratified formation of variable porosity with a constant flux at the soil surface. Relating moments of particle travel time and displacement, exact expressions for the advection and dispersion coefficients in the Focker-Planck equation are compared with the perturbation results for large distances. The first- and second-order approximations for the dispersion coefficient are robust for a lognormal velocity field. The mean Lagrangian velocity is the harmonic mean of the Eulerian velocity for large distances. This is an artifact of one-dimensional flow where the continuity equation provides for a divergence free fluid flux, rather than a divergence free fluid velocity.

On modelling hillslope water flow paths and travel times

This paper considers a Darcian representation of hillslope water flow to derive flow paths and particle travel times under transient conditions of varying degrees of saturation. A finite element solution of a porous medium flow equation is used to derive specific discharge fields. Interpolation within these fields in relation to prevailing water content allows water particles to be tracked within the slope material according to mean pore water velocities, and allows the times of such hillslope travel to be determined. Examples of specific discharge fields, particle tracks and slope-base arrival times are discussed.

Particle velocity interpolation in block‐centered finite difference groundwater flow models

A block‐centered, finite difference model of two‐dimensional groundwater flow yields velocity values at the midpoints of interfaces between adjacent blocks. Method of characteristics, random walk and particle‐tracking models of solute transport require velocities at arbitrary particle locations within the finite difference grid. Particle path lines and travel times are sensitive to the spatial interpolation scheme employed, particularly in heterogeneous aquifers. This paper briefly reviews linear and bilinear interpolation of velocity and introduces a new interpolation scheme. Linear interpolation of velocity is consistent with the numerical solution of the flow equation and preserves discontinuities in velocity caused by abrupt (blocky) changes in transmissivity or hydraulic conductivity. However, linear interpolation yields discontinuous and somewhat unrealistic velocities in homogeneous aquifers. Bilinear interpolation of velocity yields continuous and realistic velocities in homogeneous and smoothly heterogeneous aquifers but does not preserve discontinuities in velocity at abrupt transmissivity boundaries. The new scheme uses potentiometric head gradients and offers improved accuracy for nonuniform flow in heterogeneous aquifers with abrupt changes in transmissivity. The new scheme is equivalent to bilinear interpolation in homogeneous media and is equivalent to linear interpolation where gradients are uniform. Selecting the best interpolation scheme depends, in part, on the conceptualization of aquifer heterogeneity, that is, whether changes in transmissivity occur abruptly or smoothly.

MASS TRANSPORT ANALYSIS FOR FRACTURE NETWORKS CONSIDERING MATRIX DIFFUSION

A mathematical model is presented for mass transport in fracture networks. The model is based on particle tracking method and is capable of simulating interactions between rock matrix and fractures. An analytical solution of advective transport equation considering matrix diffusion is used as a probability density function to determine the travel time of particles in each fracture element. By applying the model to some test problems and comparing results with an analytical solution and a solution by finite element method, the proposed model is found to have enough accuracy and reliability. In order to examine the effect of matrix diffusion on mass transport, fracture networks are generated stochastically. The result shows that matrix diffusion has less retardation effect as the distribution of fracture apertures becomes wider.

Particle travel times of contaminants incorporated into a planning model for groundwater plume capture : Greenwald, R M; Gorelick, S M J HydrolV107, N1–4, 30 May 1989, P73–98

Particle travel times of contaminants incorporated into a planning model for groundwater plume capture : Greenwald, R M; Gorelick, S M J HydrolV107, N1–4, 30 May 1989, P73–98

Particle travel times of contaminants incorporated into a planning model for groundwater plume capture

In problems of groundwater contamination, the cost of long-term remediation can run into millions of dollars. Often the objective of the cleanup strategy involves minimizing or constraining the time it takes to remove all contaminants from the aquifer. Here, a planning model for the withdrawal of contaminated groundwater is presented that enables one to consider remediation problems in which cleanup time is an important component. An existing quasi-analytic solution for advective contaminant transport is combined with nonlinear optimization to determine the distribution of pumping and injection rates. Although the simulation model is simple, its combined use with the optimization procedure is a useful tool. Examples demonstrate how cleanup time as a management variable can dictate the design of the restoration system. The relationship between pumping rates and the ultimate cleanup time proved to be a highly nonlinear function. Inspection of the “cleanup time surface” for a reduced problem provides insight into the nature of this nonlinearity and how it influences the selected restoration design. Analysis indicated that increasing the total pumping rate can result in an increase in total cleanup time due to the deflection of contaminant travel paths.

One-dimensional kinematics of particle stream flow with application to solar wind simulation

A simple kinematic method for determining the particle velocity distribution of a model solar wind for which the spatial distribution of particles is given as a function of particle travel time has been developed by Hakamada and Akasofu (1982). Here we formalize their method mathematically and derive an inverse procedure for determining the particle distribution from a given velocity distribution. This inverse procedure is then applied to a simulated velocity distribution obtained from an MHD finite difference code.

Travel Times From Buried and Surface Infiltration Point Sources

Solutions in closed form are found for the travel times of marked particles from buried and surface infiltration point sources. The study is based on the quasi‐linear analysis of steady three‐dimensional unsaturated flow. The results are presented graphically. The mathematical analysis applies to all linear convection‐diffusion processes from continuous point sources in the appropriate infinite and seminfinite regions. In the soil water context a limitation is that travel velocities are based on a mean volumetric moisture content.