Y. W. Tsang

Usage of “Equivalent apertures” for rock fractures as derived from hydraulic and tracer tests

In the literature during the past several years there appear numerous references to the “equivalent aperture” of a rough-walled rock fracture as derived from various hydraulic and tracer tests. However, the similar or even identical terms used by different researchers for “equivalent aperture” often do not have the same meaning. This has led to some confusion in the comparison of their results. In particular, there is a serious apparent contradiction in the claims of some authors that “equivalent apertures” derived from tracer tests are much larger than those derived from hydraulic tests, and the findings of others that apertures estimated from tracer tests are consistently smaller than those estimated from hydraulic tests. This apparent contradiction of the field results in fact arises from the different definitions of the so-called “tracer aperture” as employed by different researchers. In this short technical note I have attempted to sort out the different definitions, denotations and usage of the various “equivalent apertures” and show that there are mainly three alternative definitions used in the literature. The meaning of each as related to experimental measurements is explained and their interrelationship discussed. It is shown that once the specific definition of “equivalent aperture” referred to by each researcher is identified, then the relative magnitudes of these “equivalent apertures” as reported by different groups of researchers are perfectly consistent with each other.

A variable aperture fracture network model for flow and transport in fractured rocks

A three-dimensional variable aperture fracture network model for flow and transport in fractured rocks was developed. The model generates both the network of fractures and the variable aperture distribution of individual fractures in the network. Before solving for the flow and transport of the whole network, a library of single-fracture permeabilities and particle transport residence time spectra is first established. The spatially varying aperture field within an individual fracture plane is constructed by geostatistical methods. Then the flow pattern, the fracture transmissivity, and the residence times for transport of particles through each fracture are calculated. The library of transmissivities and frequency distributions of residence times is used for all fractures in the network by a random selection procedure. The solution of flow through the fracture network and the particle-tracking calculation of solute transport for the whole network are derived from one side of the network to the other. The model thus developed can handle flow and transport from the single-fracture scale to the multiple-fracture scale. The single-fracture part of the model is consistent with earlier laboratory tests and field observations. The multiple-fracture aspect of the model was verified in the constant aperture fracture limit with an earlier code. The simulated breakthrough curves obtained from the model display dispersion on two different scales as has been reported from field experiments.

Tracer Transport in Fractures: Analysis of Field Data Based on a Variable-Aperture Channel Model

A variable-aperture channel model is used as the basis to interpret data from a 3-year tracer transport experiment in fractured rocks. The data come from the so-called Stripa-3D experiment performed by Neretnieks and coworkers. Within the framework of the variable-aperture channel conceptual model, tracers are envisioned as traveling along a number of variable-aperture flow channels, whose properties are related to the mean and standard deviation σ of the fracture aperture distribution. Two methods are developed to address the presence of strong time variation of the tracer injection flow rate in this experiment. The first approximates the early part of the injection history by an exponential decay function and is applicable to the early time tracer breakthrough data. The second is a deconvolution method involving the use of Toeplitz matrices and is applicable over the complete period of variable injection of the tracers. Both methods give consistent results. These results include not only estimates of and σ, but also the number of channels involved in the tracer transport and their Peclet numbers and dispersivities. An interesting and surprising observation is that the data indicate that for each channel the Peclet number increases with the mean travel time; i.e., dispersivity decreases with mean travel time. The meaning of this trend is discussed in terms of the strong heterogeneity of the flow system.

Effects of high variance of fracture transmissivity on transport and sorption at different scales in a discrete model for fractured rocks

Abstract A three-dimensional (3-D) variable-aperture fracture network model for flow and transport in fractured crystalline rocks has been applied to study the effects of large variability in fracture transmissivity on non-sorbing and sorbing tracer transport, and scale effects in transport distance. The variable-aperture character of the fractures is introduced into a 3-D network model through a library of single-fracture permeabilities and associated particle transport residence time spectra. Sorption onto the fracture walls is added by a mathematical model for linear sorption. The resulting variable-aperture fracture network model, VAPFRAC, can handle flow and transport from single-fracture scale to the multiple-fracture scale. The model produces multi-peak transport breakthrough curves even for relatively moderate values of the fracture transmissivity variance. These breakthrough curves display dispersion on two different scales in the same way as has been observed in several field experiments conducted in crystalline rocks. The multi-peak structure is due to so-called channeling. For high values of the fracture transmissivity variance the solute transport is unevenly distributed and the channeling effects are more prominent. The effect of linear sorption is not just a simple translation in mean residence time as in a homogeneous medium. The dispersion characteristics of the breakthrough curves also change when linear sorption is included. The degree of the change depends strongly on the fracture transmissivity variance, as does the translation. In particular, with a high fracture transmissivity variance the translation in mean residence time due to sorption is significantly smaller compared to the cases with a low fracture transmissivity variance. Finally, the high variability in the model output data suggests that extrapolation of results from a particular tracer experiment will be highly uncertain.

Reply (regarding 'Flow and tracer transport in a single fracture: A stochastic model and its relation to some field observations)

This paper provides a rebuttal to comments made regarding these authors previous writings on ground water flow modeling in geologic fractures. The authors emphasize that the earlier paper was designed in context of a typical finite difference method discretization scheme. As a result, there is inherent numerical transverse dispersion, though not numerical longitudinal dispersion since the same residence time is assumed for all the particles in each grid block. Hence the intrinsic transverse mixing both within the grid block and at the intersections for the particle-tracking method is the size of the discretized grid block. The authors also defend their calculations regarding mean residence times derived with this solute transport method.