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How should an electrical resistivity tomography laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement
Journal of Applied Geophysics 127 (2016) 45–55
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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
How should an electrical resistivity tomography laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement R. Clement ⁎, S. Moreau National Research Institute of Science and Technology for Environment and Agriculture (Irstea), Hydrosystems and Bioprocesses Research Unit, 1 rue Pierre-Gilles de Gennes, CS10030, 92761 Antony, France
a r t i c l e
i n f o
Article history: Received 4 April 2015 Received in revised form 28 January 2016 Accepted 17 February 2016 Available online 18 February 2016 Keywords: Electrical resistivity tomography Laboratory 3D complete forward modelling
a b s t r a c t Among geophysical methods, the electrical resistivity tomography (ERT) method is one of the most commonly used for the study of hydrodynamical processes. The geophysical literature relates several laboratory-scale applications of this method. Unlike the measurements taken at the field scale, few authors have taken an interest in errors associated with apparent electrical resistivity, especially in the case of ERT data acquired in the laboratory. The objective of this paper is to show that laboratory errors related to the positioning of electrodes and the geometry of cells are significant on apparent resistivity measurements. The embedment and the position of the electrode were evaluated to quantify their impact on electrical resistivity measurement. To assess these impacts, the authors propose a 3D numerical modelling investigation based on the complete design of a laboratory test cell. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Electrical resistivity tomography (ERT) is a mature geophysical method that is increasingly popular in environmental and hydrogeological studies (Barker, 1998; Binley and Kemna, 2005; Chambers et al., 2013; Loke, 2004; Loke et al., 2013; Ogilvy et al., 2002). This near-surface geophysical method provides information in various field applications: (i) for geological purposes, Van Schoor (2002) used ERT for sinkhole detection and bedrock and sand channel localization; (ii) in a hydrogeological study, Descloitres et al. evaluated the location of shallow and deep infiltrations and recharge zones with time-lapse monitoring (Descloitres et al., 2008); (iii) to monitor pollution plumes Benson (1995); Benson et al. (1997) mapped the extent of plume pollution in soil; (iv) in a municipal solid waste landfill, the leachate flow which is a key point in controlling anaerobic waste biodegradation, the ERT method was used to highlight the volume of waste mass impacted during leachate recirculation events (Audebert et al., 2014; Clement et al., 2009; Clément et al., 2009; Moreau et al., 2003). This list illustrates the various applications of the ERT method related to its many advantages: (i) it is non-destructive: the hydromechanical properties of the subsurface or hydrodynamic processes can be evaluated without digging; (ii) it is sensitive to the contrast in conductivity between the solid and liquid phases of the medium studied and (iii) it may indicate the distribution of electrical resistivity in 2D or ⁎ Corresponding author. E-mail address: [email protected] (R. Clement).
http://dx.doi.org/10.1016/j.jappgeo.2016.02.008 0926-9851/© 2016 Elsevier B.V. All rights reserved.
3D with the recent development of inversion software (Loke et al., 2013). Despite all of the advantages of the ERT method, interpretations of the electrical resistivity variations are not always obvious because the electrical resistivity can be influenced by several parameters; the most frequently cited are: water content, temperature, porosity, density and electrical conductivity of the liquid phase, (Archie, 1942; Benderitter, 1999; Clement et al., 2011; Day-Lewis et al., 2003; Moreau et al., 2011; Rhoades and Van Schilfgaarde, 1976; Salem and Chilingarian, 1999). Even if the method was developed for field applications, a great deal of research has been conducted at the laboratory scale to study one of the parameters involved in the process studied under control conditions (Brunet et al., 2011; Han et al., 2015; Kowalczyk et al., 2014; Rhoades and Van Schilfgaarde, 1976). (i) Brunet et al. (2011) have calibrated the relationship between electrical resistivity and the soil's water content to estimate water deficit at the field scale. (ii) To explain the ranges of resistivity variation observed during leachate reinjection surveys in a municipal solid waste landfill, Moreau et al. (Moreau et al., 2011) conducted a series of laboratory tests on waste samples to relate variations in density and moisture content to the electrical resistivity recorded. The authors proposed a methodology to estimate interpreted resistivity anisotropy between vertical and horizontal resistivity measurements at the laboratory scale. (iii) In another experiment, Slater et al. (2000) used a high-resolution 3D electrical resistivity tomography with crossborehole arrays to study solute transport in a large experimental tank. The authors conducted a salt tracer experiment, monitored by timelapse ERT, in a quasi-two-dimensional sandbox with the aim of determining the hydraulic conductivity distribution in the domain. They
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concluded that temporal moments of potential perturbations obtained during salt tracer tests provide a good basis for inferring the hydraulic conductivity distribution by fully coupled hydro-geophysical inversions. (iv) Binley et al. (1996) used ERT to study the internal spatial characteristics of solute transport in naturally heterogeneous soils: the analysis of the results revealed spatial variation in transport characteristics throughout the soil column. Most of the volume of the experimental cells is less than 1 m3 (Brunet et al., 2011; Moreau et al., 2011; Slater et al., 2002) and the question addressed by many authors is the position and the spacing between the electrodes to describe the entire medium studied (Moreau et al., 2011). For all the approaches presented above, two types of analysis were always encountered. The first ones consider that the distribution of resistivity is homogeneous and that apparent electrical resistivity can be considered as interpreted resistivity, as in older studies (Archie, 1942; Jackson, 1978; Kowalczyk et al., 2014; Liu et al., 2013; Rhoades and Van Schilfgaarde, 1976). The second type of analysis does not retain the same assumption and requires inversion software to calculate the interpreted resistivity distribution in the medium studied from the measured apparent resistivity (Clement and Moreau, 2012; Slater et al., 2002). In both cases, a geometric factor related to the electrodes' location is required to calculate apparent electrical resistivity. Experimental tests with solutions of known electrical conductivity are possible (as proposed by Rhoades et al. (1976)) and numerical simulation software is available to provide this evaluation. When a forward modelling algorithm is used to evaluate the geometric factor, the position, size and shape of the electrodes have to be known with the highest accuracy. However, a great deal of additional information is needed: the shape and size of the test cell must be measured and the embedment of the electrodes in the medium studied must be identified. The numerous numerical tools available now following the technical development of computers and computer languages can provide a full and accurate description of the experimental conditions. For the second type of analysis using inversion processes, electrode shape, size and embedment are difficult to take into account in the inversion software available, such as BERT (Boundless Electrical Resistivity Tomography), developed by Günther et al. (2006), or R3T, defined in Binley and Kemna (2005). Generally, the electrodes are described as a point electrode in the laboratory test cell designed in the inversion software. One of the key parameters is the position of the electrode represented by point node because even if the theoretical position of the point node is perfectly defined during the design of the laboratory test cell, in fact, there is always an error associated with the mechanical construction of the laboratory test cell. The position and the error on that position have an impact on the calculation of the geometric factor of the quadripole because the characteristics of the complete electrode cannot be ignored when spacing is short in laboratory tests. This paper proposes guidelines to design an example of a cylindrical laboratory test cell for ERT measurements using a numerical approach. The influence of electrode size and impact of the error on its position and its embedment are studied to define the accuracy needed to guarantee the computation of apparent resistivity and indirectly the distribution of interpreted resistivity using inversion software. 2. Material and methods In ERT laboratory measurements, several parameters can create errors on the apparent resistivity calculation and the evaluation of interpreted resistivity using inversion software. Considering that the instruments used for current injection and potential measurements are calibrated, we identify three major errors which could influence the geometric factor associated with a quadripole: (i) the electrode shape and size, (ii) the accurate measurement of electrode embedment and (iii) the accurate measurement of the electrode position. To estimate the impact of these parameters on the geometrical factor, we chose a classical numerical methodology based on multiple numerical forward
modelling currently applied in geophysics (Clement et al., 2009; Clément et al., 2010; Radulescu et al., 2007; Yang, 2005). We chose to base our modelling approach on our experimental laboratory test cell (LTC) presented in Fig. 1a. For all numerical modelling, the first step is the design of the LTC for different conditions: shape, size, embedment and electrode position. The second step is the calculation of the geometric factor for each combination of conditions imagined and the last one is the evaluation of the distribution of the results according to the parameters tested. 2.1. Laboratory test cell description for numerical design During the last 5 years, we conducted laboratory tests on waste samples to study relations between the electrical resistivity variations observed and different hydro-mechanical conditions such as density and water content (Moreau et al., 2011). Our experimental test cells always had the same cylindrical shape and an average volume of 0.226 m3 for 1 m height and 0.4 m diameter. These configurations were considered to design the LTC for the different numerical models tested in this paper. The LTC is made of high-density polyethylene (HDPE). Forty steel electrodes are distributed over five levels and eight vertical lines spaced 45° apart. Fig. 1a–b describes the theoretical shape, embedment and positions of the electrodes. The conventional electrodes used are cylindrical and exceed 30 mm inside the test cell. They are spaced vertically 150 mm apart and 157 mm apart horizontally on the perimeter of the test cell. 2.2. Electrical resistivity measurement The ERT method is thoroughly described in the geophysical literature (Chapellier, 2000; Dahlin, 2001; Loke et al., 2013). The apparent resistivity ρapp is calculated from a quadripole composed of two injected current electrodes A and B and two other electrodes M and N to measure a potential difference (Eq. 1). The geometric factor k depends on the position of the four electrodes called quadripole as well as the size and the shape of the electrodes when they cannot be considered to be a point in the measurement process.
ρapp ¼
k ΔVMN IAB
ð1Þ
where: ρapp is apparent electrical resistivity (Ω∙m) is the electrical potential difference measured (V) ΔVMN I is the intensity of the injected current (A) k is the geometric factor (m). The electrical resistivity arrays evaluated are the Wenner-α and the dipole–dipole array because they are the most popular (Fig. 2). From the top of the test cell, the electrodes are numbered from 1 to 40: the first level considers electrodes 1–8 and the lowest level electrodes 33–40. The vertical distance between two consecutive electrodes is equal to 150 mm, the position of the top and the bottom are, respectively, 175 mm from the upper level of the electrodes (1–8) and 125 mm from the lowest level of the electrodes (33–40). Among all imaginable quadripoles from the 40 electrodes implemented, four quadripoles were selected as being the most frequently used. The first quadripole Q1, called the horizontal, consists of four electrodes located on the same level and spaced 45° apart (Fig. 2). The second, Q2, is the vertical and considers the four electrodes on the same vertical line. The third, Q3, is the diagonal; four consecutive electrodes are all placed at different levels. The last array, Q4, is a mixed array between all of these positions. Two consecutive electrodes are at the same level and two others are also at the same level but different from the previous and on the opposite side of the LTC.
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Fig. 1. Laboratory test cell (LTC) and electrode description: a) left: picture of true laboratory test cell; middle: diagram of LTC without HDPE material; right: the numerical mesh; b) electrode type shape and size.
2.3. Elementary theory of DC conductive media modelling The governing equation for ERT forward modelling in a conductive subsurface media can be derived from Maxwell's equations. Technically ERT is based on a very-low-frequency alternating current (2–5 Hz) to limit polarization effects. In this case, we could ignore the inductive and capacitive effects in Maxwell's equation. Under static conditions, the electric field E (in V/m) can be represented by the negative gradient of the electric potential V: E ¼ −∇V:
ð2Þ
In conductive media, to simulate stationary electric current density, we consider the stationary equation of continuity and Ohm's law. In a stationary coordinate system, the point form of Ohm's law states that: J ¼ σE þ Je
ð3Þ
where, J is the current density (A/m2) equal to the current I per unit of cross-sectional area S (A), Je is the externally current density (in A/m2), E is the electric field intensity and σ is the electrical conductivity. The SI unit
for conductivity is amperes per volt-metre (A/Vm) or Siemens per metre S/m and the reciprocal of conductivity is known as resistivity in Ohmmetre. The static form of the equation of continuity then states: ∇ J ¼ −∇ ðσ∇V−JeÞ ¼ 0:
ð4Þ
The generalized form of the current sources (Qj) can be written as follows: −∇:ðσ∇V−JeÞ ¼ Q j :
ð5Þ
Considering our LTC geometry and the principle of ERT, various boundary conditions must be applied in this model (Fig. 1a). We consider that the wall of the LTC in HDPE is infinitely resistant, we applied an insulating boundary condition at the contact area between the homogenous media studied and the LTC, expressed by the Neumann condition; this equation specifies that there is no current flowing across the boundary: n J ¼ 0:
Fig. 2. Selected quadripoles and array types.
ð6Þ
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We apply the same boundary condition for the proportion of electrodes in contact with the medium studied inside the laboratory test cell and outside in contact with the air. The LTC is made with HDPE, considered to be an electrical insulator with a conductivity of 10−14 S/m, and the steel electrodes (ref: AISI 4340) are 4.32 ∗ 106 S/m of conductivity. For the ERT measurements, the current circulation is made between two current electrodes represented by a positive intensity on electrode A and a negative intensity on electrode B. According to the surface of the electrode S (m2) and the injected current I0 = 100 mA, the current density J is calculated using the following equations. For the positive electrode A, incomer current flow is considered: n J ¼ JnA :
ð7Þ
For the negative electrode B, inward current flow is considered: −n J ¼ JnB
ð8Þ
I J¼ : S
ð9Þ
distributions are built using the AC/DC module (quasi-stationary electromagnetic field) to evaluate the potential difference induced by the injected current and the electrical boundary conditions selected as infinite or insulating surfaces. This step is made automatically to calculate the electrical potential for every quadripole using F3D-lab Matlabcode (Forward 3D Laboratory) developed by Clement et al. (2011). The end of simulation calculates the geometrical factor for each quadripole and every condition for the shape, embedment and positions of the electrodes. To facilitate the building of the geometrical model and the data processing, we implemented a Matlab script and F3D-lab interface consistent with COMSOL. This script automatically creates the geometry of a LTC based on parameters selected by the user. The height and diameter of the column can be adjusted easily such as the number, shape and embedment of the electrodes. Additional refinement has been applied to obtain accurate forward calculation around the electrode. This tool calculates either the geometrical factor on a homogeneous model or apparent resistivity for the heterogeneous resistivity model; the script is available free of charge from the authors on request. 2.5. Simulation test managed
These theoretical equations are directly implemented in Comsol Multiphysics and we use the solver tools included in Comsol as the AC/DC module. 2.4. Three-dimensional electrical resistivity forward modelling The methodology of this paper is based on the use of a forward modelling tool, where the complete geometry of the LTC and the electrodes is taken into account. The general approach follows six steps presented in Fig. 3. Here, the geometry design of the electrode and the laboratory test cell is highly challenging for ERT modelling at the laboratory scale. To respond to this problem, Comsol Multiphysics 4.3b combined with Matlab 2012 are used. With Comsol, the electrical field distributions can be modelled using full 3D modelling. Electric field
The geometry of the LTC is presented above (Fig. 1), as are the electrical arrays tested. The shape, size and position chosen for the electrodes considered as the reference model are: 10 mm in diameter, 30 mm embedment inside the HDPE test cell and a vertical distance between electrodes equal to 150 mm and spaced horizontally 45° apart. The first objective is to consider different electrode positions with no associated position error and to assess the impact of various electrode diameters and embedments. Then different electrode positions around the reference positions will be studied to simulate errors that could be made during the mechanical construction of the test cell. Modelling all these parameters requires a very long computation time (i.e. about 4 weeks with an Intel Xeon CPU E5-2650 v2 2.60 GHz processor with 12 cores and 128 Gb of RAM).
Fig. 3. Methodology of full 3D modelling using Comsol Multiphysics 4.3 and F3DM Matlab script.
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2.5.1. Impact of electrode diameter and embedment as a function of electrode spacing In the first part, the impact of both electrode embedment and diameter were evaluated. We tested four different complete electrode diameters from 6, 8, 10 to 12 mm and we made different LTC numerical models with variable embedment values (10, 20, 30, 40 and 50 mm) for spacing between electrodes equal to 150 mm. For each diameter (d) and embedment (E) combination, the geometric factor KdE (m) is evaluated and compared to the reference geometric factor Kref (m) (from the electrode 10 mm in diameter and 30 mm embedment). Eq. 10 below was used to estimate the relative variations of the geometrical factor: Δk ¼
kdE −1 100 ð%Þ: kref
ð10Þ
Depending on the results of this test, we changed the unit of electrode spacing to decrease it from 150 mm to 125 mm, and then 100 mm, 75 mm and finally 50 mm, and we only tested the impact of different embedment values but always equal for the four electrodes. The same assessment was made but with a specific reference Kref geometric factor for each spacing tested. It is realistic to imagine that an embedment error can differ from one electrode to another. To conclude on this part, one quadripole was selected to compute the geometric factors for different embedment values for the four electrodes within a range of 25–35 mm with a 1 mm unit resolution. The number of models calculated for each resistivity array was 14,641: Wenner-α and dipole–dipole. 2.5.2. Electrode position accuracy In the field in most cases, the errors on electrode positions are often overlooked because they are considered to be negligible. A few centimetres of error on the electrode position is very low compared to the unit of electrode spacing greater than 1 m generally used. At the laboratory scale, this error must be much more controlled due to the small electrode spacing unit, around 150 mm or less. The aim of this part is to evaluate the accuracy required to position the electrodes on our LTC in order to reduce the error on the computation of the geometric factor. Several models have been imagined to introduce possible electrode position errors. For the four quadripoles selected and the two resistivity arrays, we varied the position of only one electrode of the four available, as proposed in Fig. 4. This number of positions allowed creating 121 models to calculate the variation of the geometric factor for one quadripole. We considered a range of variation ± 5 mm with a unit
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spacing of 1 mm along the vertical and horizontal axes (a curvilinear position) and around the location of the reference electrode. For each resistivity array, dipole–dipole and Wenner-α, 16 tests representing 1936 LTC models were computed with complete electrode geometry 10 mm in diameter and 30 mm embedment. For each test, a distribution error map using Eq. 1 was created by interpolation of the results using the Surfer software package with the gridding method: triangulation with linear interpolation. For further analysis of the impact of the electrode position error, it is realistic to consider that the error could exist on the four electrodes used for one quadripole. The solutions are so numerous, if we consider all the possible positions for the four electrodes in the range tested above, that we chose to reduce the number of positions from 121 to nine (Fig. 4). Three unit spacings for the position errors equal to 1, 2 and 3 mm around the reference position were tested separately to manage this test. Nine positions for each electrode represent 6561 combinations to evaluate the distribution of the errors in comparison to the geometric factor of the quadripole considered as the reference with no position errors. To analyse all these data, we used a classification of the geometric factor data set variation. We validated that the probability distribution of position errors on the geometric factor follows a normal distribution law, using the following normal distribution equation: f ðx; μ; σ Þ ¼
ðx−μ Þ2 1 pffiffiffiffiffiffi e− 2σ 2 σ 2π
ð11Þ
The parameter μ is the mathematic mean of the error on geometric factor distribution and σ is the standard deviation. Considering that the probability density function follows a normal distribution, we considered a 95% confidence level which corresponds to 2 standard deviations around the mean to calculate the relative error on the geometric factor for Wenner-α and dipole–dipole arrays. 3. Results and discussion 3.1. Impact of electrode diameter and embedment Fig. 5 presents the relative variation of the geometric factor according to Eq. 1 as a function of the electrode diameter and embedment. The zone where the geometric factor variation is ±3% is delimited by red and blue dotted lines and describes where the variations of the geometric factor for the parameters studied are negligible. The most important result is that the electrode diameter, in the range observed, does
Fig. 4. Electrode positions to investigate error on geometric factor: a) for one electrode (black point), b) four electrode position errors equal to 1 (red), 2 (blue) or 3 mm (purple).
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Fig. 5. Impact of electrode diameter and embedment on geometric factor for an electrode spacing unit equal to 150 mm.
not affect the geometric factor for all configurations considering the same resistivity array and the same embedment. For the dipole–dipole array, we obtained the same results for quadripoles Q1 and Q2, even though they had a different orientation. The variations of the geometric factor are lower than ±3% for embedment between 10 and 40 mm. Q3 is most influenced by the electrode embedment with a variation of the geometric factor from −5 to +8% and a range of ±3% only for embedment equal to 20 or 40 mm. The influence of the electrode embedment is completely negligible for Q4, with a relative variation between −1% and +1.5%. For the Wenner-α array, the results of the geometric factor variations are all between −3 and +3%. The overall results shown in Fig. 5 demonstrate that the geometric factor: (i) is not influenced by the electrode diameter within the range 6–12 mm and (ii) is impacted by the embedment according the quadripole and the resistivity array used. For the first reason, electrode diameter, we chose a fixed diameter of 10 mm for the next tests presented in this study. The geometric factor variations observed for different embedment values are obviously related to the unit spacing between the electrodes. The following question addressed is the impact if this unit spacing is reduced. Exactly the same test was conducted for variable units of electrode spacing from 50 to 150 mm and the results were compared in the same graph illustrated in Fig. 6 for the same four quadripoles and the two array types (Wenner-α Wα and dipole–dipole dd). For each unit of electrode spacing, the reference geometric factor considers 30 mm of embedment. As we assumed, the variation of the geometric factor increases when the unit electrode spacing decreases. Relative variations in the geometric factor are not exactly symmetrical around 30 mm of embedment because the geometry of the test cell studied is cylindrical and therefore the distance variation is not linear for electrode embedment ranging from 10 mm to 50 mm. For quadripoles Q1, Q2 and Q4 and the two resistivity arrays, the total variation of the geometrical factor in the selected range of embedment between 10 mm and 50 mm varies between +7 and −6%. Considering that the acceptable variation of the geometrical factor is between + 3 and − 3%, the minimum electrode spacing has to be 130 mm for Q1-DD, 115 mm for Q1-Wα, 120 mm for Q2-DD, 80 mm for Q2-Wα, 50 mm for Q4-DD and 70 mm for Q4-Wα and the dipole–dipole array. As an example, if the electrode spacing is the lowest tested, 50 mm for the quadripole Q1 and DD array, the electrode embedment
must be between 26 and 34 mm for a theoretical value of 30 mm to guarantee ±3% variation for the geometric factor. We consider that accuracy equal to ±4 mm to guarantee the embedment is realistic when the electrodes are implemented in the test cell. Regarding Q3, the variations are much greater, up to ±35% for the 50 mm electrode spacing. If we retain ±4 mm to guarantee the embedment around 30 mm, the unit electrode spacing has to be equal to or higher than 100 mm for Q3 and the dipole–dipole array. The results illustrated in Fig. 5 and Fig. 6 show important conclusions for laboratory surveys. For the experimental conditions described herein, the electrode diameter is not substantial for the calculation of the geometric factor. The choice of the electrode size is only a function of the media studied to have the lowest contact resistance. Embedment of the electrode is a key parameter which must be controlled if the variation of the geometric factor must be controlled. In these preliminary results, we considered a similar embedment for all electrodes, but how would the variation of the geometric factor evolve if electrode embedment was not the same on all four electrodes? We chose to test the impact of different embedment values on each electrode within a range of ±5 mm around the 30 mm reference embedment and a 50 mm fixed electrode spacing. To perform this test, all the possible combinations of electrode embedment were imagined within the range selected with a 1 mm step on all four electrodes. For each electrode, 11 embedments were possible, representing 114 combinations, for each quadruple. This test was performed only for quadripole Q1 using Wenner-α and dipole–dipole arrays. This entire modelling process requires a very long computation time (i.e. about 3 weeks with an Intel Xeon CPU E5-2650 v2 2.60 GHz processor with 12 cores and 128 Gb of RAM). It can be observed that the results of the geometric factor variations calculated (Eqs. 1 and 2) are distributed according to normal distribution and the values are: – μ(Wenner-α) = 0.2%; 2σ(Wenner-α) = 0.8%; – μ(dipole–dipole) = 0.1%; 2σ(dipole–dipole) = 3.0%;
The mathematic mean of the error on geometric factor distribution and the 2 standard deviations values demonstrate that dipole–dipole array is much more impacted by various embedments than the Wenner-α array ().
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Fig. 6. Impact of embedment on geometrical factor for several electrode spacing units.
3.2. Impact of electrode position error: Dipole–dipole and Wenner-α arrays Figs. 7 and 8 show the variation of the geometric factors for dipole–dipole and Wenner-α arrays, respectively, when an error is made on the position of only one electrode. Along the Z-axis, the 16 graphs in Figs. 7 and 8 show the vertical electrode position in the column and on the X-axis, a curvilinear position error on the edge of the cylindrical column is considered, as shown in the Materials and methods section. The positions of the modelled electrodes vary ± 5 mm with a 1 mm sampling. The black centre dot on each graph represents the perfect position of the electrode. Any variations of the geometric factor are calculated from the value obtained with a geometric factor calculated for a perfect electrode position with 10 mm embedment, 10 mm diameter and 75 mm electrode
spacing. The spatial distribution of the geometric factor variation is interpolated to obtain a map around the perfect position of the electrode. Fig. 7 Q1-A, quadripole Q1, shows the distribution of the geometric factor variations for different positions of electrode A around its theoretical position at the centre. The spatial variation along the X-axis is less than 3%, whereas the variation is negligible all along the Z-axis. The variations for − 3% and + 3% are represented by black dotted lines to clearly identify, on each map, the position errors which are not admissible to conduct such measurements. For the quadripoles Q1, Q2 and Q3, only positions of electrodes M and N used for potential measurements induced geometric factor errors greater than ± 3%. In all cases, the potential error must be reduced to ± 3 mm on one electrode to ensure that the geometric factor calculated for the theoretical position can be taken into account.
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Fig. 7. Influence of position error for one electrode error for a dipole–dipole array.
Fig. 8 illustrates a Wenner-α array. The geometric factor variations observed are less influenced by the position error of one electrode in the Wenner-α array than in a dipole–dipole array. The ±3% variation range is relevant for all position errors tested in the Wenner-α array. However, when time-lapse monitoring is managed, the use of the Wenner-α array is not always advantageous due to low-speed acquisition even with a resistivity meter equipped with several multi-channel modes. The above results illustrate the variation associated with the geometric factor calculated when an error is made on one electrode position. We demonstrate that each quadripole and resistivity array is differently affected by electrode position accuracy. If we consider that the error on the geometric factor has to be lower than 3% to use the
inversion software tools available, the position of one electrode 10 mm in diameter, with 10 mm of embedment and 75 mm of electrode spacing must not exceed ±2 mm for a dipole–dipole array and ±4 mm for a Wenner-α array. However, the error actually does not affect only one electrode in our laboratory test cell. The combination of different position errors on the four electrodes of the quadripole must be studied and follows in the next section. 3.3. Position error on four different electrodes To assess the accuracy position impact of the four electrodes on the geometric factor, we chose to produce a series of models by changing the position of the four electrodes at the same time. We cannot test
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Fig. 8. Variation of the geometric factor related to an error on the position of one electrode for a Wenner-α array.
the 121 positions, 11 mm on the X-axis and 11 mm on the Z-axis for each electrode as in the previous section because the 1214 models to calculate are not realistic. We calculated the geometric factor for a combination of nine positions for each electrode (see Fig. 4). We performed 6561 combinations (94) for each quadripole, which represents 6561 LTC geometries with full electrodes. The impact of the different positions was calculated according Eqs. 1 and 2. We cannot map the data set results as above and a data classification according to the variation in geometric factor classes was used. An example on the first data sets obtained for the Q1 quadripole and the Wenner-α array type is presented in Fig. 9.
Blackheads are the results obtained by modelling and the dotted curve represents the normal law distribution modelled using Eq. 1. In Fig. 9b, the P–P plot shows the good correlation between theoretical probability and the probability obtained with modelled data. We can consider that the error position follows a normal law distribution and a 95% confidence interval (2σ) can be calculated for each quadripole and resistivity array, which is summarized in the table in Fig. 9c. The two standard deviation value is ± 11.54% of the data calculated for quadripole Q2 and the dipole–dipole array. For a maximum position error of 3 mm and the Wenner-α array, the range of the
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Fig. 9. a) Analysis of normal distribution value for Q1 and a Wenner-α array type; b) probability–probability plot; c) error variation considering two standard deviations.
data varies from 1.22% to 5.98% and for the dipole–dipole array from 2.14% to 11.54%. Obviously, and in relation to the previous test, these results demonstrate that the accuracy of the position of the four electrodes must be better than the single electrode. Whatever the position of the electrode or the array type, quadripoles Q4 are always the least affected and quadripole Q2 the most affected. We also found a significant difference between the dipole–dipole and Wenner-α arrays, the latter being less affected by an electrode position error. The synthesis presented in Fig. 9c shows that with a 95% tolerance (2σ) and a desired error less than 3%, it is necessary to position the electrodes with an accuracy of ±2 mm for the Wenner-α and ±1 mm for the dipole–dipole array. It should be noted that these results are only valid for our test cell or a cell that has a cell diameter and interelectrode spacing greater than ours. Further knowledge and specific mechanical equipment must be implemented to achieve the millimetre accuracy for electrode positioning. 4. Conclusion The increase in tomography electrical resistivity surveys, especially at the laboratory scale, as well as the development of new software inversion tools, raises new questions that are often overlooked at the field scale, such as the influence of the size, embedment and position of the electrodes according to the electrode spacing unit. The present article shows the errors associated with these parameters when building an ERT laboratory test cell to avoid errors on geometric factor evaluation. The methodology is based on full 3D laboratory test cell modelling combined with a large number of models to extract trends in the evolution of the geometrical factor.
We have shown that the impact of electrode diameter is negligible in the range of 6–12 mm in diameter. In fact, the embedment of the electrode has the lowest impact on the geometrical factor value depending on the electrode spacing unit. The variation on the electrode position is the most influential parameter on the geometrical factor: for a cell diameter of 200 mm and 150-mm electrode spacing, ± 2 mm accuracy is required on the electrode position for a Wenner-α array and ±1 mm for a dipole–dipole array to reach a geometrical factor variation less than ± 3%. These results indicate that for our column using complete electrodes, with a 30-mm embedment, it is easy to accumulate a large number of uncertainties that may increase the error in the measurement of the geometric factors. Indeed, we did not test any errors combined on the position and embedment that could further escalate the error on the calculation of the geometric factor. The dipole–dipole array is more affected by the electrode position error than the Wenner-α array; this is also true for the vertical location of the electrodes compared to the horizontal location. It can be concluded that laboratory measurements can be complicated and require certain precautions to limit false variations during the inversion process. All the results obtained in this study are probably also valid for field measurements with a unit of electrode spacing less than 1 m. The simulation software tools available now allow a complete study of one electrical resistivity device imagined before its implementation and the beginning of the test.
References Archie, G.E., 1942. The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. Am. Inst. Min. Metall. Eng. 146, 54–61.
R. Clement, S. Moreau / Journal of Applied Geophysics 127 (2016) 45–55 Audebert, M., Clément, R., Touze-Foltz, N., Günther, T., Moreau, S., Duquennoi, C., 2014. Time-lapse ERT interpretation methodology for leachate injection monitoring based on multiple inversions and a clustering strategy (MICS). J. Appl. Geophys. http://dx. doi.org/10.1016/j.jappgeo.2014.09.024. Barker, R., 1998. The application of time-lapse electrical tomography in groundwater studies. Lead. Edge 17, 1454–1458. Benderitter, Y., 1999. Short time variation of the resistivity in an unsaturated soil: the relationship with rainfall. Eur. J. Environ. Eng. Geophys. 4. Benson, A.K., 1995. An integration of geophysical methods and geochemical analysis to map acid mine drainage — a case study. Explor. Min. Geol. 4, 411–419. Benson, A.K., Payne, K.L., Stubben, M.A., 1997. Mapping groundwater contamination using dc resistivity and VLF geophysical methods — a case study. Geophysics 62, 80–86. Binley, A.M., Kemna, A., 2005. DC resistivity and induced polarization methods. HydrogeophysicsWater Sci. Technol. Libr. Ser. 50. Springer, New York, pp. 129–156. Binley, A., Henry-Poulter, S., Shaw, B., 1996. Examination of solute transport in an undisturbed soil column using electrical resistance tomography. Water Resour. Res. 32, 763–769. http://dx.doi.org/10.1029/95WR02995. Brunet, P., Clement, R., Bouvier, C., 2011. Monitoring soil water content and deficit using Electrical Resistivity Tomography (ERT) — a case study in the Cevennes area, France. J. Hydrol. 380, 146–153. http://dx.doi.org/10.1016/j.jhydrol.2009.10.032. Chambers, J.E., Wilkinson, P.B., Penn, S., Meldrum, P.I., Kuras, O., Loke, M.H., Gunn, D.A., 2013. River terrace sand and gravel deposit reserve estimation using threedimensional electrical resistivity tomography for bedrock surface detection. J. Appl. Geophys. 93, 25–32. http://dx.doi.org/10.1016/j.jappgeo.2013.03.002. Chapellier, D., 2000. Prospection électrique de surface. In cours oneline de géophysique de l'Université de Lausanne, Suisse. Clement, R., Moreau, S., 2012. Geo-electrical forward Modelling Using Comsol Multiphysics For Laboratory Simulation. Second Workshop on Geophysical Measurements in Waste Management. Malmö 16-17 February 2012. Clément, R., Descloitres, M., Günther, T., Oxarango, L., Morra, C., Laurent, J.P., Gourc, J.P., 2009. Improvement of electrical resistivity tomography for leachate injection monitoring. Waste Manag. http://dx.doi.org/10.1016/j.wasman.2009.10.002 (in press). Clement, R., Descloitres, M., Gunther, T., Ribolzi, O., Legchenko, A., 2009. Influence of shallow infiltration on time-lapse ERT: experience of advanced interpretation. Compt. Rendus Geosci. 341, 886–898. http://dx.doi.org/10.1016/j.crte.2009.07.005. Clément, R., Descloitres, M., Günther, T., Oxarango, L., Morra, C., Laurent, J.P., Gourc, J.P., 2010. Improvement of electrical resistivity tomography for leachate injection monitoring. Waste Manag. 30, 452–464. http://dx.doi.org/10.1016/j.wasman.2009.10.002. Clement, R., Bergeron, M., Moreau, S., 2011. COMSOL multiphysics modelling for measurement device of electrical resistivity in laboratory test cell. Comsol Multiphysics Conference, p. 4. Dahlin, T., 2001. The development of DC resistivity imaging techniques. Comput. Geosci. 27, 1019–1029. Day-Lewis, F.D., Lane, J.W., Harris, J.M., Gorelick, S.M., 2003. Time-lapse imaging of salinetracer transport in fractured rock using difference-attenuation radar tomography. Water Resour. Res. 39, 1740–1752. http://dx.doi.org/10.1029/2002wr001722. Descloitres, M., Ruiz, L., Sekhar, M., Legchenko, A., Braun, J.J., Kumar, M.S.M., Subramanian, S., 2008. Characterization of seasonal local recharge using electrical resistivity tomography and magnetic resonance sounding. Hydrol. Process. 22, 384–394.
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Günther, T., Rücker, C., Spitzer, K., 2006. Three-dimensional modelling and inversion of dc resistivity data incorporating topography — II. Inversion. Geophys. J. Int. 166, 506–517. http://dx.doi.org/10.1111/j.1365-246X.2006.03011.x. Han, T., Best, A.I., Sothcott, J., North, L.J., MacGregor, L.M., 2015. Relationships among low frequency (2 Hz) electrical resistivity, porosity, clay content and permeability in reservoir sandstones. J. Appl. Geophys. 112, 279–289. http://dx.doi.org/10.1016/j. jappgeo.2014.12.006. Jackson, P.D., 1978. Resistivity–porosity–particle shape relationships for marine sands. Geophysics. Kowalczyk, S., Maślakowski, M., Tucholka, P., 2014. Determination of the correlation between the electrical resistivity of non-cohesive soils and the degree of compaction. J. Appl. Geophys. 110, 43–50. http://dx.doi.org/10.1016/j.jappgeo.2014.08.016. Liu, X., Jia, Y., Zheng, J., Shan, H., Li, H., 2013. Field and laboratory resistivity monitoring of sediment consolidation in China's Yellow River estuary. Eng. Geol. 164, 77–85. http:// dx.doi.org/10.1016/j.enggeo.2013.06.009. Loke, M.H., 2004. Tutorial : 2-D and 3-D electrical imaging surveys. Available online at http://www.geoelectrical.com. Loke, M.H., Chambers, J.E., Rucker, D.F., Kuras, O., Wilkinson, P.B., 2013. Recent developments in the direct-current geoelectrical imaging method. J. Appl. Geophys. 95, 135–156. http://dx.doi.org/10.1016/j.jappgeo.2013.02.017. Moreau, S., Bouye, J.M., Barina, G., Oberty, O., 2003. Electrical resistivity survey to investigate the influence of leachate recirculation in a MSW landfill. In: Centre, E.S.E. (Ed.), Nineth International Waste Management and Landfill Symposium. Cagliari, Italy. Moreau, S., Ripaud, F., Saidi, F., Bouyé, J.-M., 2011. Laboratory test to study waste moisture from resistivity. Proc. Inst. Civ. Eng. Waste Resour. Manag. 164, 17–30. http://dx.doi. org/10.1680/warm.900025. Ogilvy, R., Meldrum, P., Chambers, J., Williams, G., 2002. The use of 3D electrical resistivity tomography to characterise waste and leachate distribution within a closed landfill, thriplow, UK. J. Environ. Eng. Geophys. 7 (11). http://dx.doi.org/10.4133/jeeg7.1.11. Radulescu, M., Valerian, C., Yang, J.W., 2007. Time-lapse electrical resistivity anomalies due to contaminant transport around landfills. Ann. Geophys. 50, 453–468. Rhoades, J.D., Van Schilfgaarde, J., 1976. An electrical conductivity probe for determining soil salinity. Soil Sci. Soc. Am. J. 40. Rhoades, J.D., Raats, P.A.C., Prather, R.J., 1976. Effect of liquid-phase electrical conductivity, water content, and surface conductivity on bulk soil electrical conductivity. Soil Sci. Soc. Am. J. 40, 651–655. Salem, H.S., Chilingarian, G.V., 1999. The cementation factor of Archie's equation for shaly sandstone reservoirs. J. Pet. Sci. Eng. 23, 83–93. Slater, L., Binley, A.M., Daily, W., Johnson, R., 2000. Cross-hole electrical imaging of a controlled saline tracer injection. J. Appl. Geophys. 44, 85–102. Slater, L., Binley, A., Versteeg, R., Cassiani, G., Birken, R., Sandberg, S., 2002. A 3D ERT study of solute transport in a large experimental tank. J. Appl. Geophys. 49, 211–229. Van Schoor, M., 2002. Detection of sinkholes using 2D electrical resistivity imaging. J. Appl. Geophys. 50, 393–399. http://dx.doi.org/10.1016/S0926-9851(02)00166-0. Yang, J., 2005. Geo-electrical responses associated with hydrothermal fluid circulation in oceanic crust: feasibility of magnetometric and electrical resistivity methods in mapping off-axis convection cells. Explor. Geophys. 36, 281–286. http://dx.doi.org/10. 1071/EG05281.
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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo
How should an electrical resistivity tomography laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement R. Clement ⁎, S. Moreau National Research Institute of Science and Technology for Environment and Agriculture (Irstea), Hydrosystems and Bioprocesses Research Unit, 1 rue Pierre-Gilles de Gennes, CS10030, 92761 Antony, France
a r t i c l e
i n f o
Article history: Received 4 April 2015 Received in revised form 28 January 2016 Accepted 17 February 2016 Available online 18 February 2016 Keywords: Electrical resistivity tomography Laboratory 3D complete forward modelling
a b s t r a c t Among geophysical methods, the electrical resistivity tomography (ERT) method is one of the most commonly used for the study of hydrodynamical processes. The geophysical literature relates several laboratory-scale applications of this method. Unlike the measurements taken at the field scale, few authors have taken an interest in errors associated with apparent electrical resistivity, especially in the case of ERT data acquired in the laboratory. The objective of this paper is to show that laboratory errors related to the positioning of electrodes and the geometry of cells are significant on apparent resistivity measurements. The embedment and the position of the electrode were evaluated to quantify their impact on electrical resistivity measurement. To assess these impacts, the authors propose a 3D numerical modelling investigation based on the complete design of a laboratory test cell. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Electrical resistivity tomography (ERT) is a mature geophysical method that is increasingly popular in environmental and hydrogeological studies (Barker, 1998; Binley and Kemna, 2005; Chambers et al., 2013; Loke, 2004; Loke et al., 2013; Ogilvy et al., 2002). This near-surface geophysical method provides information in various field applications: (i) for geological purposes, Van Schoor (2002) used ERT for sinkhole detection and bedrock and sand channel localization; (ii) in a hydrogeological study, Descloitres et al. evaluated the location of shallow and deep infiltrations and recharge zones with time-lapse monitoring (Descloitres et al., 2008); (iii) to monitor pollution plumes Benson (1995); Benson et al. (1997) mapped the extent of plume pollution in soil; (iv) in a municipal solid waste landfill, the leachate flow which is a key point in controlling anaerobic waste biodegradation, the ERT method was used to highlight the volume of waste mass impacted during leachate recirculation events (Audebert et al., 2014; Clement et al., 2009; Clément et al., 2009; Moreau et al., 2003). This list illustrates the various applications of the ERT method related to its many advantages: (i) it is non-destructive: the hydromechanical properties of the subsurface or hydrodynamic processes can be evaluated without digging; (ii) it is sensitive to the contrast in conductivity between the solid and liquid phases of the medium studied and (iii) it may indicate the distribution of electrical resistivity in 2D or ⁎ Corresponding author. E-mail address: [email protected] (R. Clement).
http://dx.doi.org/10.1016/j.jappgeo.2016.02.008 0926-9851/© 2016 Elsevier B.V. All rights reserved.
3D with the recent development of inversion software (Loke et al., 2013). Despite all of the advantages of the ERT method, interpretations of the electrical resistivity variations are not always obvious because the electrical resistivity can be influenced by several parameters; the most frequently cited are: water content, temperature, porosity, density and electrical conductivity of the liquid phase, (Archie, 1942; Benderitter, 1999; Clement et al., 2011; Day-Lewis et al., 2003; Moreau et al., 2011; Rhoades and Van Schilfgaarde, 1976; Salem and Chilingarian, 1999). Even if the method was developed for field applications, a great deal of research has been conducted at the laboratory scale to study one of the parameters involved in the process studied under control conditions (Brunet et al., 2011; Han et al., 2015; Kowalczyk et al., 2014; Rhoades and Van Schilfgaarde, 1976). (i) Brunet et al. (2011) have calibrated the relationship between electrical resistivity and the soil's water content to estimate water deficit at the field scale. (ii) To explain the ranges of resistivity variation observed during leachate reinjection surveys in a municipal solid waste landfill, Moreau et al. (Moreau et al., 2011) conducted a series of laboratory tests on waste samples to relate variations in density and moisture content to the electrical resistivity recorded. The authors proposed a methodology to estimate interpreted resistivity anisotropy between vertical and horizontal resistivity measurements at the laboratory scale. (iii) In another experiment, Slater et al. (2000) used a high-resolution 3D electrical resistivity tomography with crossborehole arrays to study solute transport in a large experimental tank. The authors conducted a salt tracer experiment, monitored by timelapse ERT, in a quasi-two-dimensional sandbox with the aim of determining the hydraulic conductivity distribution in the domain. They
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R. Clement, S. Moreau / Journal of Applied Geophysics 127 (2016) 45–55
concluded that temporal moments of potential perturbations obtained during salt tracer tests provide a good basis for inferring the hydraulic conductivity distribution by fully coupled hydro-geophysical inversions. (iv) Binley et al. (1996) used ERT to study the internal spatial characteristics of solute transport in naturally heterogeneous soils: the analysis of the results revealed spatial variation in transport characteristics throughout the soil column. Most of the volume of the experimental cells is less than 1 m3 (Brunet et al., 2011; Moreau et al., 2011; Slater et al., 2002) and the question addressed by many authors is the position and the spacing between the electrodes to describe the entire medium studied (Moreau et al., 2011). For all the approaches presented above, two types of analysis were always encountered. The first ones consider that the distribution of resistivity is homogeneous and that apparent electrical resistivity can be considered as interpreted resistivity, as in older studies (Archie, 1942; Jackson, 1978; Kowalczyk et al., 2014; Liu et al., 2013; Rhoades and Van Schilfgaarde, 1976). The second type of analysis does not retain the same assumption and requires inversion software to calculate the interpreted resistivity distribution in the medium studied from the measured apparent resistivity (Clement and Moreau, 2012; Slater et al., 2002). In both cases, a geometric factor related to the electrodes' location is required to calculate apparent electrical resistivity. Experimental tests with solutions of known electrical conductivity are possible (as proposed by Rhoades et al. (1976)) and numerical simulation software is available to provide this evaluation. When a forward modelling algorithm is used to evaluate the geometric factor, the position, size and shape of the electrodes have to be known with the highest accuracy. However, a great deal of additional information is needed: the shape and size of the test cell must be measured and the embedment of the electrodes in the medium studied must be identified. The numerous numerical tools available now following the technical development of computers and computer languages can provide a full and accurate description of the experimental conditions. For the second type of analysis using inversion processes, electrode shape, size and embedment are difficult to take into account in the inversion software available, such as BERT (Boundless Electrical Resistivity Tomography), developed by Günther et al. (2006), or R3T, defined in Binley and Kemna (2005). Generally, the electrodes are described as a point electrode in the laboratory test cell designed in the inversion software. One of the key parameters is the position of the electrode represented by point node because even if the theoretical position of the point node is perfectly defined during the design of the laboratory test cell, in fact, there is always an error associated with the mechanical construction of the laboratory test cell. The position and the error on that position have an impact on the calculation of the geometric factor of the quadripole because the characteristics of the complete electrode cannot be ignored when spacing is short in laboratory tests. This paper proposes guidelines to design an example of a cylindrical laboratory test cell for ERT measurements using a numerical approach. The influence of electrode size and impact of the error on its position and its embedment are studied to define the accuracy needed to guarantee the computation of apparent resistivity and indirectly the distribution of interpreted resistivity using inversion software. 2. Material and methods In ERT laboratory measurements, several parameters can create errors on the apparent resistivity calculation and the evaluation of interpreted resistivity using inversion software. Considering that the instruments used for current injection and potential measurements are calibrated, we identify three major errors which could influence the geometric factor associated with a quadripole: (i) the electrode shape and size, (ii) the accurate measurement of electrode embedment and (iii) the accurate measurement of the electrode position. To estimate the impact of these parameters on the geometrical factor, we chose a classical numerical methodology based on multiple numerical forward
modelling currently applied in geophysics (Clement et al., 2009; Clément et al., 2010; Radulescu et al., 2007; Yang, 2005). We chose to base our modelling approach on our experimental laboratory test cell (LTC) presented in Fig. 1a. For all numerical modelling, the first step is the design of the LTC for different conditions: shape, size, embedment and electrode position. The second step is the calculation of the geometric factor for each combination of conditions imagined and the last one is the evaluation of the distribution of the results according to the parameters tested. 2.1. Laboratory test cell description for numerical design During the last 5 years, we conducted laboratory tests on waste samples to study relations between the electrical resistivity variations observed and different hydro-mechanical conditions such as density and water content (Moreau et al., 2011). Our experimental test cells always had the same cylindrical shape and an average volume of 0.226 m3 for 1 m height and 0.4 m diameter. These configurations were considered to design the LTC for the different numerical models tested in this paper. The LTC is made of high-density polyethylene (HDPE). Forty steel electrodes are distributed over five levels and eight vertical lines spaced 45° apart. Fig. 1a–b describes the theoretical shape, embedment and positions of the electrodes. The conventional electrodes used are cylindrical and exceed 30 mm inside the test cell. They are spaced vertically 150 mm apart and 157 mm apart horizontally on the perimeter of the test cell. 2.2. Electrical resistivity measurement The ERT method is thoroughly described in the geophysical literature (Chapellier, 2000; Dahlin, 2001; Loke et al., 2013). The apparent resistivity ρapp is calculated from a quadripole composed of two injected current electrodes A and B and two other electrodes M and N to measure a potential difference (Eq. 1). The geometric factor k depends on the position of the four electrodes called quadripole as well as the size and the shape of the electrodes when they cannot be considered to be a point in the measurement process.
ρapp ¼
k ΔVMN IAB
ð1Þ
where: ρapp is apparent electrical resistivity (Ω∙m) is the electrical potential difference measured (V) ΔVMN I is the intensity of the injected current (A) k is the geometric factor (m). The electrical resistivity arrays evaluated are the Wenner-α and the dipole–dipole array because they are the most popular (Fig. 2). From the top of the test cell, the electrodes are numbered from 1 to 40: the first level considers electrodes 1–8 and the lowest level electrodes 33–40. The vertical distance between two consecutive electrodes is equal to 150 mm, the position of the top and the bottom are, respectively, 175 mm from the upper level of the electrodes (1–8) and 125 mm from the lowest level of the electrodes (33–40). Among all imaginable quadripoles from the 40 electrodes implemented, four quadripoles were selected as being the most frequently used. The first quadripole Q1, called the horizontal, consists of four electrodes located on the same level and spaced 45° apart (Fig. 2). The second, Q2, is the vertical and considers the four electrodes on the same vertical line. The third, Q3, is the diagonal; four consecutive electrodes are all placed at different levels. The last array, Q4, is a mixed array between all of these positions. Two consecutive electrodes are at the same level and two others are also at the same level but different from the previous and on the opposite side of the LTC.
R. Clement, S. Moreau / Journal of Applied Geophysics 127 (2016) 45–55
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Fig. 1. Laboratory test cell (LTC) and electrode description: a) left: picture of true laboratory test cell; middle: diagram of LTC without HDPE material; right: the numerical mesh; b) electrode type shape and size.
2.3. Elementary theory of DC conductive media modelling The governing equation for ERT forward modelling in a conductive subsurface media can be derived from Maxwell's equations. Technically ERT is based on a very-low-frequency alternating current (2–5 Hz) to limit polarization effects. In this case, we could ignore the inductive and capacitive effects in Maxwell's equation. Under static conditions, the electric field E (in V/m) can be represented by the negative gradient of the electric potential V: E ¼ −∇V:
ð2Þ
In conductive media, to simulate stationary electric current density, we consider the stationary equation of continuity and Ohm's law. In a stationary coordinate system, the point form of Ohm's law states that: J ¼ σE þ Je
ð3Þ
where, J is the current density (A/m2) equal to the current I per unit of cross-sectional area S (A), Je is the externally current density (in A/m2), E is the electric field intensity and σ is the electrical conductivity. The SI unit
for conductivity is amperes per volt-metre (A/Vm) or Siemens per metre S/m and the reciprocal of conductivity is known as resistivity in Ohmmetre. The static form of the equation of continuity then states: ∇ J ¼ −∇ ðσ∇V−JeÞ ¼ 0:
ð4Þ
The generalized form of the current sources (Qj) can be written as follows: −∇:ðσ∇V−JeÞ ¼ Q j :
ð5Þ
Considering our LTC geometry and the principle of ERT, various boundary conditions must be applied in this model (Fig. 1a). We consider that the wall of the LTC in HDPE is infinitely resistant, we applied an insulating boundary condition at the contact area between the homogenous media studied and the LTC, expressed by the Neumann condition; this equation specifies that there is no current flowing across the boundary: n J ¼ 0:
Fig. 2. Selected quadripoles and array types.
ð6Þ
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We apply the same boundary condition for the proportion of electrodes in contact with the medium studied inside the laboratory test cell and outside in contact with the air. The LTC is made with HDPE, considered to be an electrical insulator with a conductivity of 10−14 S/m, and the steel electrodes (ref: AISI 4340) are 4.32 ∗ 106 S/m of conductivity. For the ERT measurements, the current circulation is made between two current electrodes represented by a positive intensity on electrode A and a negative intensity on electrode B. According to the surface of the electrode S (m2) and the injected current I0 = 100 mA, the current density J is calculated using the following equations. For the positive electrode A, incomer current flow is considered: n J ¼ JnA :
ð7Þ
For the negative electrode B, inward current flow is considered: −n J ¼ JnB
ð8Þ
I J¼ : S
ð9Þ
distributions are built using the AC/DC module (quasi-stationary electromagnetic field) to evaluate the potential difference induced by the injected current and the electrical boundary conditions selected as infinite or insulating surfaces. This step is made automatically to calculate the electrical potential for every quadripole using F3D-lab Matlabcode (Forward 3D Laboratory) developed by Clement et al. (2011). The end of simulation calculates the geometrical factor for each quadripole and every condition for the shape, embedment and positions of the electrodes. To facilitate the building of the geometrical model and the data processing, we implemented a Matlab script and F3D-lab interface consistent with COMSOL. This script automatically creates the geometry of a LTC based on parameters selected by the user. The height and diameter of the column can be adjusted easily such as the number, shape and embedment of the electrodes. Additional refinement has been applied to obtain accurate forward calculation around the electrode. This tool calculates either the geometrical factor on a homogeneous model or apparent resistivity for the heterogeneous resistivity model; the script is available free of charge from the authors on request. 2.5. Simulation test managed
These theoretical equations are directly implemented in Comsol Multiphysics and we use the solver tools included in Comsol as the AC/DC module. 2.4. Three-dimensional electrical resistivity forward modelling The methodology of this paper is based on the use of a forward modelling tool, where the complete geometry of the LTC and the electrodes is taken into account. The general approach follows six steps presented in Fig. 3. Here, the geometry design of the electrode and the laboratory test cell is highly challenging for ERT modelling at the laboratory scale. To respond to this problem, Comsol Multiphysics 4.3b combined with Matlab 2012 are used. With Comsol, the electrical field distributions can be modelled using full 3D modelling. Electric field
The geometry of the LTC is presented above (Fig. 1), as are the electrical arrays tested. The shape, size and position chosen for the electrodes considered as the reference model are: 10 mm in diameter, 30 mm embedment inside the HDPE test cell and a vertical distance between electrodes equal to 150 mm and spaced horizontally 45° apart. The first objective is to consider different electrode positions with no associated position error and to assess the impact of various electrode diameters and embedments. Then different electrode positions around the reference positions will be studied to simulate errors that could be made during the mechanical construction of the test cell. Modelling all these parameters requires a very long computation time (i.e. about 4 weeks with an Intel Xeon CPU E5-2650 v2 2.60 GHz processor with 12 cores and 128 Gb of RAM).
Fig. 3. Methodology of full 3D modelling using Comsol Multiphysics 4.3 and F3DM Matlab script.
R. Clement, S. Moreau / Journal of Applied Geophysics 127 (2016) 45–55
2.5.1. Impact of electrode diameter and embedment as a function of electrode spacing In the first part, the impact of both electrode embedment and diameter were evaluated. We tested four different complete electrode diameters from 6, 8, 10 to 12 mm and we made different LTC numerical models with variable embedment values (10, 20, 30, 40 and 50 mm) for spacing between electrodes equal to 150 mm. For each diameter (d) and embedment (E) combination, the geometric factor KdE (m) is evaluated and compared to the reference geometric factor Kref (m) (from the electrode 10 mm in diameter and 30 mm embedment). Eq. 10 below was used to estimate the relative variations of the geometrical factor: Δk ¼
kdE −1 100 ð%Þ: kref
ð10Þ
Depending on the results of this test, we changed the unit of electrode spacing to decrease it from 150 mm to 125 mm, and then 100 mm, 75 mm and finally 50 mm, and we only tested the impact of different embedment values but always equal for the four electrodes. The same assessment was made but with a specific reference Kref geometric factor for each spacing tested. It is realistic to imagine that an embedment error can differ from one electrode to another. To conclude on this part, one quadripole was selected to compute the geometric factors for different embedment values for the four electrodes within a range of 25–35 mm with a 1 mm unit resolution. The number of models calculated for each resistivity array was 14,641: Wenner-α and dipole–dipole. 2.5.2. Electrode position accuracy In the field in most cases, the errors on electrode positions are often overlooked because they are considered to be negligible. A few centimetres of error on the electrode position is very low compared to the unit of electrode spacing greater than 1 m generally used. At the laboratory scale, this error must be much more controlled due to the small electrode spacing unit, around 150 mm or less. The aim of this part is to evaluate the accuracy required to position the electrodes on our LTC in order to reduce the error on the computation of the geometric factor. Several models have been imagined to introduce possible electrode position errors. For the four quadripoles selected and the two resistivity arrays, we varied the position of only one electrode of the four available, as proposed in Fig. 4. This number of positions allowed creating 121 models to calculate the variation of the geometric factor for one quadripole. We considered a range of variation ± 5 mm with a unit
49
spacing of 1 mm along the vertical and horizontal axes (a curvilinear position) and around the location of the reference electrode. For each resistivity array, dipole–dipole and Wenner-α, 16 tests representing 1936 LTC models were computed with complete electrode geometry 10 mm in diameter and 30 mm embedment. For each test, a distribution error map using Eq. 1 was created by interpolation of the results using the Surfer software package with the gridding method: triangulation with linear interpolation. For further analysis of the impact of the electrode position error, it is realistic to consider that the error could exist on the four electrodes used for one quadripole. The solutions are so numerous, if we consider all the possible positions for the four electrodes in the range tested above, that we chose to reduce the number of positions from 121 to nine (Fig. 4). Three unit spacings for the position errors equal to 1, 2 and 3 mm around the reference position were tested separately to manage this test. Nine positions for each electrode represent 6561 combinations to evaluate the distribution of the errors in comparison to the geometric factor of the quadripole considered as the reference with no position errors. To analyse all these data, we used a classification of the geometric factor data set variation. We validated that the probability distribution of position errors on the geometric factor follows a normal distribution law, using the following normal distribution equation: f ðx; μ; σ Þ ¼
ðx−μ Þ2 1 pffiffiffiffiffiffi e− 2σ 2 σ 2π
ð11Þ
The parameter μ is the mathematic mean of the error on geometric factor distribution and σ is the standard deviation. Considering that the probability density function follows a normal distribution, we considered a 95% confidence level which corresponds to 2 standard deviations around the mean to calculate the relative error on the geometric factor for Wenner-α and dipole–dipole arrays. 3. Results and discussion 3.1. Impact of electrode diameter and embedment Fig. 5 presents the relative variation of the geometric factor according to Eq. 1 as a function of the electrode diameter and embedment. The zone where the geometric factor variation is ±3% is delimited by red and blue dotted lines and describes where the variations of the geometric factor for the parameters studied are negligible. The most important result is that the electrode diameter, in the range observed, does
Fig. 4. Electrode positions to investigate error on geometric factor: a) for one electrode (black point), b) four electrode position errors equal to 1 (red), 2 (blue) or 3 mm (purple).
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Fig. 5. Impact of electrode diameter and embedment on geometric factor for an electrode spacing unit equal to 150 mm.
not affect the geometric factor for all configurations considering the same resistivity array and the same embedment. For the dipole–dipole array, we obtained the same results for quadripoles Q1 and Q2, even though they had a different orientation. The variations of the geometric factor are lower than ±3% for embedment between 10 and 40 mm. Q3 is most influenced by the electrode embedment with a variation of the geometric factor from −5 to +8% and a range of ±3% only for embedment equal to 20 or 40 mm. The influence of the electrode embedment is completely negligible for Q4, with a relative variation between −1% and +1.5%. For the Wenner-α array, the results of the geometric factor variations are all between −3 and +3%. The overall results shown in Fig. 5 demonstrate that the geometric factor: (i) is not influenced by the electrode diameter within the range 6–12 mm and (ii) is impacted by the embedment according the quadripole and the resistivity array used. For the first reason, electrode diameter, we chose a fixed diameter of 10 mm for the next tests presented in this study. The geometric factor variations observed for different embedment values are obviously related to the unit spacing between the electrodes. The following question addressed is the impact if this unit spacing is reduced. Exactly the same test was conducted for variable units of electrode spacing from 50 to 150 mm and the results were compared in the same graph illustrated in Fig. 6 for the same four quadripoles and the two array types (Wenner-α Wα and dipole–dipole dd). For each unit of electrode spacing, the reference geometric factor considers 30 mm of embedment. As we assumed, the variation of the geometric factor increases when the unit electrode spacing decreases. Relative variations in the geometric factor are not exactly symmetrical around 30 mm of embedment because the geometry of the test cell studied is cylindrical and therefore the distance variation is not linear for electrode embedment ranging from 10 mm to 50 mm. For quadripoles Q1, Q2 and Q4 and the two resistivity arrays, the total variation of the geometrical factor in the selected range of embedment between 10 mm and 50 mm varies between +7 and −6%. Considering that the acceptable variation of the geometrical factor is between + 3 and − 3%, the minimum electrode spacing has to be 130 mm for Q1-DD, 115 mm for Q1-Wα, 120 mm for Q2-DD, 80 mm for Q2-Wα, 50 mm for Q4-DD and 70 mm for Q4-Wα and the dipole–dipole array. As an example, if the electrode spacing is the lowest tested, 50 mm for the quadripole Q1 and DD array, the electrode embedment
must be between 26 and 34 mm for a theoretical value of 30 mm to guarantee ±3% variation for the geometric factor. We consider that accuracy equal to ±4 mm to guarantee the embedment is realistic when the electrodes are implemented in the test cell. Regarding Q3, the variations are much greater, up to ±35% for the 50 mm electrode spacing. If we retain ±4 mm to guarantee the embedment around 30 mm, the unit electrode spacing has to be equal to or higher than 100 mm for Q3 and the dipole–dipole array. The results illustrated in Fig. 5 and Fig. 6 show important conclusions for laboratory surveys. For the experimental conditions described herein, the electrode diameter is not substantial for the calculation of the geometric factor. The choice of the electrode size is only a function of the media studied to have the lowest contact resistance. Embedment of the electrode is a key parameter which must be controlled if the variation of the geometric factor must be controlled. In these preliminary results, we considered a similar embedment for all electrodes, but how would the variation of the geometric factor evolve if electrode embedment was not the same on all four electrodes? We chose to test the impact of different embedment values on each electrode within a range of ±5 mm around the 30 mm reference embedment and a 50 mm fixed electrode spacing. To perform this test, all the possible combinations of electrode embedment were imagined within the range selected with a 1 mm step on all four electrodes. For each electrode, 11 embedments were possible, representing 114 combinations, for each quadruple. This test was performed only for quadripole Q1 using Wenner-α and dipole–dipole arrays. This entire modelling process requires a very long computation time (i.e. about 3 weeks with an Intel Xeon CPU E5-2650 v2 2.60 GHz processor with 12 cores and 128 Gb of RAM). It can be observed that the results of the geometric factor variations calculated (Eqs. 1 and 2) are distributed according to normal distribution and the values are: – μ(Wenner-α) = 0.2%; 2σ(Wenner-α) = 0.8%; – μ(dipole–dipole) = 0.1%; 2σ(dipole–dipole) = 3.0%;
The mathematic mean of the error on geometric factor distribution and the 2 standard deviations values demonstrate that dipole–dipole array is much more impacted by various embedments than the Wenner-α array ().
R. Clement, S. Moreau / Journal of Applied Geophysics 127 (2016) 45–55
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Fig. 6. Impact of embedment on geometrical factor for several electrode spacing units.
3.2. Impact of electrode position error: Dipole–dipole and Wenner-α arrays Figs. 7 and 8 show the variation of the geometric factors for dipole–dipole and Wenner-α arrays, respectively, when an error is made on the position of only one electrode. Along the Z-axis, the 16 graphs in Figs. 7 and 8 show the vertical electrode position in the column and on the X-axis, a curvilinear position error on the edge of the cylindrical column is considered, as shown in the Materials and methods section. The positions of the modelled electrodes vary ± 5 mm with a 1 mm sampling. The black centre dot on each graph represents the perfect position of the electrode. Any variations of the geometric factor are calculated from the value obtained with a geometric factor calculated for a perfect electrode position with 10 mm embedment, 10 mm diameter and 75 mm electrode
spacing. The spatial distribution of the geometric factor variation is interpolated to obtain a map around the perfect position of the electrode. Fig. 7 Q1-A, quadripole Q1, shows the distribution of the geometric factor variations for different positions of electrode A around its theoretical position at the centre. The spatial variation along the X-axis is less than 3%, whereas the variation is negligible all along the Z-axis. The variations for − 3% and + 3% are represented by black dotted lines to clearly identify, on each map, the position errors which are not admissible to conduct such measurements. For the quadripoles Q1, Q2 and Q3, only positions of electrodes M and N used for potential measurements induced geometric factor errors greater than ± 3%. In all cases, the potential error must be reduced to ± 3 mm on one electrode to ensure that the geometric factor calculated for the theoretical position can be taken into account.
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Fig. 7. Influence of position error for one electrode error for a dipole–dipole array.
Fig. 8 illustrates a Wenner-α array. The geometric factor variations observed are less influenced by the position error of one electrode in the Wenner-α array than in a dipole–dipole array. The ±3% variation range is relevant for all position errors tested in the Wenner-α array. However, when time-lapse monitoring is managed, the use of the Wenner-α array is not always advantageous due to low-speed acquisition even with a resistivity meter equipped with several multi-channel modes. The above results illustrate the variation associated with the geometric factor calculated when an error is made on one electrode position. We demonstrate that each quadripole and resistivity array is differently affected by electrode position accuracy. If we consider that the error on the geometric factor has to be lower than 3% to use the
inversion software tools available, the position of one electrode 10 mm in diameter, with 10 mm of embedment and 75 mm of electrode spacing must not exceed ±2 mm for a dipole–dipole array and ±4 mm for a Wenner-α array. However, the error actually does not affect only one electrode in our laboratory test cell. The combination of different position errors on the four electrodes of the quadripole must be studied and follows in the next section. 3.3. Position error on four different electrodes To assess the accuracy position impact of the four electrodes on the geometric factor, we chose to produce a series of models by changing the position of the four electrodes at the same time. We cannot test
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Fig. 8. Variation of the geometric factor related to an error on the position of one electrode for a Wenner-α array.
the 121 positions, 11 mm on the X-axis and 11 mm on the Z-axis for each electrode as in the previous section because the 1214 models to calculate are not realistic. We calculated the geometric factor for a combination of nine positions for each electrode (see Fig. 4). We performed 6561 combinations (94) for each quadripole, which represents 6561 LTC geometries with full electrodes. The impact of the different positions was calculated according Eqs. 1 and 2. We cannot map the data set results as above and a data classification according to the variation in geometric factor classes was used. An example on the first data sets obtained for the Q1 quadripole and the Wenner-α array type is presented in Fig. 9.
Blackheads are the results obtained by modelling and the dotted curve represents the normal law distribution modelled using Eq. 1. In Fig. 9b, the P–P plot shows the good correlation between theoretical probability and the probability obtained with modelled data. We can consider that the error position follows a normal law distribution and a 95% confidence interval (2σ) can be calculated for each quadripole and resistivity array, which is summarized in the table in Fig. 9c. The two standard deviation value is ± 11.54% of the data calculated for quadripole Q2 and the dipole–dipole array. For a maximum position error of 3 mm and the Wenner-α array, the range of the
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Fig. 9. a) Analysis of normal distribution value for Q1 and a Wenner-α array type; b) probability–probability plot; c) error variation considering two standard deviations.
data varies from 1.22% to 5.98% and for the dipole–dipole array from 2.14% to 11.54%. Obviously, and in relation to the previous test, these results demonstrate that the accuracy of the position of the four electrodes must be better than the single electrode. Whatever the position of the electrode or the array type, quadripoles Q4 are always the least affected and quadripole Q2 the most affected. We also found a significant difference between the dipole–dipole and Wenner-α arrays, the latter being less affected by an electrode position error. The synthesis presented in Fig. 9c shows that with a 95% tolerance (2σ) and a desired error less than 3%, it is necessary to position the electrodes with an accuracy of ±2 mm for the Wenner-α and ±1 mm for the dipole–dipole array. It should be noted that these results are only valid for our test cell or a cell that has a cell diameter and interelectrode spacing greater than ours. Further knowledge and specific mechanical equipment must be implemented to achieve the millimetre accuracy for electrode positioning. 4. Conclusion The increase in tomography electrical resistivity surveys, especially at the laboratory scale, as well as the development of new software inversion tools, raises new questions that are often overlooked at the field scale, such as the influence of the size, embedment and position of the electrodes according to the electrode spacing unit. The present article shows the errors associated with these parameters when building an ERT laboratory test cell to avoid errors on geometric factor evaluation. The methodology is based on full 3D laboratory test cell modelling combined with a large number of models to extract trends in the evolution of the geometrical factor.
We have shown that the impact of electrode diameter is negligible in the range of 6–12 mm in diameter. In fact, the embedment of the electrode has the lowest impact on the geometrical factor value depending on the electrode spacing unit. The variation on the electrode position is the most influential parameter on the geometrical factor: for a cell diameter of 200 mm and 150-mm electrode spacing, ± 2 mm accuracy is required on the electrode position for a Wenner-α array and ±1 mm for a dipole–dipole array to reach a geometrical factor variation less than ± 3%. These results indicate that for our column using complete electrodes, with a 30-mm embedment, it is easy to accumulate a large number of uncertainties that may increase the error in the measurement of the geometric factors. Indeed, we did not test any errors combined on the position and embedment that could further escalate the error on the calculation of the geometric factor. The dipole–dipole array is more affected by the electrode position error than the Wenner-α array; this is also true for the vertical location of the electrodes compared to the horizontal location. It can be concluded that laboratory measurements can be complicated and require certain precautions to limit false variations during the inversion process. All the results obtained in this study are probably also valid for field measurements with a unit of electrode spacing less than 1 m. The simulation software tools available now allow a complete study of one electrical resistivity device imagined before its implementation and the beginning of the test.
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