On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions

Abstract. Steady flow between a fully penetrating recharging and pumping well (doublet) takes place in a heterogeneous aquifer. The spatially variable hydraulic conductivity is modeled as a lognormal stationary random function, of anisotropic two-point covariance. The latter is characterized by the horizontal and vertical integral scales I and Iv respectively. A tracer, or a reactive solute obeying first-order kinetics, is injected as a pulse or continuously in the recharging well. Our aim is to determine the flux-averaged concentration (the breakthrough curve) in the pumping well as a function of tim, e and of the various parameters of the problem, i.e., o-• (the logconductivity variance), l = l/I (l is the distance between wells), and e = IvI (the anisotropy ratio). A simple solution of this difficult problem is achieved by adopting a few simplifying assumptions: (1) the wells are fully penetrating, of length much larger than I vand of radius r w much smaller than I, (2) a first-order solution in o-• of the flow and transport equations is sought, (3) the anisotropy ratio is small, say e < 0.2, and (4) neglect of the effect of pore-scale dispersion. After determining the travel time, mean and variance, the mean flux-averaged concentration is found by assuming that, is lognormal. In a homogeneous medium there is a large spreading of the solute signal in the pumping well owing to the variation of the travel time among the streamlines connecting the two wells. The effect of heterogeneity is similar to that of pore-scale dispersion; that is, it leads to enhanced spreading and in particular to an early breakthrough. The solution has potential applications to aquifer tests and to evaluation of efficiency of remediation schemes and may serve as a benchmark for numerical models. IntroductionWe consider steady aquifer flow between a recharging well and a pumping one (briefly a doublet, Figure 1), operating under a constant head difference. A solute is injected either for a short period (pulse) or continuously in the recharging well, and the concentration is monitored in the pumping one. Such a configuration is quite common in many applications, for example, as a testing method to determine aquifer properties or in remediation schemes. Therefore it is of interest to model the flow and transport for this type of application. The simplest case is that of a homogeneous formation, which was investigated in the past. Thus, in the recent work of Koplik et al. [1994], the problem is solved for transport of a tracer by advection and by pore-scale dispersion. For the high Peclet numbers characterizing natural formations, the spread of the pulse in the pumping well is dominated by the advective effect. This large effect is due to the nonuniformity of the flow, resulting in differences in the travel times along the streamlines connecting the two wells, the quickest and slowest paths being the ones in the wells plane [Kurowski et al., 1994]. With neglect of pore-scale dispersion, the transport problem can be solved In reality, natur...

“…In the present study we are interested in determining the fluxaveraged [Kreft and Zuber, 1978] concentration in the pumping well, defined by…”

Section: The Latter Is Modeled As Anisotropic With the Horizontal Inmentioning

confidence: 99%

Reactive solute transport in flow between a recharging and a pumping well in a heterogeneous aquifer

Dagan¹,

Indelman²

1999

“…The above solutions for flux concentrations can be transformed to solutions representing resident concentrations using (2) [Kreft and Zuber, 1978;. It should be emphasized that the above solutions diverge only for media possessing a large dispersivity relative to the distance from the inlet to the sampling point.…”

Section: It Is Easily Verified That Substitution Of (2) Into (1) Yielmentioning

confidence: 99%

Boundary effects on solute transport in finite soil columns

Schwartz¹,

McInnes²,

Juo³

et al. 1999

Abstract. This study investigates the influence of inlet and outlet disturbances and formulated boundary conditions on the estimation of the dispersion coefficient and retardation factor for short soil columns. Unsaturated miscible displacement experiments utilizing a Br-tracer were carried out on undisturbed columns of a fine-textured Ultisol. Solutions were applied using either a fritted plate or an array of dispensing tips that produced droplets at a prescribed flow rate. One-and two-layer analytical solutions of the advective-dispersive equation were fitted to effluent concentrations using nonlinear least squares parameter optimization. Comparison of two-layer simulations with experimental data indicated that the analytical solution with a semi-infinite interface boundary best approximated effluent concentrations under the conditions of this study. This solution corresponds to a continuous flux concentration and a macroscopically discontinuous resident concentration at the interface between the soil and porous plates. Parameter estimates were not significantly different with respect to the application method used at the inlet. This may be attributed to a less uniform distribution of solution onto the soil surface by the drip apparatus and/or by the presence of stagnant regions within the inlet reservoir and hence increased dispersion within the inlet platen apparatus. Two-layer simulations indicated that the dispersion coefficient was underestimated by 14-27% when the influence of the inlet and outlet apparatus were not included in the fitted solution of the advective-dispersive equation. In addition, use of one-layer analytical solutions caused the retardation factor to be overestimated by no more than the fractional increase in pore volume imparted by the platen apparatus. IntroductionStudies of solute transport are often limited to the analysis of breakthrough curves over relatively short longitudinal distances. For instance, soil vertical heterogeneity complicates the analysis of transport, and a complete description of the mechanisms often requires that each soil horizon be examined separately. In modeling the data obtained, one uses a mechanistic model with appropriate boundary and initial conditions to fit experimental breakthrough curves and quantify model parameters. When dispersivity is large in comparison to the distance from the inlet to the sampling point, the boundary conditions . Second, the use of reactive media with a wide range in particle size may introduce a certain degree of local nonequilibrium at high pore water velocities. Parker [1984] has shown that the analytical solution that prescribes a finite boundary at both the inlet and outlet fails to describe measured concentrations in the extreme case where local equilibrium between pore regions is not attained. In this paper we are principally concerned with slow pore water velocities where such effects are of minor importance, and hence the one-region advective-dispersive equation (ADE) should provide a good description to experimental...

“…It is hoped that the present discussion of the physical meanings of different formulations of boundary conditions for stochasticadvective models will make it more accessible for general purpose transport modeling. Paraphrasing Krefi and Zuber [1978], a difficulty in the use of mathematical models consists either in choosing a proper set of initial and boundary conditions describing a physical system and finding a solution for these conditions or in finding a The two modes of injection and detection were recognized and discussed in the context of deterministic advective systems, i.e., laminar flow in tubes, by Levenspiel and Turner [1970]. Levenspiel andTurner [1970, p. 1605] defined two particular injection cases when "the fluid velocity is not uniform through the cross section at the injection point": one in which mass is injected in proportion to the local advective velocity and the other in which mass is injected uniformly.…”

Section: Introductionmentioning

confidence: 99%

“…It was already clear to them that the injection mode had a profound impact on subsequent transport. Krefi and Zuber [1978] discussed the correspondence of different boundary conditions (BC) and dependent variables for use with the advective-dispersive equation (ADE) to different injection and detection modes. In the case of the ADE the detection mode determines whether the governing differential equation will be written in terms of a resident concentration (for resident detection) or a flux-averaged concentration (detection in fluid flux).…”

Section: Introductionmentioning

confidence: 99%

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

Demmy¹,

Berglund²,

Graham³

1999

Abstract. The effect of solute injection mode is examined in the stochastic-advective' framework. For three-dimensional heterogeneous aquifers, uniform resident injection of solute results in nonlinear propagation of mass arrival time mean and variance with distance. Injection in flux results in a linear propagation of mean mass arrival time and mass arrival time variance that is both lower and appears to reach a linear regime more rapidly than uniform resident injection. Implementation of instantaneous injections for both modes as boundary conditions for mathematical and numerical models is discussed.