marscot et al 2006 code

Journal of Applied Geophysics 60 (2006) 55 67 www.elsevier.com/locate/jappgeo

A general approach for DC apparent resistivity evaluation on arbitrarily shaped 3D structuresLaurent Marescot a,b,, Stphane Rigobert c , Srgio Palma Lopes a , Richard Lagabrielle a , Dominique Chapellier bb

Laboratoire Central des Ponts et Chausses, 44341 Bouguenais, France Institute of Geophysics, University of Lausanne, 1015 Lausanne, Switzerland c Laboratoire Central des Ponts et Chausses, 75732 Paris, France Received 6 January 2005; accepted 15 December 2005

a

Abstract This paper presents a general and comprehensive way to evaluate the geometric factors used for the computation of apparent resistivities in the context of DC resistivity mapping and non-destructive investigations, in laboratory or in the field. This technique enables one to consider 3-dimensional objects with arbitrary shape. The expression of the geometric factor results from the early definition of apparent resistivitiy. It is expressed as the ratio of the resistances obtained from measurements to the resistances induced in the medium with unitary resistivity considering the same object geometry and electrode set-up. In this work, a finite element code is used for the computation of the geometric factor. In this code, the electrodes do not need to be located on the nodes of the mesh. This option makes the finite element mesh generation task easier. A first synthetical example illustrates how the present approach could be applied to apparent resistivity mapping in an environment with a complex underground topography. A second example, based on real data in a water tank, illustrates the simulation of a resistivity survey on a structure with finite extent, e.g. a laboratory sample. In both examples, topographic artefacts and effects of material sample shapes are successfully taken into account and reliable apparent resistivity descriptions of the structures are obtained. The effectiveness of the method for the detection of heterogeneities in apparent resistivity maps is highlighted. 2006 Elsevier B.V. All rights reserved.Keywords: Apparent resistivity; Geometric factor; Cell constant; Topography correction; Finite element modelling; Laboratory measurements

1. Introduction Recent improvements in data acquisition combined with the development of powerful computer workstations have encouraged the use of resistivity techniques Corresponding author. Currently: Institute of Geophysics, ETHSwiss Federal Institute of Technology, HPP O7 ETH Hoenggerberg, CH-8093 Zurich, Switzerland. Tel.: + 41 44 633 75 61; fax: +41 44 633 10 65. E-mail address: [email protected] (L. Marescot). 0926-9851/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jappgeo.2005.12.003

for non-conventional problems. Resistivity methods are of great interest in civil or mining engineering investigations to examine the structure of tunnels, underground quarry columns or mine galleries, as a helping tool for the planning of safe gallery extensions. In these applications, the terrains may have an uneven topography and a half-space approximation may not be applicable to the overall geometry of the problem (e.g. Sasaki and Matsuo, 1990; Dobroka et al., 1991; Draskovits and Simon, 1992; Hering et al., 1995; Maillol et al., 1999; Yaramanci, 2000; Yaramanci and Kiewer,

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2000). In non-destructive investigations and petrophysical analysis, the electrical properties of material samples (e.g. soil, rock or concrete core samples) or architectural ornamentations (Lataste, 2002; Taylor and Barker, 2002; Giao et al., 2003) are studied. Electrical techniques are also used in medical and biological engineering to study the structures and properties of human body (Lionheart, 2004) or biological samples from a set of measurements made around, sometimes inside, the investigated object (Faes et al., 1999; Linderholm et al., 2004). In these examples, the domain investigated has a finite extent (e.g. a laboratory test-cell) and a complex shape or topography. Resistivity methods are employed on complicated 3dimensional (3-D) models and a large number of parameters is required to describe the geometry accurately. Measured apparent resistivity data need to be inverted using an iterative algorithm to give a clearer image of the investigated structures. Nevertheless, there are still some fields of interest where apparent resistivities are directly used to infer information on the properties of an object, like for laboratory petrophysical measurements (e.g. Lataste, 2002; Taylor and Barker, 2002; Giao et al., 2003), or at a larger scale in resistivity mapping or profiling (e.g. VanGemert et al., 1996; Marescot et al., 2003a) or in well-logging and for borehole measurements (e.g. Le Masne and Poirmeur, 1988; Poirmeur and Vasseur, 1988; Leroux, 2000). Resistivity mapping techniques currently meet new expectations with the developments of mobile galvanic or electrostatic arrays (Panissod et al., 1997). Apparent resistivities are also used in processing electrical anisotropy for fracture and karst detection or petrophysical determination (Watson and Barker, 1999; Busby, 2000) or in monitoring of complex structures like volcanoes (Utada, 2003). Apparent resistivity rather than electrical resistance is used by geophysicists and engineers when investigating the electrical properties of an object. By relating the electrical resistance to the array dimension, each measurement will depend more on the electric structure of the object than on the array length. The electrical resistance values are traditionally transformed into apparent resistivity values using the geometric factor of each array, which can be calculated only for simple geometric models, such as a half space or a cylindrical sample. Since the global electrical measurements are influenced by the outline of the sample, this frequently implies to reshape the object or to apply some crude approximations, which are not always satisfactory solutions. There is therefore a need in having a totally versatile technique to evaluate apparent resistivities in

any situation and especially for the cases outlined above. When the investigated object features an arbitrary shape, however well known, the determination of apparent resistivity can be carried out, referring to the most general definition of apparent resistivity as recalled here after. In this paper, a procedure is presented for an easy and reliable computation of the apparent resistivity parameter on any 2-D or 3-D structure of arbitrary shape, in the laboratory or in the field, and using any electrode layout. The apparent resistivity formulation is first detailed. The following section concerns an application of the method to the apparent resistivity mapping of a synthetic model with strong underground topography. Finally, laboratory tank measurements are presented to illustrate the effectiveness of the method for petrophysical characterization and resistivity mapping. 2. Approaches to defining apparent resistivities 2.1. Analytical approach The potential values are traditionally (Kunetz, 1966) transformed into apparent resistivities by multiplying the measured resistance by the array geometric factor. The following relation is generally used, with G the geometric factor, expressed in meters, and R the electrical resistance, expressed in ohms: qapp GR with G1 AN 1 BM

1

1 AM

1 BN

4k A1 A1N B1 B1N M V V M V V

2

where A and B are the current electrodes and M and N the potential electrodes. A and B are the images of A and B with respect to the ground surface (see Fig. 1). It has to be emphasized that this expression for G is only strictly correct for a flat earth. This user friendly expression of G (Eq. (2)) originates from the early geoelectrical prospecting schemes. It is very popular and has become sort of a standard used by most geophysicists. Nevertheless, direct current resistivity methods are now applied to a wide variety of fields where the flat earth model is clearly not the standard reference anymore. As a conventional choice, Eq. (2) could still be used, but it is a well known fact that it brings strong artefacts in the case of a more arbitrary geometry or topography (e.g. resistivity mapping along a cliff, a canyon or in a tunnel). Therefore, a more


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Fig. 1. The half-space earth model: a continuous infinite medium, separated from air by a plane interface.

versatile convention is suggested as described in the following section. 2.2. General approach The approach presented herein is based on the early definition of apparent resistivity (Stefanescu et al., 1930; Kunetz, 1966; Lagabrielle, 1977). Kunetz (1966, page 10) notes: In order to facilitate interpretation of the results, the first important step is to choose the desirable data from those taken, to organize it properly, and to present it in a convenient form. In the beginning, it was noted that, in place of the potential itself, it was preferable to consider directly the difference between the real potential and the potential that would exist under the same conditions in a homogeneous earth. Normally, one takes the ratio of the measured potential to the theoretical potential, or the actual field strength to the theoretical field strength, at a given point. This ratio has become the fundamental parameter of electrical prospecting and is known as the apparent resistivity, when the resistivity of the reference medium equals unity. The apparent resistivity becomes the real resistivity if the earth in question is homogeneous. The fundamental parameter suggested by Kunetz is therefore simply the ratio V / V0, where V is a potential drop measured in the field and V0 is the corresponding potential drop evaluated in the equivalent homogeneous earth in the same conditions. Obviously, in the frame of mind of early applications of direct current resistivity to subsurface prospecting, these conditions could be understood as the earth topography, the electrode set-up and the current intensity. But in fact this definition genuinely describes the most general

case, for which the conditions extend to: the medium complete geometry, whether it is of infinite or finite dimensions, the natural boundary conditions (e.g. insulation or ground conditions), the volume and the shape of electrodes (e.g. point, ring, plate, line or spherical electrodes), as well as the number of current sources and the respective current intensities (e.g. focussing arrays). This basically includes any field of application. Therefore we suggest that this most general definition is restored, in place of the popular analytical formulation whenever it proves to fail. The following sections illustrate some powerful advantages of this definition as a practical tool. As was underlined by Kunetz, the ratio V / V0 can only be considered as the apparent resistivity if the resistivity of the reference homogeneous medium equals unity. Indeed, the V0 potential value is directly proportional to the resistivity of the homogeneous medium. Furthermore, the V / V0 ratio is dimensionless. In order to make this early definition clearer, it is suggested to rewrite it in a form that naturally brings the proper dimension back: qapp DV q DV0 0

1

3

the V0 potential drop being calculated for an homogeneous medium of arbitrary resistivity 0. In this case, 0 / V0 does not depend on 0 and in practice 0 can be set to any value (for instance 1 m). The constraint of having to use the same current intensity for the evaluation of V and V0 on the homogeneous model can be overcome by writing: qapp R q R0 0 4

where R is the resistance measured in the field and R0 is the resistance calculated for the homogenous medium with resistivity 0 and the same electrode set-up. In this paper, this approach (Eq. (4)) shall be referred to as the GGF (General Geometric Factor) approach. Considering the classical expression of apparent resistivity given by Eq. (1), one obtains the expression of geometrical factor as: G q0 R0 5

G has the dimension of a length (m) and only depends on the object geometry and the electrode layout. Let us remind that R0 is proportional to 0. Therefore, G does not depend on 0 which has an arbitrary value.

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A single calculation performed on a homogenous model brings the geometrical factor to fully take the electrode set-up and site geometry into account. Numerical methods, such as boundary integral formulations, finite differences or the finite element method, may be used for the computation of V0. In practice, the reference homogeneous model includes any well known geometry input (e.g. a natural border like a cliff or well located inner galleries or voids) and exclude any searched structures (unknown geological targets like faults or cavities). Apparent resistivities defined either by Eq. (1) or Eq. (4) embrace two different aspects of the problem. First, geometrical factors account for the electrode set-up used in the experiment and for the complete geometry of the investigated object. Second, measured electrical resistances (V / I) are representative of local properties of the material, namely the true resistivity distribution. Hence, it must be emphasized that an apparent resistivity is not an intrinsic characteristic of the material. Apparent resistivities can even be negative when the electric fields in the homogeneous and heterogeneous structures are in opposite directions (i.e. V and V0 have opposite signs). Furthermore, V0 may occasionally tend towards zero (or G tends towards infinity) thus leading to undetermined apparent resistivity values, although the resistivity of the medium is well defined. In fact, an apparent resistivity plot or map is barely a convenient way, and in some applications a useful one, to represent the measured resistances (Kunetz, 1966). When the data set contains enough information, an inverse process is always desirable to get a correct image of a heterogeneous structure. Fox et al. (1980) proposed the use of a percent correction factor in case of terrain topography effects for 2-D resistivity profiles. In their approach, the authors calculate the apparent resistivity values for a homogeneous earth with the observed topography. The ratio of the true resistivity to the calculated apparent resistivity values for the homogeneous model is then multiplied by the measured apparent resistivity values, both apparent resistivities being calculated with the usual analytical geometric factor. They came to the conclusion that the adjusted data more accurately show the effects of the actual earth resistivity structures. Using this approach, some authors suggested that the corrected data could be approximatively interpreted as if they were the responses of a flat earth model (Holcombe and Jiracek, 1984; Tong and Yang, 1990; Tsourlos et al., 1999; Hennig et al., 2005). It is clear that the resistivity anomalies due to topography cannot be completely separated from those

due to subsurface structures by using a simple linear correction. Therefore, the approximation does not come from the way these authors take the topographic effect into account but rather from the fact they interpret the corrected data as if they were the responses of a flat earth model. Hence, the term correction may not be appropriate since it suggests that the apparent resistivity calculated with this approach still contains geometric errors. This approach can be shown to be indeed more than a correction and is in fact the most general way to express and to compute accurately the apparent resistivity, fully considering the non-trivial geometry or topography of the earth rather than trying to approximate it as flat. 2.3. Evaluation of the geometric factor The evaluation of apparent resistivities for arbitrarily shaped structures, as defined by Eq. (4), requires the virtual calculation of the responses of the homogeneous medium (namely V0). Obviously, one would hope that this evaluation is as accurate as possible in order to keep the relative errors on apparent resistivities not much larger than the ones on the measured V. These errors could originate from the evaluation method itself, or from poor information on the model geometry or even from errors on electrode locations. For all these reasons, the authors would not recommend experimental methods which try to reproduce the geometry, on a full scale or on a reduced one, then filling up the model with some resistivity-controlled medium (e.g. electrolyte of well known concentration). This approach may be difficult to use due to the complicated shape of the objects or may bring geometrical errors as well as experimental troubles on controlling the medium resistivity. We would therefore suggest that the General Geometric Factor (GGF) are numerically evaluated. In the present paper, the potential V0 are obtained with a finite element code. The finite element (FE) method (Zienkiewicz and Taylor, 2000) has been used by several authors in the context of direct current forward modelling to calculate the response of surveys on complex earth structures (Coggon, 1971; Fox et al., 1980; Pridmore et al., 1981; Holcombe and Jiracek, 1984; Sasaki, 1994; Tsokas et al., 1997; Marescot, 2004). This numerical tool is well adapted to our approach since the FE method makes it possible for the user to precisely represent any complex structure in order not to artificially perturb the electrical field with an inaccurate model geometry. Most commercial FE codes require to locate sources and potential electrodes on nodes of the mesh. This implies


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Fig. 2. The finite element model used for the synthetic example. A series of known cavities and a narrow gallery (all filled with air) are included into the FE model (background resistivity 100 m) and a fractured zone (500 m) is simulated in the vicinity of these two voids. The vertical axis is oriented upwards. The complete modelling domain (in m) is x [0, 140], y [0, 140] and z [0, 67.5], as shown in the upper left part of the figure. The fractured zone is approximately cylindrical with an axis of centre coordinates (74, 66) in the xy plane. It has an 8-m radius and a vertical extension ranging from z = 63.5 to 65.5. A series of depth slices through the central part of the mesh is shown. The fractured zone is in grey colour.

to design a mesh which fits the sources and electrodes locations and which is refined enough for an accurate description of the geometry and for the numerical convergence. In all cases, the mesh is closely related to the electrode layout and is rarely suitable when considering another set-up. To alleviate this problem, the most general approach is to use an electrodeindependent mesh for the FE model. This idea has already been investigated in the context of a finite difference modelling (Spitzer et al., 1999) and a FE

modelling (Pain et al., 2002; Marescot et al., 2003b) but is still not widespread used. The choice for the mesh size is then solely governed by the need for accurate results. This option makes the finite element mesh generation task easier, especially when the structures of interest have complicated shapes. For the following examples, the finite element code CESAR-LCPC was used (Humbert et al., 2005) and the mesh-independent electrode location was accounted for by the finite element formalism (Marescot et al., 2003b).

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3. Synthetic example: resistivity mapping in presence of known cavities Research works published in the literature clearly state that topography can have a significant effect on resistivity measurements (e.g. Kunetz, 1966; Cecchini and Roccroi, 1980; Fox et al., 1980; Oppliger, 1984; Tsourlos et al., 1999; Hennig et al., 2005). These papers show that the influence of topography on noninverted resistivity surveys (resistivity mappings, vertical electrical soundings or mise--la-masse surveys) can lead to serious misinterpretations of the data. When resistivity data can be inverted, the topography is directly incorporated into the inversion model (e.g. Tong and Yang, 1990; Sasaki, 1994). Useful reviews and illustrations on the possible schemes to implement the topography into an inversion model can be found in Loke (2000). Nevertheless, when no inversion is carried out (e.g. in resistivity mapping), topography effects must be taken into account to allow for meaningful interpretation based on apparent resistivities. Fig. 2 shows the model for a synthetic example of a surface resistivity mapping survey. This example is used for the comparison of the GGF approach (General Geometric Factor) to the half-space analytical geometric factor. Each method leads to apparent resistivities using the corresponding definition of the geometric factor and the same set of potential differences V (figuring field measurements) simulated with the finite element method on the heterogeneous medium. A series of known cavities and a narrow gallery (all filled with air) are included in a model of background resistivity 100 m. A fractured zone (500 m) is simulated in the vicinity of these galleries (Fig. 2). The vertical extension of the galleries ranges from 65 to 66.5 m. The fractured zone is roughly a vertical cylinder with a radius of about 8 m and a vertical extension ranging from 63.5 to 65.5 m. A set of depth slices through the central part of the model mesh is shown in Fig. 2. The mesh is refined in a central zone composed of 6-node pentahedral elements and is coarser in the outer padding zone composed of 8node hexahedral elements for a total of 42 372 nodes and 62 354 elements. The dimension of elements in the inner region is approximately 1 1 0.5 m3 near the surface. Zero-value potentials are imposed at the vertical and bottom bounding planes of the domain. Boundaries were set 50 m away from the central zone in each direction. The model used to simulate measured field data therefore includes both gallery and the fractured zone. A surface resistivity mapping is

simulated with a square array (side length is 4 m) like in a mobile array survey (Panissod et al., 1997) to give an overview on the lateral extension of the fractured zone. A data point is collected each meter in the x- and y-directions thus leading to a data set of 1369 apparent resistivities. The use of an electrode-independent mesh greatly facilitates the mesh generation process since an automatic meshing algorithm can be used without having to fit the locations of the electrodes (there are 40 40 = 16 000 electrode locations). Fig. 3 shows the apparent resistivity map obtained by multiplying the half-space analytical geometric factor Eq. (2) by the simulated field data V. Each data point is plotted at the centre of the corresponding square array. The apparent resistivity map mainly reflects the location of the underground galleries and the effect of the fracture zone can be hardly observed. The geometry of the known galleries must therefore be taken into account. The potentials V0 were calculated on the same mesh but for a homogeneous medium with a resistivity of 1 m. The apparent resistivity map calculated with the GGF approach (Eq. (4)) is shown Fig. 4. The presence of the fractured zone is enhanced and is not masked by the underground void structure. In this example, the GGF approach allowed to include explicit geometrical information on the complex topography in a resistivity mapping survey and to focus on an unknown geological target in the subsurface. 4. Real data example from a 3-D laboratory model In laboratory, when it is possible to give a sample of material a particular shape, such as a cylinder or any shape of constant section, a basic geometric factor (or cell constant) is given by: G S L 6

where a plate electrode of same surface S is placed on each end of the cylinder and L the length of the sample. A current is then applied, that flows through the sample, and the voltage drop between the current electrodes is measured. This measurement device is nevertheless difficult to use in practice because of the large frequency dependent contact resistance, of electrochemical origin, that appears between the sample and the electrodes. Better-adapted laboratory apparatus can be used in practice (Taylor and Barker, 2002) but the difficulty to reshape the sample into a suitable geometry still remains. Moreover, for in situ measurements on structures with complicated geometries (Lataste, 2002), no destructive reshaping can be


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Fig. 3. Apparent resistivity plot for the mapping survey using the half-space analytical geometric factor approach. The known galleries are outlined with a plain line and the fractured zone with a dashed line. The dimension of the mobile array is also shown. Each data point is conventionally plotted at the centre of the corresponding square array.

carried out and point electrodes are used. In this case, the GGF approach allows an accurate evaluation of the resistivity of the original sample without any reshaping.

This section presents an experimental resistivity survey performed on a water tank (Fig. 5). The tank is composed of a parallelepipedic structure with two vertical cylindrical cavities. This model figures material

Fig. 4. Apparent resistivity plots for the mapping survey using the general geometric factor (GGF) approach. The known galleries are outlined with a plain line and the fractured zone with a dashed line. The dimension of the mobile array is also shown. Each data point is conventionally plotted at the centre of the corresponding square array.

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Fig. 5. xy view of the finite element mesh used for the tank simulations (top). The locations of the 24 electrodes are represented (black dots) and the plastic heterogeneity is in grey colour. 3-D view of the FE mesh (bottom). The modelling domain (in cm) is x [0, 56], y [0, 36] and z [0, 7] and the node on which a null potential condition is imposed is indicated.

with complicated structure (e.g. rock or concrete sample) although its geometry is well known. This example has a general significance for the reader since it could represent a variety of situations frequently met at a larger scale in the field and not only in a laboratory experiment. For instance, this geometry could stand for a complex topography area with cliffs and two well-known vertical pits, or two horizontal galleries. In the experimental set-up, potential field values are measured with a Wenner array and 24 electrodes along a profile which runs next to the vertical pits and the edges of the tank (black dots on Fig. 5). A 2-cm constant electrode spacing is used. The complexity of the geometry, namely the tank edges and the pits, is expected to have a strong influence on the spatial distribution of the potential field. Let us remind that the GGF approach first requires the numerical solution of the forward problem considering an homogenous medium with 0 = 1 m. A finite element mesh is therefore created with the same geometry as the material tank, including both

pits. The mesh is composed of 60 498 6-node hexahedral elements and is represented in Fig. 5. The dimension of elements is about 1 1 0.5 cm3 in the upper layer and about 1 1 1 cm3 in the lower part. A zero potential value is imposed on a single node at the bottom of the model and the current density flux perpendicular to all air-medium boundaries is set to zero. As for the preceding synthetic example, the use of an electrode-independent mesh greatly facilitates the mesh generation process. In the first part of this example (model 1), the tank is filled up with water except for the two pits which remain void. The global resistivity of the tank is investigated, simulating measurements on a laboratory sample using point electrodes. Potential field values are measured according to the experimental set-up described above. Two sets of apparent resistivities are then computed to compare the GGF approach to the half-space analytical approach. The corresponding apparent resistivity pseudosections are represented respectively on Figs. 7 and 6. The background


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Fig. 6. Apparent resistivity contoured pseudosections using the half-space analytical geometric factor approach for model 1. Model 1 is a tank model of parallelepipedic shape in which two cylindrical pits are retained (Fig. 5). The background filling medium is water (resistivity 36.6 m at 20 C) and the cylindrical shapes remain void (air-filled). The conventional data point positions are shown (black crosses).

resistivity of the water, measured with a conductivimeter, is 36.6 m at 20 C. The values obtained with the half-space analytical geometric factor approach are strongly overestimated compared to the true background resistivity (Fig. 6): apparent resistivities range from 40 to over 300 m. Moreover, this pseudosection shows increasing apparent resistivities with array length that is neither homogeneous nor realistic, especially for the largest electrode spacings. Obviously, the tank bottom has a large influence and dominates the information content. Due to the low lateral sensitivity of the Wenner array, the two vertical pits do not seem to have a large influence on the measurements (slight resistivity increase around x = 32 cm on Fig. 6). This would have been different with another electrode configuration or if the pits were closer to the centre of the profile. Oppositely, the apparent resistivity values obtained with the GGF approach agree well with the measured resistivity of

the water and a homogeneous image of the object can be seen (Fig. 7). We can observe that the GGF approach guarantees the integrity of the object and greatly facilitates the determination of the electrical properties of the structure. It also alleviates technical problems related to the destructing and reshaping of the object (use of plate electrodes for example). Moreover, working with the entire structure enables one to evaluate the degree of homogeneity of the material by varying the location and the size of the electrode array. This is not possible on the basis of a small sample extracted from the structure. It should be stressed that the numerical calculation of the geometric factor must be carried out only once, as far as the shape of the sample and the electrode configuration remain the same. This operation does not require extensive computations. In the second part of the example (model 2), a small parallelepipedic plastic structure is added in the tank as

Fig. 7. Apparent resistivity contoured pseudosections using the GGF approach for the model 1. Model 1 is a tank model of parallelepipedic shape in which two cylindrical pits are retained (Fig. 5). The background filling medium is water (resistivity 36.6 m at 20 C) and the cylindrical shapes remain void (air-filled). The conventional data point positions are shown (black crosses).

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Fig. 8. Apparent resistivity contoured pseudosections using the half-space analytical geometric factor approach for model 2. Model 2 is the same as model 1 (water with resistivity 36.6 m at 20 C) with an additional parallelepipedic plastic structure laid in the tank (Fig. 5). The conventional data point positions are shown (black crosses) and the location of the plastic structure is outlined.

an unknown gallery neighbouring the well-known ones and the same resistivity profiling is carried out simulating an attempt to localize this underground structure. Measurements performed on the top surface of the tank are used in the GGF and the half-space analytical geometric factor approaches for the computation of apparent resistivities. Since the plastic structure is considered as an heterogeneity in a well-known background, it was not included in the FE model used to evaluate the potential values V0 which therefore are identical to the ones used for model 1. The pseudosection for model 2 using the half-space analytical approach (Fig. 8) shows only minor deviations from the model 1 containing only water and no plastic heterogeneity (Fig. 6). The prevailing effect from the tank edges masks the occurrence of the resistive target (plastic structures). Oppositely, the location of the unknown gallery is

clearly seen on the pseudosection using the GGF approach (Fig. 9). Finally, we tested the effect of geometric discrepancies between the real sample or the medium and the FE model used to evaluate the potential values V0. This effect mainly depends on the relative amplitude of this error with respect to the array size and on the sensitivity of the array used. An erroneous water thickness (6 cm instead of 7 cm, i.e. about 15% error) was introduced in the FE model used for the computation of the GGF. As the potential values V0 globally increase, the resulting apparent resistivity values show a decrease (Figs. 10 and 11), especially for large array lengths, thus reaching a minimum of about 30 m at the bottom of the pseudosections for both model. Nevertheless, the resistive target still yields a visible anomaly, although of lesser amplitude. This example

Fig. 9. Apparent resistivity contoured pseudosections using the GGF approach for the model 2. Model 2 is the same as model 1 (water with resistivity 36.6 m at 20 C) with an additional parallelepipedic plastic structure laid in the tank (Fig. 5). The conventional data point positions are shown (black crosses) and the location of the plastic structure is outlined.


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Fig. 10. Effect of a geometric discrepancy between the real sample and the FE model used to evaluate the potential values V0. Apparent resistivity contoured pseudosections using the GGF approach for model 1 (water-filled tank with two air-filled pits included).

emphasizes the importance of using a FE model that precisely reproduces the real geometry of the investigated structure and that fully takes into account the topographic effects when carrying out apparent resistivity mapping surveys. 5. Conclusions A generalized approach for the calculation of geometric factors has been presented. The GGF approach is comprehensive: it facilitates the direct use of apparent resistivities for qualitative interpretation and it broadens the area of applications of electrical prospecting and testing to any geometry. This method enables the evaluation of apparent resisitivities when no analytical expression for the geometric factor can be calculated. In this paper, a finite element method using electrodes that are not necessarily located on mesh nodes was used: the designing of the mesh is solely governed by the need

for accurate simulations and an automatic meshing algorithm could be used. Since the full geometry of the studied objects is accounted for, the approach described in this work is especially adapted for non-destructive testing (e.g. walls, trees, dams, concrete piles, concrete samples) as well as for petrophysical, industrial or biological data analysis, particularly when no inversion can be carried out. Note that the GGF approach offers some valuable improvements to techniques based on the direct use of apparent resistivity, but is not intended as a substitute for an inversion process in any case. However, the GGF approach allows a quick insight into the resistivity mapping of a structure or a soil and can give useful a priori information that may be used in an inversion scheme. As it was shown, this approach can for example be used to account for strong topographic effects in resistivity mapping surveys (e.g. measurements along cliffs, dams or in galleries). It can also be undertaken for measurements on soil samples of which shape depends

Fig. 11. Effect of a geometric discrepancy between the real sample and the FE model used to evaluate the potential values V0. Apparent resistivity contoured pseudosections using the GGF approach for model 2 (same as model 1 with an additional plastic structure laid nearby the pits).

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