Abstract. Extended tailing of tracer breakthrough is often observed in pulse injection tracer tests conducted in fractured geologic media. This behavior has been attributed to diffusive exchange of tracer between mobile fluids traveling through channels in fractures and relatively stagnant fluid between fluid channels, along fracture walls, or within the bulk matrix. We present a field example where tracer breakthrough tailing apparently results from nondiffusive transport. Tracer tests were conducted in a fractured crystalline rock using both a convergent and weak dipole injection and pumping scheme. Deuterated water, bromide, and pentafluorobenzoic acid were selected as tracers for their wide range in molecular diffusivity. The late time behavior of the normalized breakthrough curves were consistent for all tracers, even when the pumping rate was changed. The lack of separation between tracers of varying diffusivity indicates that strong breakthrough tailing in fractured geologic media may be caused by advective transport processes. This finding has implications for the interpretation of tracer tests designed to measure matrix diffusion in situ and the prediction of contaminant transport in fractured rock. IntroductionPulse injection tracer tests conducted in fractured rocks typically result in a recovered concentration history (breakthrough) that is highly skewed to later times, particularly when compared to advection and dispersion in unconsolidated porous media. The most common explanation for this "breakthrough tailing" is that while some of the tracer moves quickly through open channels, a significant fraction of the tracer is delayed by diffusive exchange with the rock matrix. Tracer mass that moves primarily through open channels results in an early peak in concentration, while the tracer that is heavily influenced by diffusive exchange results in a low concentration "tail" over an extended period of time. The exchange of mass between relatively mobile fluid in the fracture and relatively immobile fluid in the rock matrix is usually called "matrix

It is common to represent solute tranport in heterogeneous formations in terms of the resident concentration C(x, t), regarded as a random space function. The present study investigates the alternative representation by q, the solute mass flux at a point of a control plane normal to the mean flow. This representation is appropriate for many field applications in which the variable of interest is the mass of solute discharged through a control surface. A general framework to compute the statistical moments of q and of the associated total solute discharge Q and mass M is established. With x the direction of the mean flow, a solute particle is crossing the control plane at y = η, z = ζ and at the travel (arrival) time τ. The associated expected solute flux value is proportional to the joint probability density function (pdf) g1 of η, ζ and τ, whereas the variance of q is shown to depend on the joint pdf g2 of the same variables for two particles. In turn, the statistical moments of η, ζ and τ depend on those of the velocity components through a system of stochastic ordinary differential equations. For a steady velocity field and neglecting the effect of pore‐scale dispersion, a major simplification of the problem results in the independence of the random variables η, ζ and τ. As a consequence, the pdf of η and ζ can be derived independently of τ. A few approximate approaches to derive the statistical moments of η, ζ and τ are outlined. These methods will be explored in paper 2 in order to effectively derive the variances of the total solute discharge and mass, while paper 3 will deal with the nonlinear effect of the velocity variance upon the moments of η, ζ and τ

[1] Conceptual and mathematical models are presented that explain tracer breakthrough tailing in the absence of significant matrix diffusion. Model predictions are compared to field results from radially convergent, weak-dipole, and push-pull tracer experiments conducted in a saturated crystalline bedrock. The models are based upon the assumption that flow is highly channelized, that the mass of tracer in a channel is proportional to the cube of the mean channel aperture, and the mean transport time in the channel is related to the square of the mean channel aperture. These models predict the consistent À2 straight line power law slope observed in breakthrough from radially convergent and weak-dipole tracer experiments and the variable straight line power law slope observed in push-pull tracer experiments with varying injection volumes. The power law breakthrough slope is predicted in the absence of matrix diffusion. A comparison of tracer experiments in which the flow field was reversed to those in which it was not indicates that the apparent dispersion in the breakthrough curve is partially reversible. We hypothesize that the observed breakthrough tailing is due to a combination of local hydrodynamic dispersion, which always increases in the direction of fluid velocity, and heterogeneous advection, which is partially reversed when the flow field is reversed. In spite of our attempt to account for heterogeneous advection using a multipath approach, a much smaller estimate of hydrodynamic dispersivity was obtained from push-pull experiments than from radially convergent or weak dipole experiments. These results suggest that although we can explain breakthrough tailing as an advective phenomenon, we cannot ignore the relationship between hydrodynamic dispersion and flow field geometry at this site. The design of the tracer experiment can severely impact the estimation of hydrodynamic dispersion and matrix diffusion in highly heterogeneous geologic media. INDEX TERMS:1829 Hydrology: Groundwater hydrology; 1832 Hydrology: Groundwater transport; 5104 Physical Properties of Rocks: Fracture and flow; KEYWORDS: fractured rock, matrix diffusion, tracer tests, contaminant transport, advection dispersion, channeling Citation: Becker, M. W., and A. M. Shapiro, Interpreting tracer breakthrough tailing from different forced-gradient tracer experiment configurations in fractured bedrock, Water Resour.

Uncertainty in the mass flux for advection dominated solute movement in heterogeneous porous media is investigated using the Lagrangian framework developed in paper 1 by Dagan et al. (this issue). Expressions for the covariance of the mass flux and cumulative mass flux are derived as functions of the injection volume and sampling area size relative to the scale of heterogeneity. The result is illustrated for solute advection in three types of heterogeneous porous media: stratified formations, two‐ and three‐dimensional porous media; small perturbation approximation is used for the two‐ and three‐dimensional cases. Variances of the mass flux and cumulative mass flux are evaluated as functions of the injection volume (area) scale versus log‐hydraulic conductivity integral scale. The greatest decrease in coefficient of variation (CV) of the mass flux is for the source scale 1–5 times the hydraulic conductivity integral scale; further increase in the source size decreases CV comparatively less. The variance of the cumulative mass flux (or total discharge) indicates that for the source size of 20 hydraulic conductivity integral scales, the transport conditions are almost ergodic. The present results also indicate that the cumulative mass flux is a relatively robust quantity for describing field‐scale solute transport.

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.