Stochastic analysis of solute arrival time in heterogeneous porous media

Abstract

Longitudinal advective solute movement in heterogeneous porous media is investigated by considering the solute arrival time at a plane perpendicular to the mean fluid velocity. The moments of the solute arrival time are defined in terms of the stochastic properties of a statistically anisotropic hydraulic conductivity field. The flux-averaged concentration is specified by introducing the moments of the arrival time into a probability density function for the arrival time. The quadratic dependence of the arrival time variance on position in the vicinity of an injection point is indicative of a nondiffusive process. For the assumed spatial correlation of the hydraulic conductivity, the variance of the arrival time asymptotically approaches a linear dependence on the position from the injection point similar to a diffusion process. The impact of assuming solute movement to be a diffusive process from the onset of the solute injection causes erroneous estimates of flux-averaged concentrations at distances from the injection point that are of the order of the correlation length of the hydraulic conductivity. The arrival time analysis and the particle position analysis given in Dagan (1982, 1984) are complementary interpretations of advective solute movement that yield different definitions of the solute concentration; the position analysis intrinsically defines the resident or volume-averaged concentration, while the flux-averaged concentration is defined from the arrival time analysis. The temporal variation of the resident and flux-averaged concentration are similar at a given position for small values of the variance of the hydraulic conductivity, or at distances from the solute injection point that are large relative to correlation length of the hydraulic conductivity. 18, 1193-1214, 1982. Todorovic, P., A stochastic model of longitudinal diffusion in porous media, Water Resour. Res., 6, 211-222, 1970.

Injection mode implications for solute transport in porous media: Analysis in a stochastic Lagrangian Framework

“…The temporal moments in Table 1 Table 2). The mean arrival time for the uniform injection mode is the travel distance divided by the harmonic mean velocity, a result commonly associated with "instantaneous" injections into stratified systems [e.g., Shapiro and Cvetkovic, 1988]. However, injection in flux yields a mean arrival time equal to travel distance divided by arithmetic mean velocity.…”

“…In particular, spatial variability in groundwater velocity caused by hydraulic conductivity heterogeneity is generally regarded as the most important mechanism that induces spreading at the field scale [e.g., Dagan, 1989]. This advection-dominated nature of nonreactive solute transport has lead to the development of stochastic-advective approaches to transport modeling, which have followed richly diverse paths [e.g., Simmons, 1982;Dagan, 1984;Shapiro and Cvetkovic, 1988;Charbeneau, 1988;Jury and Roth, 1990]. These different development paths share the notion that larger-scale velocity variability dominates spreading and that local dispersion may be neglected in modeling average longitudinal transport.…”

Lateral solute mixing processes — A key for understanding field-scale transport of water and solutes

“…If the flow regions are non-interacting, meaning that the boundaries of each region are impermeable for particles, an individual particle remains for its entire travel time within the region it entered. This mechanisms leads to a process which is called stochastic-advective and has often been used to analyze solute transport (e.g., Dagan and Bresler, 1979;Jury, 1982;Simmons, 1982;Shapiro and Cvetkovic, 1988). If a particle can move laterally through the boundary of its flow region, it experiences the velocities of other flow regions.…”

“…Stochastic-advective transport models can be regarded as special cases of multiple region models with noninteracting regions (e.g., Dagan and Bresler, 1979;Jury, 1982;Simmons, 1982;Shapiro and Cvetkovic, 1988). Although lateral mass exchange is neglected, longitudinal spreading of a pulse is achieved by choosing an appropriate shape of the velocity distribution as shown in Fig.…”

Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments

“…Temporal moments have been used for the characterization of conservative transport in heterogeneous aquifers by various authors [Shapiro and Cvetkovic, 1988 [1997] showed that the impact of local-scale dispersion on the second central moment of the integrated breakthrough curve is small for most practical cases and may be even less for kinetically sorbing solutes.…”