On the use of apparent hydraulic diffusivity as an indicator of connectivity
Introduction
Considerable attention has been devoted to the characterization and representation of the complex heterogeneity found in most geologic media. Connected features have received part of this attention due to their substantial impact on subsurface flow and transport. However, the concept of connectivity – how to define it, how to measure it, and under what conditions and how it affects various types of hydrological response – is still being debated (e.g. Western et al., 2001, Grayson et al., 2002, Bruderer-Weng et al., 2004, Knudby and Carrera, 2005). In hydrogeology, “connectivity” is most often used as a reference to the physical presence of connected zones of either high or low conductivities. The closely related concepts of “channeling”, “preferential flow paths”, and “early solute arrival” are in general used as references to the effect that connectivity has on hydrological response. In this paper we use “flow connectivity” as a reference to a flux increase caused by connected features. Similarly, we use “transport connectivity” to refer to the early solute arrival as compared to the average arrival time. It is important to realize that these effects, expressed through the hydrologic response of a geological medium, may be observed even in the absence of continuous high permeability structures transversing the domain. Continuous structures are in general sufficient, but not strictly necessary for concentrated flow and early solute arrival (e.g. Sánchez-Vila et al., 1996, Fogg et al., 2000, Lee, 2004).
In a recent paper (Knudby and Carrera, 2005), we presented and analyzed indicators of statistical, flow and transport connectivity. No significant correlation was found between the three types of indicators. In other words, the assumption that a statistical measure captures the relevant aspects of connectivity should be verified before it is employed. Our results furthermore suggested that the presence of connected features can influence flow and transport differently. This points at the usefulness of treating connectivity as a process-dependent concept. Scheibe and Yabusaki (1998) present results which illustrate the potential benefits of such an approach. In short, it seems relevant to investigate the character of the relationship between flow and transport connectivity, and more specifically what role connected features play in this context.
The magnitude and spatial distribution of properties which control groundwater flow and transport are commonly estimated using pumping and tracer tests (e.g. Walton, 1987, Fetter, 2000, Meier et al., 2001, Ptak et al., 2004). Classical analytical interpretation of pumping and tracer tests is usually based on simplifying assumptions such as homogeneity, perfect layering, statistical stationarity, etc. Such assumptions facilitate interpretation, but may lead to biased parameter estimates (e.g. Gómez-Hernández et al., 1995, Meier et al., 1999, Beckie and Harvey, 2002, Sánchez-Vila and Carrera, 2004). More importantly, they do not allow extraction of all the information contained in the test results. However, if interpreted jointly or with specific attention to connectivity (e.g. Alabert et al., 1992, Herweijer, 1996a, Herweijer, 1996b, Gómez-Hernández et al., 1997, Meier et al., 1998, Butler et al., 1999, Guimerá and Carrera, 2000, Fernandez-Garcia et al., 2002, Martinez-Landa and Carrera, 2005, Illman and Tartakovsky, 2005), pumping and tracer tests can improve the precision of estimates of effective flow and transport parameters and at the same time provide valuable information on heterogeneity (Vandenbohede and Lebbe, 2003). Hydraulic tomography, which consists of the joint inversion of multiple hydraulic tests while assuming the permeability to be a random function, illustrates the role of connectivity on effective parameters (Vasco et al., 2000, Yeh and Liu, 2000, Meier et al., 2001, Vesselinov et al., 2001, Bohling et al., 2002, Brauchler et al., 2003).
Sánchez-Vila et al. (1996) showed that for two-dimensional univariate log-Gaussian fields for which the high-transmissivity (T) zones are better connected than the low-T zones, the large scale effective transmissivity, Teff, is consistently larger than the geometric average of the point values, TG, which is the expected value for two-dimensional isotropic multilog-Gaussian fields (Matheron, 1967). Meier et al., 1999, Sánchez-Vila et al., 1999a, Sánchez-Vila et al., 1999b illustrated that even in the presence of connected features, estimates of Teff based on interpretation of pumping tests of long duration in heterogeneous media using Jacob’s method (Cooper and Jacob, 1946) compare well with estimates of Teff based on an assumption of mean uniform flow. In other words it is possible to get good estimates of Teff from pumping tests even in the presence of significant connectivity. In this aspect, Teff stands in contrast to the other parameter commonly estimated from pumping tests, the storativity, S. Schad and Teutsch (1994) used three-phased transient pumping test response to estimate the length of high-permeable lenses. They found that estimates of S obtained from Theis type curve fitting were lower for observation wells located in high-T zones than for observation wells in low-T zones. Subsequently, Herweijer and Young, 1991, Herweijer, 1996a, Herweijer, 1996b, Meier et al., 1998 found that estimates of S obtained from Jacob’s method contain information not only on near-well materials, but also on the degree of hydraulic interconnectedness between the pumping and the observation well.
All of the above studies employ an undefined and visual notion of connectivity (e.g. Grayson et al., 2002). Despite the lack of quantification, there is nevertheless substantial evidence that the storativity, S, and therefore also the hydraulic diffusivity, D (D = T/S), as interpreted from Jacob’s method are related to connectivity, and that the relationship has to do with the presence of connected features.
In short, Jacob’s method consists of plotting drawdown vs. time on a semi-logarithmic paper. Estimates of Teff and subsequently S are then found by drawing a straight line through late-time points and computingwhere Q is the constant pumping rate, m is the slope of the straight line, t0 is the time axis intercept, and r is the radial distance to the observation well (see e.g. Freeze and Cherry, 1979, for further details). When the observation and the pumping wells are connected by high-T features, the drawdown signal will be observed early, i.e. t0 will be small. In other words, high connectivity is expressed through an artificially small estimate of S and consequently a large estimate of D. Heterogeneity causes estimates of storativity obtained from Jacob’s method to be different from their actual values. The corresponding values of diffusivity are also affected by heterogeneity in a similar way, although they may be considered more representative of overall system behavior. We therefore refer to the values obtained from Jacob’s method as apparent storativity, Sa, and apparent diffusivity, Da (Da = Teff/Sa). The effect that flow connectivity has on Sa is analogous to the effect that transport connectivity has on apparent porosity estimates from tracer tests (Guimerá and Carrera, 2000, Fernandez-Garcia et al., 2002).
Solute transport related support of the conjecture that Sa and Da contain information on connectivity can be found in studies of Herweijer and Young, 1991, Herweijer, 1996a, Herweijer, 1996b, Paris, 2002. Herweijer (1996a) analyzed pumping test data from the MADE site and numerically simulated pumping and tracer tests for the site. Linear correlation was found between the logarithm of the travel time of the drawdown signal and the logarithm of the tracer travel time. In a similar investigation for three-dimensional fracture networks, Paris (2002) found that tracer and drawdown breakthrough time show strong correlation over distances comparable to the size of the disc-shaped fractures. The strength of the correlation was found to decrease with increasing matrix diffusion.
Based on the above, it is reasonable to conjecture that apparent storativity, Sa, and therefore also the apparent hydraulic diffusivity, Da, as interpreted from Jacob’s could be useful indicators of flow and/or transport connectivity. The objective of the present paper is to use defined and therefore quantifiable measures of connectivity to test the relevance of Sa and Da as indicators or measures of connectivity. Also, we investigate how the two parameters are related to two indicators of flow and transport connectivity analyzed by Knudby and Carrera (2005). Sa and Da are related through Teff, which as shown by Meier et al. (1998) can be estimated even in the presence of significant heterogeneity.
Section snippets
Procedure
In order to test to what degree Da can be considered an indicator of connectivity, we use the type of approach employed in several recent studies focusing on connectivity (e.g. Wen and Gómez-Hernández, 1998, Western et al., 2001, Zinn and Harvey, 2003, Knudby and Carrera, 2005). In short, we generate transmissivity fields and subsequently rearrange their T-values so as to enhance the presence of connected features. We thereby obtain sets of fields for which one field is (visually) better
Results
To test the relevance of apparent diffusivity as a measure of connectivity, we computed the values of CF, CT, and DR for a total of 1200 fields comprised of eight series of 50 MG-fields, 50 F-fields, and 50 C-fields. Table 1 shows that rearrangement method C is less consistent than rearrangement method F when it comes to increasing the arithmetic averages of CF, CT, and DR. In fact, rearrangement method C causes a small, but consistent decrease in average CT for series 1–2 (for ranges equal to
Discussion
The analysis of the correlation between arrival times showed that log(DR) is strongly, and approximately linearly, correlated to log(CF · CT). This explains the observation that DR reacts to differences in connectivity in a way that is intermediate to CF and CT. Since CF accounts for the flow rate increase due to connected features, whereas CT accounts for early arrival as compared to the average arrival time, DR can be seen as accounting in an integrated way for both flow and transport
Conclusions
Several previous studies have found that the storativity estimated from interpretation of pumping tests using Jacob’s method contain information on the degree of hydraulic interconnectedness between the pumping and the observation well rather than represent actual storativity. As a consequence, both the apparent storativity, Sa, and the related dimensionless apparent hydraulic diffusivity, DR (DR = Da/(TG/S)) could potentially be valuable as indicators of connectivity. However, only visual (and
Acknowledgements
Funding for the above work was provided by The Danish Research Agency and The Spanish Nuclear Waste Management Agency (ENRESA).
References (64)
- et al.
Relating flow channeling to tracer dispersion in heterogeneous networks
Advances in Water Resources
(2004) - et al.
Convergent-flow tracer tests in heterogeneous media: combined experimental–numerical analysis for determination of equivalent transport parameters
Journal of Contaminant Hydrology
(2002) - et al.
To be or not to be multi-Gaussian. A reflection on stochastic hydrogeology
Advances in Water Resources
(1998) - et al.
A comparison of hydraulic and transport parameters measured in low-permeability fractured media
Journal of Contaminant Hydrology
(2000) Applications of percolation theory to porous media with distributed local conductances
Advances in Water Resources
(2001)- et al.
Asymptotic analysis of three-dimensional pressure interference tests: A line-injection line-monitoring solution
Advances in Water Resources
(2005) - et al.
On the relationship between indicators of geostatistical, flow and transport connectivity
Advances in Water Resources
(2005) - et al.
Binary upscaling – the role of connectivity and a new formula
Advances in Water Resources
(2006) - et al.
Tracer tests for the investigation of heterogeneous porous media and stochastic modelling of flow and transport – a review of some recent developments
Journal of Hydrology
(2004) - et al.
On the striking similarity between the moments of breakthrough curves for a heterogeneous medium and a homogeneous medium with a matrix diffusion term
Journal of Hydrology
(2004)
Scale effects in transmissivity
Journal of Hydrology
Effects of the investigation scale on pumping test results in heterogeneous porous aquifers
Journal of Hydrology
Scaling of flow and transport behavior in heterogeneous groundwater systems
Advances in Water Resources
The importance of the third dimension on transport through saturated porous media: case study based on transport of particles
Journal of Hydrology
Combined interpretation of pumping and tracer tests: theoretical considerations and illustration with a field test
Journal of Hydrology
Upscaling hydraulic conductivities in heterogeneous media: An overview
Journal of Hydrology
Numerical modeling of macrodispersion in heterogeneous media – a comparison of multi-Gaussian and non-multi-Gaussian models
Journal of Contaminant Hydrology
Hopping conductivity in disordered systems
Physical Review B
Applied Groundwater Modeling
What does a slug test measure: An investigation of instrument response and the effects of heterogeneity
Water Resources Research
Steady shape analysis of tomographic pumping tests for characterization of aquifer heterogeneities
Water Resources Research
A travel time based hydraulic tomographic approach
Water Resources Research
Pumping tests in networks of multilevel sampling wells: Motivation and methodology
Water Resources Research
Conduction of Heat in Solids
Shortest paths algorithms: Theory and experimental evaluation
Mathematical Programming
A generalized graphical method for evaluating formation constants and summarizing well field history
American Geophysical Union Transactions
GSLIB. Geostatistical Software Library
A note on two problems in connection with graphs
Numeriche Mathematik
Applied Hydrogeology
Groundwater flow and sand body interconnectedness in a thick, multiple-aquifer system
Water Resources Research
Automatic generation of flow nets with conventional ground-water modeling algorithms
Groundwater
Cited by (97)
Effect of fault activation on the hydraulic connectivity of faults in mudstone
2022, Geomechanics for Energy and the EnvironmentComparison of transfer function models for well-aquifer system response to atmospheric loading
2020, Journal of HydrologyHydraulic characterization and identification of flow-bearing structures based on multi-scale investigations applied to the Lez karst aquifer
2019, Journal of Hydrology: Regional StudiesCitation Excerpt :Compared to seasonal responses, the lower values of storativity estimated for the Triadou and Terrieu boreholes from daily responses also confirm a better hydraulic connectivity to the Lez pumping station. Indeed, the storativity obtained using a radial flow model can be used as an indicator of the hydraulic connectivity in heterogeneous media (Meier et al., 1998, Sánchez-Vila et al., 1999; Knudby and Carrera, 2006). The use of simple classical solutions of Theis (1935) and its approximation of Cooper and Jacob (1946) to analyze short (daily pumping) and late times of the regional drawdown data provided a high variability of storativity, which ranges from 4.2 × 10−5 to 6.3 × 10−2.
Convergent radial transport in three-dimensional heterogeneous aquifers: The impact of the hydraulic conductivity structure
2019, Advances in Water Resources
- 1
Tel.: +34 934016852; fax: +34 934017251.