Abstract

Fracture geometric characteristics (FGC) such as fracture intensity, fracture network connectivity and aperture distribution are crucial features controlling the hydraulic and geotechnical properties of rock formations. Geophysical methods have been used to detect contrast in subsurface material. The Electrical Resistivity (ER) method is one of such methods. The method was applied in an experimental test to investigate ER response with varying fracture intensity I_f and aperture size Aper width in an experimental set-up at Kwame Nkrumah University of Science and Technology (KNUST, Kumasi, Ghana, geotechnical laboratory. The concepts of electrical resistivity variation with rock mass fracture intensity, and aperture were used to obtained experimental data. ER profiles, each measuring 0.9 m long were recorded using 4 electrodes deployed using the Dipole-dipole, Wenner and Schlumberger configurations from the experimental setup. To quantify the relationship between apparent resistivity and fracture intensity, scatter-plots were drawn with apparent resistivity as the abscissa and fracture intensity as well as aperture width as ordinate. There were strong positive linear and regressive correlations between ρa and fracture intensity. Mathematical relationships are established that relate the ER and fracture intensity, and ER with aperture width. The highest coefficient of determination R² of 0.924 was represented for best fit equation, for relationship between apparent resistivity and fracture intensity. For aperture width, the best fit model was given a logarithmic relation as ρ_a =-465.46(Aperture width)² +4880.5(Aperture width)+1295.5. The study demonstrates the potential usefulness of the ER approach in rock fracture characterisation investigations, which is economic, efficient and less time consuming compared to other methods, of subsurface fracture characterization such core drilling.

1. Introduction

The characterisation of shallow subsurface material and the spatial variability at large scale is a crucial issue in many research studies and fields of application, ranging from agriculture, mineral and petroleum exploration, geology, to civil and environmental engineering. Investigations on the movement of fluid and contaminants through rock mass and the factors that control such movements are of great concerns in the hydrogeological field.

Fracture properties such as intensity, fracture network connectivity and aperture size distribution are crucial features controlling the hydraulic and geotechnical properties of rock formation. In this experimental study the focus is on the response of electrical resistivity to fracture density and variation in aperture width.

Information on subsurface fracture properties are often obtained traditionally through site characterisation which rely on sparse network of intrusive sampling points which consist of test pitting, coring and monitoring wells ^{1, 20}. Electrical resistivity methods have a wide range of applications. It has been used extensively in the search for suitable groundwater sources ², for soil characterisation ³, landslide modelling ⁴, monitoring of soil moisture content ⁵ and detecting salt–water intrusion ⁶ among others.

Although these techniques have been widely implemented at various sites, there are a number of limitations in the use of these techniques. These direct observation approaches of charactering the subsurface are cumbersome, expensive and sometimes impossible. They are time-consuming and suffer from low sampling density, relying on data from a limited number of locations, sometimes, widely spaced ⁷. However if properly designed and implemented, geophysical methods are capable of imaging detailed subsurface structures ⁸.

Studies have shown that geophysical approaches can be used to delineate subsurface geological structures ^{9, 22}.

A generalised respond to variations in physical properties of the earth’s subsurface including its rocks, fluids and voids have been published ^{2, 8, 10, 11, 12}, However, the influence of fracture density or intensity, and variation of fracture aperture and subsequently, variation in volume and concentration of contaminants in fractures is not well understood. The focus of this study is to help overcome the shortcomings of the conventional approaches in obtaining subsurface fracture properties by advancing electrical resistivity measurement technique.

In this paper, two sets of experimental studies using 6 sets of rock block samples representing different fracture intensity and aperture widths were carried out. Attempts are made separately to discuss the influences of fracture intensity and aperture size on the electrical resistivity of the different rock samples, soil types and attempt to model the relationship between ER and fracture intensity, and also between ER and aperture size.

2. Theoretical Background

The fundamental physical law used in resistivity surveys that governs the flow of electrical current through earth material (soil and/or rock) is Ohm’s law The equation for Ohm’s law in vector form for current flow in a continuous medium is given by ^{13, 14}:

(1)

where, σ is the electrical conductivity of the medium, J_c is the current density and E is the electric field intensity.

The resistivity technique is based on the principle that, when a current, I is passing through an electrically uniform cube of side length L (Figure 1), the material within the cube resists the conduction of electricity through it, resulting in a potential drop, V between opposite faces. The resistance R (Ω) is proportional to the length L (m) of the resistive material and inversely proportional to the cross-sectional area, A (m²):

(2)

where V is voltage, I is current, and R is resistance.

The value of the R varies according to the dimensions of the material such as, length (L) and cross-sectional area (A), measured in meters. The electrical resistivity value, ρ can be acquired by the relationship:

(3)

In homogeneous media, the value represents the true resistivity of the medium ¹⁴. However, the geological environment is composed of heterogeneities and therefore the resistivity measures in this case represent a weighted average of true resistivities. This is referred to as the apparent resistivity ρ_a, expressed as:

(14)

where ΔV is the measured potential difference and I is the current passed through the ground.

The arrangement of the current and potential electrodes defines the geometric factor (K), measured in meters, in the determining of resistivity. Thus, the primary objective of generating and measuring an electric potential field is, to determine the spatial distribution of ρ_a (Ω.m) in the subsurface ¹⁵.

The process of measuring subsurface resistivity involves, placing four electrodes in the ground in a line at equal spacing, applying a measured direct current (DC) or low frequency alternating current (AC) to the outer two electrodes, and measuring the AC voltage between the inner two electrodes. A measured resistance is calculated by dividing the measured voltage by the measured current. This resistance is then multiplied by the geometric factor that depends on the spacing (a) between each electrode (Figure 2) to determine the apparent resistivity (ρa). the geometric factor for the Wenner array is, K = 2πa, and K = (n -1)(n +1)πa/2 for the Schlumberger. For the dipole–dipole configuration, the current electrodes AB and the potential electrodes MN have the same spacing a but the two pairs are widely separated by a distance na, where n >> 1. The geometric factor for the dipole–dipole array is, Κ = πn(n + 1)(n + 2)a.

3. Methodology

The experimental procedures and methodologies employed in this study to investigate ER response to fracture intensity and aperture width are presented in detail in the following sections.

The experimental work was in two parts, profiling of electrical response to: (1) fracture intensity, and, (2) aperture variation. Prior to the experimental exercise, model rock blocks were prepared to be used as replicas of fractures, fracture spacing, intensity, and aperture variations.

From field outcrops and core observations, typical fractures in tight sedimentary and crystalline rocks, consists of fractures with aperture width between 0.1 to 10.0 mm, and filled with air, water, clay material, quartz or other cementing material ¹⁶. To investigate the effect of fracture density and aperture variation on ER response, a number of rock model blocks were prepared. The rock blocks were obtained from a quarry site near Ablekuma in the Greater Accra Region, Ghana. The rocks are lightgrey, fine-medium grained, metaporphosed quartzite of the Togo structural units (TSU) ¹⁷. They are composed of quartz, apatite, sericite and alkaline-feldspar. The rocks were selected for sample preparation based on their grain size, quality (competent and unweathered) and low matrix permeability. This was done mainly with the view that, flow and transport takes place principally only through the fractures and not the matrix ^{18, 19}.

The rock blocks were trimmed into regular geometrically defined samples at the Geotechnical Laboratory of Kwame Nkrumah University of Science and Technology (KNUST). Due to challenges in cutting specimen rock blocks, the largest rock block, which was also used as intact rock sample block was 35 cm x 20 cm x 30 cm. The sample rock block IDs and their dimensions are presented in Table 1.

An apparatus was constructed for measuring the horizontal electrical resistivity of intact and representative fractured rock. The apparatus consists of a wooden case of dimension 90 x 22 x 27 cm, four copper electrodes, MiniRes Ultra40 resistivity meter, connecting cables (Figure 3), calipers, and measuring tape. The wooden case is an electrical insulator (electrical resistivity ~ 1.00×10¹⁴ to 1.00×10¹⁶ ohm-m, (Anon 2018).

Electrical resistivity investigation was in two parts, profiling of electrical response to: 1) fracture intensity, to infer the variations of resistivity along horizontal as fracture intensity increases, and 2) aperture width variation. Apparent resistivity measuring procedure involves, spreading fine grained soil at the bottom (up to 10 cm thick), placing the rock block or blocks sample in the wooden case (at the central portion), spreading fine grained soil top of the rock block up to a thickness of 15 cm and driving the electrodes into the soil on top of the rock block sample.

To collect resistivity data using the MiniRes Ultra40 imaging system, the electrodes were deployed along an array in the Dipole-Dipole, Wenner, and Sclumberger configurations. For a typical measurement, the MiniRes meter reading, together with the spacing of the electrodes were recorded. If the electrodes were spaced at uniform intervals a along the line, then the apparent resistivity is calculated by the relation:

(4)

where, ρ is the resistance read on the MiniRes.

One complete profile was performed for each sample.

Table 2 presents the combinations of the different blocks and the resulting number of fractures and fracture area.

For aperture variation, the arrangement of electrodes and current flow system remain unchanged except that, in this case, two rock blocks (15 x 20 x 25 cm) were used (Figure 4), and the spacing between the faces in contact varied from 0.2 mm to 10 mm (Table 3). Measurements of change in aperture was done using the calipers. A pieces of A-4 paper was placed in between the spacing to avoid the opposite fracture faces coming into contact or being filled by the soil particles. Resistivity data was collected using the MiniRes Ultra40 imaging system. Electrodes were deployed along an array in the Sclumberger configuration.

There were 6 experiments in total and, for each experiment, current flow data were recorded.

Fundamental principle to all resistivity methods is the concept that, current (I) can be impressed into the ground and the effects of the current within the ground can be measured (Van Nostrand and Cook, 1996). The apparent resistivity ρ_a, as measured by the electrical imaging (EI) system is, the product of the entire rock block under the soil cover responding to the impressed current. Due to the presence of fractures, the rock block is no more homogeneous, it is important therefore, to model the apparent resistivities at discrete locations in order to make a more quantitative interpretation. Inverse modelling was performed using RES2DINV^TM software 21] to produce a two dimensional resistivity model based on the applied resistivity data.

Following the inverse modelling, the EI electrostructural information was used to interpret features of interest (geometry of the rock blocks, effects of the fractures and aperture size).

4. Results and Discussions

The resistivity profiles defined the geometry of rock block sample (Figure 5) and indicates medium range resistivity values 800 to 1000 Ohm-m).

Figure 5 shows some of the resistivity imaging profiles acquired across the wooden case with rock samples located within the central portion. For the samples studied, the apparent resistivity ranged from about 1,200 Ohm-m (intact rock) to about 12,500 Ohm-m, at the highest fracture intensity (5 fractures) (Table 4). As expected, fracture surface area increased with the increase of number of fractures.

Maximum and minimum values of ρ_a were 1000-1300 and 10000-12500 Ωm respectively.

Changes in ρ_a values indicates the variation of electrical response. To quantify the relationship between apparent resistivity and fracture intensity, a scatter-plot (Figure 6) was drawn with apparent resistivity as the abscissa and fracture intensity as ordinate.

The aim of this part of the experimental work was to investigate effects of fractures on apparent resistivity. Thus, only measured apparent resistivity values for fractured rock samples were plotted and modelled.

The 2D images of the Schlumberger array approach resistivity data appear to define the geometry of the rock blocks much better than the Wenner and Dipole-Dipole resistivity data. The Dipole-Dipole method was much time consuming.

There were strong positive linear and regressive correlations between ρa and fracture intensity.

With increasing number of fractures, the apparent resistivity increases in a power law relationship (exponential distribution) as:

(5)

With coefficient of determination, R² = 0.924, i.e.

(6)

Where ρ_a is apparent resistivity, and f_n is the number of fractures

Generally the relationship between apparent resistivity and fracture show strong linear and regression, however the highest coefficient of determination R² of 0.924 was represented in Equation 5.

The probability density function of the exponential distribution involves a parameter, which is a positive constant λ whose value determines the density function’s location and shape.

where λ is the value at which plot of the probability density function intercepts the y-axis.

Figure 7 shows the electrical image across the vertical fracture with aperture of 6 mm between the comparatively lower resistivity quartzite rock block.

For the samples studied, the apparent resistivity ranged from about 2620 Ohm-m (intact rock) to about 14050 Ohm-m. Measured apparent values where plotted for model construction. Table 5 is a summary of the experimental results.

The aim of this part of the experimental work was to investigate effects of fractures on apparent resistivity. Thus, only measured apparent resistivity values for fractured rock samples were plotted and modelled.

The best fit model was given a logarithmic relation as follows_

(7)

i.e.

(8)

Where ρ_a is apparent resistivity.

The resistivity data also indicate a well-defined high resistivity anomaly at central portion of the rock block samples, which could be interpreted as wood and near to an airfilled apace.

The effects of fractures and aperture size on apparent resistivity in fractured crystalline rocks have been investigated. The laboratory measurement results indicates, apparent resistivity increases with increase in fracture intensity and aperture size and are modelled with a power laws, suggesting that, fractures and fractured zones can be detected by locating the highest resistivity contrast regions.

This experimental investigation provides further evidence for the identification of rock contrasts using electrical resistivity approach reported by, Sandberg et al., 2002; Skinner and Heinson, ⁹, and Falgas et al., 2011.

Fractures and fractured zones often constitute a discontinuity creating an anomaly within an otherwise homogeneous rockmass. They can be identified geophysically by the contrasting properties of the presence of a fracture or fracture zone, irrespective of the rock types.

5. Conclusions

The experimental results indicates that the measurements of electrical resistivity depend on electrical heterogeneity and that variations in the signal were detectable, even at the laboratory scale. For each rock sample with the same rock type and infill material, apparent resistivity increases as fracture intensity and aperture width increases.

The measurements enabled a relationship between electrical resistivity and fracture intensity on one hand, and resistivity and aperture width to be established. The mathematical models established in the investigation can be explored to aid future investigation into field applicability of ER methods to characterise fracture density and aperture size distribution in near-surface crystalline rocks.

References