Comment on “Derivation of effective hydraulic parameters of a karst aquifer from discharge hydrograph analysis” by S. J. Baedke and N. C. Krothe

[2] The application of the above procedure by BK2001 incurs two major mistakes. First, the conceptual model used by Rorabaugh to obtain equation (1) is not applicable to a spring hydrograph, mainly because equation (1) is derived for parallel flow from the aquifer to the stream, and the flow to a spring is fundamentally radial. Second, there is no valid justification to apply that procedure to the different slopes observed in a single hydrograph with the aim of obtaining diffusivities of different portions of the aquifer, as we discuss below. [3] While the first mistake does not need further comment, the second one deserves a more detailed explanation. We devote the remainder of this text to clarify the basis of our statement. [6] Table 1 shows how the ρi2 values, and therefore the αi values, depend on the degree of connectivity between aquifer and river; these coefficients are lower the smaller the connection is. Particularly α1 can be significantly lower for a poor connection than the corresponding value for a perfectly connected river. It is therefore incorrect to use Rorabaugh's method to determine the diffusivity of an aquifer from the value of α1 derived from the late slope of a spring hydrograph, without accounting for the degree of connection between the aquifer and the river. It is also incorrect to apply it to the outflow coming from a spring hydrograph. The incorrect application of Rorabough's approach to the outflow hydrograph from an imperfectly connected river may lead to a diffusivity that can be much lower than the real one. [7] The exponents αi in (4) can also be determined numerically for the case of heterogeneous aquifers with complex geometries and arbitrary initial and boundary conditions by using either a finite difference or finite element approximation of the flow partial differential equation [Andreu and Sahuquillo, 1987]. Estrela and Sahuquillo [1997] use a finite element model to compute the exponents αi for the aquifer containing the Arteta spring in Navarra (Spain), a spring that can produce flows as high as 40 m3 s−1. The aquifer is discretized as seen in Figure 1 using two types of elements, to which two distinct hydraulic conductivities are assigned. Some elements represent potential karst conduits by narrow radial and circular bands of high transmissivity, and the remaining elements are larger and of low transmissivity. To understand the relationship between the coefficients αi and the shape of the hydrograph, Estrela and Sahuquillo [1997] determine combinations of two hydraulic conductivity values, one for the narrow elements and another for the larger elements, that result in a hydrograph with a late time slope in semilogarithmic paper of 0.0038 (log(m3 day−1)/day) (This slope is the one observed in the Arteta spring). Table 2 shows the resulting αi values for three different transmissivity contrasts between the two types of elements. Figure 2 displays the hydrographs associated with these transmissivity combinations. Table 2 shows three alternative distributions of transmissivities that result in α1 = 0.0038 day−1: (1) a homogeneous aquifer with transmissivity of 5.1 × 103 m2/day, and two heterogeneous cases, (2) low transmissive elements of 1.0 × 103 m2/day, and high transmissive elements of 1.0 × 105 m2/day, and (3) low transmissive elements of 1.0 × 103 m2/day, and high transmissive elements of 1.6 × 106 m2/day. [8] From the results of Table 2 and the representation of the hydrographs in Figure 2 the following comments can be made. The homogeneous aquifer that is able to reproduce the observed slope has a transmissivity of 5.1 × 103 m2day−1, an order of magnitude larger than the value of 7.5 × 102 m2day−1 that results from the application of equation (2), following BK2001, with L equal to 7000 m: the average distance from the spring to the impervious boundary. This discrepancy is due to the difference in flow patterns between the model assumed by (2), essentially parallel flow, and the aquifer, essentially radial flow. The next comment is that to reproduce the effect of karstification through the inclusion of high flow zones, the transmissivities of the less pervious zones have to be reduced to 1.0 × 103 m2day−1, whereas the high values could be as high as 1.0 × 105 m2day−1 or 1.6 × 106 m2day−1; it is only when the contrast between transmissivities is as large as three orders of magnitude (Series 3) that two distinct slopes are clearly appreciated in the hydrograph. As we have said, these different slopes are not associated with specific portions of the aquifer, but they are a combined result of the geometry of the aquifer and the relative distribution between the two types of elements. If these slopes had any relevance to the aquifer transmissivities, their ratio should be about the transmissivity contrast; however, the ratio of slopes in Series 3 is about 0.1/0.0038 = 26, while the contrast of transmissivities is 1.6 × 106/1.0 × 103 = 1.6 × 103. This is a clear example that the slopes of the hydrograph cannot be used to identify the transmissivities of the aquifer in the way applied by BK2001. [9] Aquifer outflows, whether to a river or through a spring, independently of the degree of heterogeneity of the aquifer, its geometry or its boundary conditions, can be described by the sum of an infinite series of decreasing exponentials. Identification of these exponentials does not warrant the existence of distinct and disjoint zones of different transmissivities in the aquifer as implied by the analysis of BK2001, since the coefficients characterizing these exponentials are complex functions of the aquifer as a whole that can only be treated with a rigorous analysis of the eigenvalues and eigenfunctions associated with the groundwater flow partial differential equation. [10] The authors wish to thank the anonymous reviewers for their helpful comments and suggestions. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.