Numerical modelling of direct current electrical resistivity for the characterisation of cracks in soils
Introduction
As fractures or cracks play an important role in the mass transfer of rocks and soils, their detection and hydraulic conductivity characterisation are major objectives in several branches of Earth science. However, although they create high physical contrasts, the direct detection of fractures by geophysical methods is difficult, due to their comparatively small volume. In geothermal studies, for example, several electromagnetic tools have been proposed for the detection of fractures in single boreholes (Tabbagh and Giannakopoulou, 1995) or cross boreholes (Giannakopoulou et al., 1997). However, only the borehole wall imaging tools, the Formation Micro Scanner (FMS) and the Borehole Televiewer (BHTV), have been used operationally (Genter et al., 1991). Ground Penetrating Radars can be used in quarries where hard rocks can be measured directly (Grandjean and Gourry, 1996, Derobert and Abraham, 2000). In surface geophysics, the direct detection of fractures is exceptional and researchers tend to look for fracture corridors or fracture zones where clusters of fractures exist, and can be detected as a whole but not individually (Benderitter, 1997).
In soil studies, the importance of characterising the soil structure on great details has attracted researcher's attention to various non-invasive techniques. The measurement of DC electrical resistivity has been shown to be a promising tool for the characterisation of the soil structure of cultivated horizons (Besson et al., 2004). In the context of reduced tillage or non-tillage, soil macroporosity is regenerated in compacted zones by several processes. Among them, swelling and shrinkage (resulting from wetting/drying cycles) induce soil cracks, whose development and geometry need to be characterised. The first reported laboratory experiment on a compacted block of soil, containing one man-made crack, demonstrated the applicability of DC electrical characterisation using Electrical Resistivity Tomography (ERT), by means of microelectrode panels on the block's surface (Samouëlian et al. 2003). A more complex 3D experiment was carried out on a compacted block of soil where cracks developed through drying (Samouëlian et al., 2004). Apparent resistivity pseudo-sections calculated for this block clearly revealed the presence of cracks. However, relevant inversion of the data, to provide their position, thickness and geometry, remained impossible because of the absence of suitable numerical codes adapted to the description of thin planar features with variable orientations. To fill this gap, the present study proposes to apply the Method of Moments, MoM, to model such features. Following a brief description of the principles of the method, a study of the sensitivity of DC data to the different characteristics of a fracture is described. This is followed by an example of the inversion of experimental laboratory data.
Section snippets
Principle of the method of moments applied to the static DC case
This method, which belongs to the category of volume integral equation techniques was proposed by Harrington (1968). It was first used in geophysical prospecting to model 3D electromagnetic datasets (Raiche, 1974, Hohmann, 1975, Tabbagh, 1985) and was later extended to electrical prospecting (Dabas et al., 1994) which corresponds to the assumption where time derivatives are supposed to be null. It establishes the equivalence between the presence of heterogeneities and the presence of
Experimental and numerical data
All experimental measurements were carried out with soil blocks collected by the INRA Experimental Unit of Mons-En-Chaussée (Somme, France). The soil is an orthic Luvisol, with a silty texture (clay: 20%, silt: 75%, sand: 5%; organic matter: 1.7%) developed from loess (Richard et al., 2001). Soil blocks with a density of 1.60g cm− 3 were sampled in the surface horizon, following soil compaction by traffic under wet conditions. The resistivity of the compacted soil blocks was measured to be 41 Ω
Comparison with experiments
Results from the first laboratory experiment and the corresponding numerical simulation are compared in Fig. 3. The pseudo sections obtained by direct modelling are seen to be in good agreement with the experimental results, for both the apparent resistivity distribution and the variation of amplitude as a function of increasing crack depth. Results from the second laboratory experiment and their corresponding numerical simulations are compared in Fig. 4. The effect of the orientation, width
Inversion procedure
The aim of an inversion algorithm is to determine unknown quantities, which in the present case correspond to 8 geometrical parameters, which generate a best-fit theoretical response to known experimental values. As the influence of these parameters is non-linear, a linearised least squares iterative approach is used, which starts from a first guess at the value of each of the parameters, and then calculates the respective increments to be added at each iteration. From the various applicable
Conclusion
For both for assessing and simulating the electrical response and the main physical parameters of cracks in soils, the model presented here constitutes a light, it can be implemented on a simple PC computer, and efficient, the time required for a complete inversion is less than 3 min, tool. This results from the limited number of unknown parameters used to characterise the cracks.
This tool will allow both laboratory and field studies to be developed with the aim of monitoring the self
Acknowledgements
This work was achieved within a cooperative research program established between INRA ‘Science du sol’ and UMR 7619 Sisyphe (UPMC-CNRS-ENSMP-EPHE). This program continues and is now supported by DST program (GESSOL2 and ADD).
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