# Innovative Hydrogeophysical Approaches as Aids to Assess Hungarian Groundwater Bodies
## 2 Materials and Methods
### The Applied Geophysical Methods
#### 2.3.2 Hydrogeophysical Logging Methods
In the knowledge of grain sizes, pore-water resistivity (\(R_{w}\)) and resistivity of groundwater formations (\(R_{0}\)) corrected to the effect of drilling fluid invasion; one can estimate hydraulic conductivity solely from well logging measurements. The formation factor (\(F\)) as the ratio of \(R_{w}/R_{0}\) and the Hazen's effective grain diameter (\(d_{10}\)) determined from sieve analysis are directly connected in freshwater formations [55]. For the dominant grain diameter, this connection leads to the following equation:  
\[d=1.671\cdot C_{d}\lg\frac{R_{0}}{R_{w}}, \tag{5}\]  
where \(C_{d}\) = 5.22 \(\times\) 10\({}^{-4}\) is proposed for not too poorly sorted sediments with a resistivity formation factor less than 10. This condition normally satisfies Hungarian porous (unconsolidated) groundwater bodies. In freshwater formations, the electric current is hardly conducted through the spaces between the grains, rather mainly on the surfaces of particles. Thus, the resistivity is inversely proportional to the specific surface of rock grains. By assuming that the sedimentary rock is composed of spherical particles, the specific surface can be calculated as \(S\) = \(\theta(1-\Phi)/d\). For better separation of good storage capacity aquifers, the specific surface (\(S\)) can be derived from (5) as:  
\[S=\Bigg{[}36\left(1-\Phi\right)^{2}\Big{/}\left(1.671\cdot C_{d}\cdot\lg\frac {R_{0}}{R_{w}}\right)^{2}\Bigg{]}^{1/2}. \tag{6}\]  
By coupling [PERSON]'s tortuosity factor [56] with porosity and the formation factor, (2) can be properly modified to give a continuous (in situ) estimate for hydraulic conductivity (in m/s) along a borehole:  
\[K=C_{k}\frac{\Phi^{3}}{\left(1-\Phi\right)^{4}}\Bigg{[}\left(\lg\frac{R_{0}} {R_{w}}\right)^{2}\Big{/}\left(\frac{R_{0}}{R_{w}}\Phi\right)^{1.2}\Bigg{]}, \tag{7}\]  
where \(C_{k}\) = 855.7-\(C_{l}C_{d}\)\({}^{2}\) is a proportionality factor and \(C_{t}\) is a temperature-dependent constant calculated as \(C_{t}\) = 1 + 3.37 \(\times\) 10\({}^{-2}\)\(T\) + 2.21 \(\times\) 10\({}^{-4}\)\(T^{2}\) (where \(T\) is given in units of \({}^{\circ}\)C). Aquifers are characterized by hydraulic conductivities \(K\) (m/s) \(>\) 10\({}^{-6}\), while aquitards are indicated by \(K\) (m/s) \(<\) 3 \(\times\) 10\({}^{-8}\). An added advantage of the Csokas method is that it can give an estimate also to the critical velocity of flow and the volume capacity of the well [54]. Multivariate statistical techniques for processing well-logging datasets are proposed for a more reliable characterization of porous water bodies. For rock typing and separating the aquifers from impervious intervals, \(N\)-dimensional non-hierarchical cluster analysis is suggested [57]. In \(k\)-means clustering, the number of observed variables (\(N\)) is chosen arbitrarily, which may depend on detailed sensitivity analysis. Suitable well logs measured at different depths form objects of the data space that are separated into \(k\) number of groups in an iterative process by using a similarity criterion (defined as some vector norm of the differences of data vectors). The number of clusters must be a priori given, which must be harmonized with preliminary geological knowledge. Too few clusters provide poor vertical resolution, while too many groups provide unrealistic lithological categories. By calculating the overall deviation of data objects from their cluster centers (centroids) at different cluster numbers (sum of squared error), the well-known elbow method specifies the optimal number of clusters. It must be emphasized that this pure mathematical consideration must be verified by the available geological information. The position of centroids is of high importance because the solution is not independent of the selection of their initial values. Cluster analysis of incomplete datasets requires even more advanced algorithms. In case of having missing values of some variables, traditional clustering may fail because one loses too many objects from the analysis.