## 2 Electrical Resistivity/Impedance Measurement: Is It Really that Simple?
### AC Measurement of Electrical Resistivity
#### 2.2.2 4-Electrode Configuration  
The AC 4-point probe method, also known as Wenner array method, derived from soil mechanics to evaluate the possible corrosion of buried structures, is one of the most applied methodologies for field concrete structures [35; 92]. Four electrodes equally spaced are used, guaranteeing their electric contact with the surface e.g., by water pre-wetting, contact graphite gel or pastes; the two internal electrodes measure the voltage created by the application of an electric current between the two external electrodes (Figure 2). The frequency range commonly used is wide (Table 4). The Wenner-array method appears a suitable and reliable test method for in-field measurements and performance-based quality control of concrete durability [4]; the electrical resistivity measured by this approach is the average electrical resistivity of a hemisphere of radius equal to the electrode spacing (\(a\)), as reported in Equation (2):  
\[\rho\ =\ 2\pi a\frac{V}{I} \tag{2}\]  
This relationship is valid with the hypothesis of homogeneous medium and semi-infinite dimensions. In real conditions and with the use of surface electrodes, the apparent resistivity (\(\rho_{a}\)) is measured; in fact, because of non-homogeneity and finite geometric dimensions of the specimen under test, a cell constant correction factor (\(K\)) needs to be applied. The correction factor \(K\) can be obtained in two different ways: from theoretical considerations or from calibration by means of standard concrete samples or electrolytes of known resistivity. The relationship between the electrical resistivity and its apparent value is reported in Equation (3):  
\[\rho\ =\ \frac{\rho_{a}}{K} \tag{3}\]  
There are no widely accepted standards for the measurement of electrical resistivity using a Wenner array (ASTM G57-06 \"Standard Test Method for Field Measurement of Soil Resistivity Using the Wenner Four-Electrode Method\" [93] is used in soil mechanics). A Spanish standard (UNE 83988-2 \"Concrete durability. Test methods. Determination of the electrical resistivity. Four points or Wenner method\") suggests the use of cylindrical samples (100 mm diameter, 200 mm height); measurements should be carried out in saturated surface dry (s.s.d.) conditions, in order to consider the means homogeneous (since the inter-particle voids are water-saturated) [94]. Another provisional standard, namely AASHTO T 358 [95], was made in 2015 by the Department of Transportation (DoT) community of the American Association of State Highway and Transportation Officials (AASHTO); this standard suggests to apply current (flat-topped trapezoidal wave at approximately 13 Hz and a nominal peak-to-peak voltage of 25 V) by a Wenner probe located on the specimen surface (electrode spacing of 38 mm) and measure the voltage drop to calculate the impedance of the specimen. [PERSON] et al. [96] applied this standard to cylindrical specimens (102 mm diameter, 205 mm height) containing different SCMs. They found a very strong linear correlation with the bulk resistivity measured according to  
Figure 2: Wenner array method. the AASHTO TP119 \"Standard Method of Test for Electrical Resistivity of a Concrete Cylinder Tested in a Uniaxial Resistance Test\" [70] (y = 1.93x, with R\({}^{2}\) = 0.99). Also [PERSON] et al. [81] performed measurement according to AASHTO T 358, using stainless steel plate electrodes, ensuring a good electric contact with the specimen by lime-water saturated sponges. They compared their results with the surface resistivity method described in FM 5-578 [97], obtaining very good correlation (R\({}^{2}\) = 0.9986); moreover, they observed variation coefficients of approximately 4% and 13% for intra- and inter-operator measurements, respectively. A resistivity profile can be obtained by moving the electrodes along a horizontal section at constant depth, whereas a vertical electric sounding can be achieved by progressively widening the electrode spacing and keeping the quadrupole center fixed.