Acta Geodyn. Geomater., Vol. 16, No. 4 (196), 465–471, 2019

DOI: 10.13168/AGG.2019.0039

journal homepage: https://www.irsm.cas.cz/acta

ORIGINAL PAPER

RESISTOR NETWORK AS MODELING TOOL FOR FRACTURE DETECTION IN CRYSTALLINE ROCKS

Jaroslav JIRKŮ 1, 2) *and Jan VILHELM 1)

1) Charles University, Faculty of Science, Institute of Hydrogeology, Engineering Geology and Applied Geophysics, Albertov 6, Prague, Czech Republic

2) G IMPULS Praha spol. s r.o., Pristavni 24, Prague, Czech Republic

*Corresponding author‘s e-mail: jirku4@natur.cuni.cz

ABSTRACT

The rock massif with sparse fracture network can be considered as highly inhomogeneous. Thefractures enable electric current to flow in otherwise relatively non-conductive country rock. Themeasured apparent resistivity in such a medium strongly depends on the mutual position of theindividual electrodes in relation to the conductive fractures. Commonly used 2D resistivitymodels are built up by rectangular blocks with finite dimensions and characterized by theirresistivity values. The fractures are however very thin compared to the model block size.Electrical resistivity of fractures can be in this case modelled only by reduction of resistivityvalue in the individual blocks containing fractures. We have developed a resistor-network modelwhere the medium is replaced by discrete 2D orthogonal resistors lattice. Instead of resistivityblocks this model is built up by individual resistors which are characterized by their resistancevalue. The proposed resistor network model can be used for evaluating the electric resistivitymeasurements in underground storage monitoring projects for the interpretation of the ERTmeasurements carried out directly on the surface of crystalline rocks with individual fractures.The model can be used for modelling the artificial conductors as well. The effect of fractures andfracture zones is presented together with an example from real measurement in an undergroundtunnel.

ARTICLE INFO

Article history:

Received 13 September 2019 Accepted 5 November 2019 Available online 13 November 2019

Keywords: ERT Resistor network Forward problem Res2Dmod Geophysics Resistivity

uncertainty of the fracture position and the resistivitydistribution is significantly smoothed and blurredthen.

After first initial attempts (Vilhelm and Jirků,2018) we developed a resistor-network model to solvethis issue. In this case the medium is replaced bya discrete 2D orthogonal resistors lattice. Instead ofresistivity value [Ωm] which represents continuousmaterial property this model is built up by individualresistors which are characterized by their resistancevalues [Ω].

Resistor network model is used for evaluatingapparent resistivity pseudosections in the ERTmethod. We demonstrate the possibilities of thismodel for Wenner-alpha and Wenner-Schlumbergerelectrode arrays. Some basic model calculationsdemonstrate agreement between the finite-elementcalculation (Res2DMod by Loke) and proposedresistor network model. Performed model calculationsdemonstrate also the new possibility for modelling ofanisotropic resistivity medium. The resistor networkmodel is also suitable for modelling thin artificialconductors’ effect such as metallic wells, anchors,pipes or rebars (Rucker, 2012; Ronczka et al., 2015;Dahlin and Loke, 2018).

INTRODUCTION

The rock massif with sparse fracture network canbe considered as highly inhomogeneous medium fromthe point of view of its electrical resistivity properties.The fractures enable electric current to flow inotherwise relatively non-conductive country rock.Fractures can be very thin and highly conductive.These fractures can be of an irregular shape and theycan be mutually interconnected. The measuredapparent resistivity in such a medium stronglydepends on the mutual position of the individualelectrodes in relation to the conductive fractures. Thisis especially pronounced in the case of resistivitymeasurement performed directly on the rock surfacewith fractures.

Commonly used 2D resistivity models are madeof rectangular blocks with finite dimensions. Theindividual blocks are characterized by their resistivityvalues. The size of the blocks is usually increasingwith increasing depth which corresponds todecreasing value of the so-called sensitivity function(Loke, 2019; Butler, 2017). The fractures are howeververy thin compared to the model block size. Electricalresistivity of fractures can be in this case modelledonly by reduction of the resistivity value in theindividual blocks containing fractures. The

Cite this article as: Jirků J, Vilhelm J: Resistor network as modeling tool for fracture detection in crystalline rocks. Acta Geodyn. Geomater.,16, No. 4 (196), 465–471, 2019. DOI: 10.13168/AGG.2019.0039

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Fig. 1 The array of the resistance laboratory tests on the rock samples.

sidewalls – Figure 1. The size of the electrodes wasthe same as size of the sample sidewall. To ensureelectrical connection of the electrodes with thespecimen a highly conductive contact medium basedon water suspension of CuSO4.5H2O mixed with flourwas used. The rock sample was connected in serieswith the 10 kΩ resistor and DC stabilized electricalsource of 60V. Using precise Hewlett–Packard 3466Amicro-voltmeter a voltage drop on the sample wasmeasured. Together with the known value of electriccurrent in the circuit it is possible to calculateresistance of the sample by means of the Ohm's law.The sample resistance can be transformed intomaterial’s electrical resistivity using the known cross-section area of a rock specimen.

The electric resistivity measurements on the drysamples and on the samples with different levels ofwater saturation were then performed. The volumetricweight and open porosity were estimated according tothe modified ČSN EN 1936 (2007) standard. Openporosity is defined by mutually connected systems ofpores that are water-permeable and allow the flow, so-called effective porosity, and tiny non-connectedpores and microcracks on a sample's surface, whichcan be saturated with water (Norton and Knapp,1977).

The electrical resistance was always measured inthree perpendicular directions on each cubical sample.First, the samples were completely saturated bymineralized water (conductivity of approximately150 µS/cm). Samples saturation took 24 hours and itwas carried out at ambient temperature and under lowpressure conditions using vacuum pump. Then thesamples were slightly dried out to two decreasedsaturation levels and their electrical resistance was re-measured. Finally, the samples were completely dried-out during the temperature of 105 °C for 24 hours.

The proposed resistor network model can be forinstance used for evaluation of the electric resistivitymeasurements in the underground storage monitoringprojects for the interpretation of the ERTmeasurements carried out directly on the crystallinerocks surface with individual fractures. Sucha conclusion was confirmed by an extensive study ofrelation between joints’ frequency and hydraulicpermeability in the Czech massif granites down todepths of 300 meters (Rukavičková, 2008). Theauthors found out that granites show the indirectrelation between number of joints and permeability,i.e. the connection seems to be more or less random.The direct relation (being crucial for consideringgranites a geo-barrier) occurs in case of sparse butinterconnected fractures. The resistor network modelseems to be a tool for founding out the presence of aninterconnected fracture in a form of thin localized areathat can change electrical conductivity according tothe saturation swiftly and strongly.

FRACTURES IN CRYSTALLINE ROCKS

Disruption of unweathered and compactcrystalline rocks by fractures is very often of a localcharacter, even in case of a disrupted zone presence.In a rock massif one can observe separate fractures.The electrical resistivity of such otherwise non-conductive rocks then strongly depends on thefractures’ presence in a close vicinity of a place wherethe resistivity is measured and also on the fractureswater saturation together with the joint water’smineralization. This can be documented on anexperiment we carried out – we measured thedisrupted (by fractures) rock samples’ resistance inthe lab conditions. The samples had a cubical shape of8 cm-long sides. Two stainless metallic electrodeswere attached on the pairs of opposite specimen

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10 [kOhm]60 [V]

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Fig. 2 Resistivity of a series of samples with different connected porosity at different saturation levels. Cubicspecimens, resistivity values measured in three different directions for each specimen.

Model using the blocks of preset value ofresistivity is suitable mainly in cases of relativelyhomogenous environment or when it does not containthin bodies of significantly different resistivity. Whileusing blocks sizes in range of meters during modellingenvironment with fractures or thin fracture zones ofdecimeters a significant blurring of the resistivitymodel occurs, in terms of geometry and the resistivityvalues as well. Contrary to this the model based on theresistor network allows natural implementation ofa thin conductive structure that is characterized mainlyby its high conductivity and length (its width does nothave instant meaning). The calculation on the resistornetwork model is displayed in Figure 4.

The calculation of the resistor network model isbased on the Kirchhoff laws. In particular it is basedon calculation of all loop currents, marked in Figure 4.The calculation leads to the linear equations system.According to calculated loop currents one can get thevoltage between any two points on the surface, i.e. tomodel a resistance measurement via arbitraryelectrodes array. It is also possible to use the loopcurrents themselves to characterize the currentdistribution in an environment which shows forinstance the influence of a single model’s parts to thefinal apparent resistivity.

Contrary to the resistivity distribution model theresistor model does not represent a real physicalenvironment’s model but, while used suitably, it canbe used for the model calculation of apparentresistivity distribution. The essence is mainly in factthat the real physical property is the materialresistivity while the value of resistance representingan electronic component does not say anything aboutits physical dimensions. It is known that during

Heating up the samples to desired temperature wasgradual with exponentially increasing temperaturefrom 20°C for approximately 4 hours (about onedegree per three minutes). The sense of such gradualheating is preventing from new microcracksgeneration due to high thermal gradient during theprocess (according to Zeng et al. (2019) and Reuschleet al. (2006)).

Measured resistance values of all samples in alldirections are displayed in Figure 2. Separate samplesare characterized by their connected porosity. Foreach sample all three values of resistance aredisplayed (i.e. three orthogonal directions). Colorsdistinguish different saturation levels – 100 %saturation, short and long drying and 0 % saturationeventually. The saturation level in case of partlysaturated samples is expressed by the symbol size andvaries for separate samples. The meaning of Figure 2graph is proving the fact that crystalline rocksdisrupted by fractures can change the resistance inseveral orders due to connected porosity and differentsaturation. The places with disruptions representa strong inhomogeneity of a rock environment butsuch an inhomogeneity can be very thin and local atthe same time. Its influence is strongly based on thepresence of a physical contact with measuringelectrode. Due to position between the electrode andfracture the value of apparent resistivity may changerapidly.

RESISTOR NETWORK MODEL

The resistor network model is an alternative tocommonly used 2D model that is based on divisionthe medium into the blocks of a constant apparentresistivity (Fig .3).

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Fig. 3 Comparison of 2D model with distribution of blocks with given resistivity (in Ωm) and resistor networkmodel formed by resistances of different resistors (in Ω). Both models are completed with scheme ofelectrode arrays.

Fig. 4 Resistor network model with horizontal and vertical resistors. The red arrows denote loop currents.

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Fig. 5 Thin conductive fracture: a) resistor network model, b) resistivity pseudo-section.

via 5000 Ω resistors of 1-meter length, the conductivefracture is made by the 500 Ω resistors. Consideringthe chosen 1-meter size of the resistors such a modelcan be compared to the model that would consist of5000 Ωm and 500 Ωm resistivity blocks. Chosenmeshes model and resulting apparent resistivity cross-section is in Figure 5.

BUKOV UNDERGROUND LABORATORY – AN EXAMPLE OF MODEL CALCULATION

As an example of the resistor network model’sfunctionality we state a simulated apparent resistivitycross-section calculation corresponding to themeasurement on the mine gallery’s wall in the Bukovunderground laboratory that works as an experimentalanalogue and test site for the future nuclear wastegeological repository. The electrical resistivitytomography (ERT) measurement was carried outdirectly on the wall. The electrodes spacing was50 cm with 38 electrodes in total. While proposing themodel we also took in account the steel floorreinforcing rebar working as a deeper conductivemedium during the measurement. Figure 6 showsrelatively decent match between measured apparentresistivity cross-section and the calculated one madeby using the resistor network model.

making the electric circuits one can use an equivalentcircuit to replace selected resistor by several otherresistors while its final resistance will equal to theoriginal one. Such a property of the electric circuitsproves a general principal ambiguity of the inverseproblem’s solution. However, we can set basicgeometrical assumptions about the resistor network orconsider the resistor network analogical to the block-like environment’s model with set resistivity values.As an example we can state an assumption ofa resistor network that has geometrical dimensionsof the side of 1 m. In that case the network willbehave analogically as the resistivity distributionmodel of 1x1 m square blocks. A thin conductivefracture may be modelled via resistors with lowresistance value. Then the necessity of simultaneoussolution of the fracture dimension and its resistivityvanishes – it is only necessary to set an adequatelylow resistance value matching the measurement withan electrode connected directly into the fracture.

As an example of the apparent resistivitypseudo-cross section calculated via resistor networkone can show an inclined conductive fracture model.For the sake of simplicity the size of the resistornetwork’s mesh was chosen as 1 m (side’s length).Surrounding non-conductive environment is modeled

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Fig. 6 Resistivity data and resistivity model from the Bukov underground laboratory. a) Measured apparent resistivity section. b) Resistor network model. c) Apparent resistivity section calculated for the resistor network model.

methods. Using the resistor networks might increasethe level of a detail we can get from suchmeasurements like geometry of the fracture or itsphysical picture. Such an approach may be usefuleven in case of a geophysical long-term monitoringwhen narrowing down the rate of change is veryimportant too.

An inverse problem is always ambiguous andautomatically generated inversion is not alwayssufficient. This is caused by the fact that the inverseproblem usually includes searching for a model ofresistivity distribution which would generate anidentical apparent resistivity distribution that would bemeasured. To narrow down principal ambiguity ofresistivity methods, we can work with simplifiedmodels such as vertical electrical sounding (VES)where we are trying to approximate the geologicalmedium with a 1D layered model. Even in this case,this includes the ambiguity caused by the equivalenceprinciple. Solving more complex 2D or 3D forwardgeological problems is usually carried out viacalculation of apparent resistivity values by themethods of finite-differences or finite-elements like incase of the Res2Dmod program. We proved that usingthe resistor network approach may be more reliablewhen expecting strong changes in measuredenvironment. In such a case the finite-differencesapproach may be less-sufficient in getting rid of theambiguity constraints.

In Figure 7 we show the result of the Res2Dinvinversion made of the same data. The iteration processhad to be carried out twice to cut down the RMS errorof the inversion (via elimination of the poor data dueto high contact resistance) to current 13.7 percent atthe 4th iteration. One can see that the inverted pictureof the environment is generally similar but representsseveral differences – absolute values of theresistivities are generally higher for the surface zonesas such values tend to increase with every iterationstep. The inverted picture also distinguishes thesubsurface conductive zone into two parts –conductive narrow blue zone (first hundreds Ωm) andsemi-conductive neighboring zone of around1000 Ωm that was not distinguished in Figure 6.

DISCUSSION OF RESULTS

The presence of a fracture in the geologicalenvironment can be a crucial issue. Among major taskthen belong – set the position and slope of thefracture, quantify the volume of it and guess on thephysical parameters of the fracture, mainly resistivityin our case and eventually causes of such parameters’changes. The influence of the water saturation may beextremely strong as proven in our laboratory tests –difference between partly and completely saturatedfracture can vary in order-ranges. It was proven thatsolving these types of issues of strongly anisotropicenvironment is generally challenging for direct current

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Fig. 7 Res2Dinv inverted cross-section from the Bukov underground laboratory.

REFERENCES

Butler, S.L.: 2017, Analysis of the moments of thesensitivity function for resistivity over a homogeneoushalf-space: Rules of thumb for pseudoposition, offlinesensitivity and resolution. J. Appl. Geophys., 143,149–155. DOI: 10.1016/j.jappgeo.2017.06.005

ČSN EN 1936: 2007, Natural stone test methods -Determination of real density and apparent density,and of total and open porosity. Standard is the Czechversion of the European Standard EN 1936:2006.

Dahlin, T. and Loke, M.H.: 2018, Underwater ERTsurveying in water with resistivity layering withexample of application to site investigation for a rocktunnel in central Stockholm. Near Surf. Geophys., 16,3, 230–237. DOI: 10.3997/1873-0604.2018007

Loke, M.H.: 2019, Tutorial: RES2DINVx64 ver. 4.09. Norton, D. and Knapp, R.: 1977, Transport phenomena in

hydrothermal systems; the nature of porosity. Am. J.Sci., 277, 913–936. DOI: 10.2475/ajs.277.8.913

Tullborg, E.L. and Larson, S.Å.: 2006, Porosity incrystalline rocks – A matter of scale. Eng. Geol., 84,1, 75–83. DOI: 10.1016/j.enggeo.2005.12.001

Reuschlé, T., Gbaguidi Haore, S. and Darot, M.: 2006, Theeffect of heating on the microstructural evolution ofLa Peyratte granite deduced from acoustic velocitymeasurements. Earth Planet. Sci. Lett., 243, 692–700.DOI: 10.1016/j.epsl.2006.01.038

Ronczka, M., Voss, T. and Günther, T.: 2015, Cost-efficientimaging and monitoring of saltwater in a shallowaquifer by using long electrode ERT. J. Appl.Geophys., 122, 202–209. DOI: 10.1016/j.jappgeo.2015.08.014

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Rukavičková, L.: 2008, Hydraulic conductivity and fracturefrequency dependence in granite massifs. GeoscienceResearch Reports for 2007, 41, Czech GeologicalSurvey, Prague, 232–235, (in Czech).

Vilhelm, J. and Jirků, J.: 2018, Forward modeling by meansof resistor network – new tool for fracture detection incrystalline rocks. Geophysical Research Abstracts, 20,EGU2018-10381, EGU General Assembly.

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The calculation of an apparent resistivity cross-section was solved only as a forward problem.However, every forward problem may have its inverseproblem. If we were able to prepare an adequateinverse problem to the resistor network’s model itmight work better than in case of finite differences.This might be caused by more contrast and lesssmoothened cross-section of resistor networkcomparing to one of the Res2Dmod. A thinconductive fracture’s manifestation is localized farmore precisely. This will make the inverse problembetter conditioned and more stable.

CONCLUSIONS

In this paper we describe a tool that was madefor modeling the electrical resistivity forwardproblem. It was proven that the resistor network canbe associated with geometrical dimensions andperforms calculations of 2-D subsurface resistivitymodels. The resistor network uses generated cross-sections similarly to Res2Dmod´s approach but seemsbeing more stable and detailed in some of theenvironments. Using the resistor network, we createda program that is equivalent to the finite-differencesapproach but allows working even with special caseslike high-contrast fracture modeling. This can beuseful in solving difficult applied problems such as inengineering geology or in geotechnics – our approachworks more stable and reliable in case of very thinconductors, high resistivity contrasts or in stronglyanisotropic environment. The resistor network may bepromising when using data measured directly on therock tunnel surface (repositories, galleries etc.). ACKNOWLEDGMENT

The research was supported by the projectCzechGeo/EPOS-Sci (CZ.02.1.01/0.0/0.0/16 013/0001800).The field data was gathered within the Radioactivewaste repository authority (RAWRA) project "Long-term monitoring of the rock massif in the URF Bukovvia non-destructive geophysical methods.“