USGS Groundwater Information: Hydrogeophysics Branch

by *J.W. Lane, Jr.*, *F.P. Haeni*,
and *W.M. Watson*

U.S. Geological Survey, 11 Sherman Place, U-5015, Storrs, CT 06103

jwlane@usgs.gov; phaeni@usgs.gov

- Abstract
- Introduction
- The Square-Array Method
- Field Measurements
- Data Analysis
- Azimuthal Resistivity
- Comparison of Square Array and Collinear Array
- Use of Square Array to Detect Fractures at Grafton County, New Hampshire
- Interpretation of Square-Array Data and Crossed Square-Aray Data
- Comparison of Interpreted Data and Other Supporting Data
- Summary
- References

Azimuthal square-array direct-current (dc) resistivity soundings were used to detect fractures in bedrock in the Mirror Lake watershed in Grafton County, New Hampshire. Soundings were conducted at a site where crystalline bedrock underlies approximately 7 m (meters) of glacial drift. Measured apparent resistivities changed with the orientation of the array. Graphical interpretation of the square-array data indicates that a dominant fracture set and (or) foliation in the bedrock is oriented at 030° (degrees). Interpretation of crossed square-array data indicates an orientation of 027° and an anisotropy factor of 1.31. Assuming that anisotropy is due to fractures, the secondary porosity is estimated to range from 0.01 to 0.10.

Interpretations of azimuthal square-array data are supported by other geophysical data, including azimuthal seismic-refraction surveys and azimuthal Schlumberger dc-resistivity soundings at the Camp Osceola well field. Dominant fracture trends indicated by these geophysical methods are 022° (seismic-refraction) and 037° (dc-resistivity). Fracture mapping of bedrock outcrops at a site within 250 m indicates that the maximum fracture-strike frequency is oriented at 030°.

The square-array dc-resistivity sounding method is more sensitive to a given rock anisotropy than the more commonly used Schlumberger and Wenner arrays. An additional advantage of the square-array method is that it requires about 65 percent less surface area than an equivalent survey using a Schlumberger or Wenner array.

Research on the application of surface-geophysical methods to detect bedrock fractures and to estimate hydraulic properties of fractured bedrock is being conducted by the U.S. Geological Survey (USGS) as part of a bedrock research investigation in the Mirror Lake watershed of the Hubbard Brook Experimental Forest in Grafton County, New Hampshire (fig. 1).

One part of the study assesses the ability of direct-current (dc) resistivity methods to detect bedrock fractures. Dc-resistivity methods have been successfully used by a number of investigators to detect bedrock fractures (Risk, 1975; McDowell, 1979; Palacky and others, 1981; Soonawala and Dence, 1981; Taylor, 1982; Mallik and others, 1983; Leonard-Mayer, 1984a, 1984b; Ogden and Eddy, 1984; Taylor, 1984; Taylor and Fleming, 1988; Lieblich and others, 1991, 1992; Ritzi and Andolsek, 1992). Most of these investigations have used collinear arrays (Wenner or Schlumberger) rotated about a fixed centerpoint to measure azimuthal variations in apparent resistivity that are related to sets of similarly oriented, steeply dipping fractures (Lewis and Haeni, 1987).

Habberjam (1972) showed that a square array is more sensitive to anisotropy in the subsurface and requires less surface area than collinear arrays. Recently, dc-resistivity surveys using a square array have been conducted to detect productive fracture zones in crystalline bedrock for ground-water supply (Darboux-Afouda and Louis, 1989; Sehli, 1990). These studies verified Habberjam's earlier work, showing that the square array has a greater sensitivity to a given bedrock anisotropy and requires less surface area than collinear arrays.

The square array was tested at the Mirror Lake research site. This paper describes the square-array dc-resistivity method, outlines a simplified method of data analysis to determine fracture strike and secondary porosity, and presents the results of the field test.

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Figure 1. Location of U.S. Geological Survey fractured bedrock research site, showing square-array test area, Mirror Lake, Grafton County, New Hampshire. (Modified from Haeni and others, 1993, fig. 1.)

The square array was originally developed as an alternative to Wenner or Schlumberger arrays when a dipping subsurface, bedding, or foliation was present (Habberjam and Watkins, 1967). A complete discussion of the square array and methods of data analysis is provided by Habberjam (1979). Techniques for analyzing directional-resistivity data provided by the square-array method have been developed (Habberjam, 1972), but the method has not been widely used. Few case studies or interpretive methods specifically applied to the square array are found in the literature, although commercial software is available for layered earth interpretations.

A dc-resistivity survey using the square-array method is conducted in a manner similar to that for traditional collinear arrays. The location of a measurement is assigned to the centerpoint of the square. The array
size (A) is the length of the side of the square. The array is expanded symmetrically about the centerpoint, in increments of A(2)^{½} (Habberjam and Watkins, 1967), so that the sounding can be interpreted as a
function of depth.

For each square, three measurements are made - two perpendicular measurements (alpha, a; and beta, b) and one diagonal measurement (gamma, g)
(fig. 2). The a and b measurements provide information on the directional variation of the subsurface apparent resistivity (r_{a}).
The azimuthal orientation of the a and b measurements is that of the line connecting the current electrodes. The g measurement serves as a
check on the accuracy of the a and b measurements.

Figure 2. Electrode positions for square-array measurements.

In an isotropic medium,

in a homogeneous anisotropic medium,

where r_{a} = apparent resistivity, in ohm-meters.

Apparent resistivity is determined using the equation:

where r_{a} = apparent resistivity; K= geometric factor for the array; DV= potential difference, in volts; and I= current magnitude, in amperes.

For the square array,

where A= square-array side length, in meters. In practice, multiple square-array data are collected at small angular intervals. For example, six square arrays separated by a rotational angle of 15° will provide three independent crossed square-array data sets (two square arrays separated by an angle of 45°; fig. 2) for analysis, as well as sufficient data for graphical display and interpretation of individual square-array data on a rosette diagram using Taylor and Fleming's methods (1988).

The data were analyzed using a method described by Habberjam (1972). The following discussion is a brief summary of the method and outlines the potential uses of the square-array method for hydrologic investigations.

*Depth Sounding*

Habberjam and Watkins (1967) demonstrated that apparent-resistivity data obtained with a square array can be interpreted using published methods for Wenner or Schlumberger soundings. This is done by translating apparent-resistivity data obtained from the square array to those of equivalent Wenner or Schlumberger arrays.

First, the square-array resistivity measurements are reduced to a single measurement (r_{m}) by

r_{m} = [(r_{a a} ) (r_{a b})]^{½},(5)

Habberjam and Watkins, 1967

where r_{m} = mean geometric resistivity;

The more rigorous relation between the size of a square array and the size of an equivalent Wenner or Schlumberger array is given by

where A = square-array side length, in meters; r = AM = current electrode and nearest potential electrode separation; and s = MN = potential electrode separation (fig. 3).

Figure 3. Common collinear arrays equivalent to a square array of side length A.

Later work (Habberjam, 1972) showed that an azimuthal-independent value for that spacing can be obtained by calculating the geometric mean of r_{m} obtained from two square arrays
separated by a rotational angle of 45° (about the centerpoint) (the crossed square array). However, reduction of the square-array resistivity measurements to yield a single directionally stable measurement for layered
earth interpretation removes the information about directional resistivity differences contained within the individual square-array measurements.

Variations in azimuthal resistivity can be caused by many factors such as slope of the bedrock surface, or dip of bedding or foliation. Experiments by Sauck and Zabik (1992) have shown that azimuthal resistivity data can be affected by overburden thickness. The following discussion assumes that the resistivity differences are only caused by similarly oriented, steeply dipping fractures (fig. 4).

Figure 4. Model of homogeneous anisotropic Earth. (Modified and reprinted from Habberjam, 1972, fig. 2, and published with permission.)

Habberjam (1972) derived the expression for the variation of apparent resistivity with square-array orientation over a homogeneous anisotropic earth. For fractured rock that approximates such a medium, the predicted square-array apparent resistivity in a given orientation is

where r_{m} = [(r_{aa} ) (r_{ab})]^{½} apparent
resistivity perpendicular to fractures; r_{a1} = apparent resistivity parallel to fractures; q = angle measured from azimuth of current electrodes to fracture
strike; N = effective vertical anisotropy,=[(1+(l^{2} - 1)sina^{2})]½; l = coefficient of anisotropy; l
= r_{at}/r_{a1})^{½ }; a = dip of fractures; N = l for a = p/2.

When variations in azimuthal resistivity are detected over an anisotropic earth, and the variations are caused by fractures, the interpretive methods of Habberjam (1972; 1975) and Taylor (1984) can be used to determine fracture strike and to estimate secondary porosity.

*Determination of Fracture Strike*

The fracture strike can be determined graphically or analytically. To interpret strike graphically, the apparent resistivity for azimuthal square arrays is plotted against the azimuth of that measurement. The principal
fracture strike direction is perpendicular to the direction of maximum resistivity.

Crossed square-array data can be interpreted analytically in order to yield fracture strike (Habberjam, 1975):

where q = fracture strike, measured from r_{a1};

A = [(r_{a3} + 3r_{a1})/2 + (r_{a4} + r_{a2})/(2)^{½}][(2 + (2)^{½}];

B = [(r_{a1} + 3r_{a3})/2 + (r_{a2} + r_{a4})/(2)^{½}][(2 + (2)^{½}];

C = [(r_{a4} + 3r_{a2})/2 + (r_{a1} + r_{a3})/(2)^{½}][(2 + (2)^{½}];

D = [(r_{a2} + 3r_{a4})/2 + (r_{a3} + r_{a1})/(2)^{½}][(2 + (2)^{½}];

r_{a1}, r_{a2}, r_{a3}, and r_{a4} are constituent resistivity measurements from
a crossed square array (fig. 2).

*Estimation of Secondary Porosity*

Using the crossed square-array measurements, the secondary porosity (f) can be estimated by modifying Taylor's method developed for collinear arrays in saturated, clay-free, non-shale rocks
(1984). To calculate secondary porosity, it is first necessary to calculate the anisotropy (N) from the field data, using Habberjam's method (1975):

where T = A^{-2} + B^{-2} + C^{-2} + D^{-2}; S = 2[(A^{-2} - B^{-2})^{2} + (D^{-2} - C^{-2})^{2}]^{½}; and A, B, C, and D are
defined previously.

The secondary porosity is then estimated by:

where f = secondary porosity; r_{max} = maximum square-array apparent resistivity; r_{min} = minimum square array
apparent resistivity; and C = specific conductance of ground water in microsiemens per centimeter.

**Comparison of Square Array and Collinear Array**

*Sensitivity to Anisotropy*

To correctly interpret azimuthal dc-resistivity data over fractured rock, the bedrock must behave as an anisotropic medium. Satisfying this requires the electrode spacings to be large with respect to the fracture spacing
(Taylor, 1982). The square array has a ratio of potential electrode to current electrode spacing of 1:1. The Wenner array ratio is 1:3, and the Schlumberger ratio is generally less than 1:10. As the above arrays are
expanded, the square array, with the largest electrode- spacing ratio, will most rapidly satisfy the above condition.

The square array has been shown to be more sensitive to anisotropy than the Schlumberger or Wenner array (Habberjam, 1972; LeMasne, 1979; Darboux-Afouda and Louis, 1989). For the square array, the ratio of apparent
resistivity measured perpendicular to fracture strike (r_{at}) to apparent resistivity parallel to fracture strike (r_{a1}) is called the apparent
anisotropy (l_{a}) and is given by the ratio:

(Darboux-Afouda and Louis, 1989)

The apparent anisotropy for the Schlumberger or Wenner array is given by:

where N = effective vertical anisotropy.

The apparent anisotropy of the square array and the Schlumberger array for a true rock anisotropy is shown in figure 5. The higher apparent anisotropy measured by the square array is an advantage, because the anisotropy is less likely to be obscured by heterogeneities in bedrock or overburden, bedrock relief, cultural noise, electrode placement errors, or other sources of noise.

Figure 5. Anisotropy measured by the square array and the Schlumberger array for a given bedrock anisotropy. (Reprinted from Darboux-Afouda and Louis, 1989, fig. 2 and published with permission.)

*Survey Area Requirements*

The square-array geometry is more compact than Schlumberger or Wenner arrays for azimuthal surveys. The square array requires 65 percent less surface area than the equivalent collinear arrays (Habberjam and Watkins,
1967). This is an advantage in an area with significant lateral heterogeneities or when the area available to conduct a survey is limited.

*Maximum Resistivity in Relation to Fracture Strike*

The direction of maximum apparent resistivity measured by the square array will be perpendicular to fracture strike. This is a function of the cosine term in the denominator of equation 7.

The direction of maximum apparent resistivity measured by the collinear array will be parallel to the fracture strike (Habberjam, 1972). This is a consequence of the "paradox of anisotropy" (Keller and Frischknecht, 1966) and is determined by the sine term in the denominator of the following equation showing apparent resistivity in relation to orientation:

where K = geometric factor for the array, N = effective vertical anisotropy, and r_{m} = mean geometric resistivity.

**USE OF SQUARE ARRAY TO DETECT FRACTURES AT GRAFTON COUNTY, NEW HAMPSHIRE**

The square-array dc-resistivity method was field tested in Priest Field, near the Camp Osceola (CO) well cluster, located at the Mirror Lake research site (fig. 1). At this site, approximately 7 m
(meters) of stratified glacial drift is underlain by crystalline bedrock. The square-array survey was conducted during the summer of 1992. The survey consisted of six square-array soundings separated by a 15° rotational
angle about the array centerpoint. The A-spacings of the arrays were expanded from 5 m to 50 m (in increments of (2)^{½}) for each sounding. The data were collected using an ABEM Multimac dc-resistivity system.
This computer-controlled data acquisition and storage system (fig. 6) allowed a complete sounding at a given azimuth to be collected automatically through remotely accessed addressable switchers,
which connected electrodes for a given measurement. Software provided with the system was modified for the square array to control the measurement sequence.

Figure 6. Block diagram of direct-current resistivity data-collection system.

**Interpretation of Square-Array Data and Crossed Square-Array Data**

Data collected in Priest Field show a significant variation of apparent resistivity for different azimuthal array orientations for all A- spacings. Apparent-resistivity data collected at the site are shown in table 1. The data from the largest arrays were analyzed to minimize possible overburden effects. The apparent resistivity plotted against azimuth are shown for the 40- and 50-m A-spacings, which are least affected by the overburden (fig. 7). For both A-spacings shown, the maximum resistivity measured is parallel to a trend of 120°. A graphical interpretation of the data is that a primary fracture set is present with a strike of 030°. The 50-m data show a secondary trend of high resistivity at 060°. Assuming that these data do not reflect heterogeneities in the bedrock, this peak could indicate the presence of a secondary fracture set oriented at 150°.

Table 1. Azimuthal Apparent Resistivities Collected at Priest Field, Mirror Lake Watershed, New Hampshire

Figure 7. Square array apparent resistivity plotted again azimuth for the 40- and 50-meter A-spacings, collected at Priest Field, U.S. Geological Survey fractured bedrock research site, Mirror Lake, New Hampshire. (Modified from Haeni and others, 1993, fig. 5).

**Interpretation of Square-Array Data and Crossed Square-Array Data**

Because data for six square arrays separated by 15° were collected, there are three independent crossed square arrays available for analysis. Using the analytical methods described in the section "Field Measurements" for the 50-m data, a strike estimate of 027° and an anisotropy value of 1.31 was obtained.

Using these analytical results for the strike and anisotropy values, the apparent resistivity values predicted using equation 6 were calculated and the results superimposed on the field data (fig. 7). This shows that the field data closely approximates the data for an isotropic homogeneous earth.

Using the anisotropy of 1.31, a maximum resistivity of 3,040 ohm-m (ohm- meters), a minimum resistivity of 1,132 ohm-m, and a range of specific conductance for ground water in the CO well field of 30 to 315 µS/cm (P.T. Harte, U.S. Geological Survey, written commun., 1991), and equation 10, the secondary porosity due to fracturing is estimated to range from 0.01 to 0.10. This demonstrates the sensitivity of the secondary porosity estimate to the specific conductance of ground water.

**Comparison of Interpreted Data and Other Supporting Data**

*Fracture Strike* An outcrop located on the center median of Interstate 93 (within 250 m of the site) consists of schist, massive granitic dikes, granitic dikes with schist xenoliths, and pegmatite (C.C. Barton,
U.S. Geological Survey, written commun., 1989). The graphically interpreted fracture strike from the square array data of 030° and the crossed square- array data of 027° correlates with fracture-strike frequency data
measured at the I-93 outcrop. Analysis of the measured fracture-strike data yields a mean fracture-strike frequency maximum of 030° with major peaks at 005° and 040°. Most of the fractures are steeply dipping (C.C.
Barton, U.S. Geological Survey, written commun., 1993; fig. 8). The secondary resistivity maximum seen in the 50-m square- array data indicates the possible presence of another fracture set oriented at 150°. This might
correlate with a minor fracture strike maximum at the outcrop oriented at 135° or indicate a change in foliation orientation.

Figure 8. Equal area plot of fractures mapped from Route I-93 outcrops near the U.S. Geological Survey fractured bedrock research site, West Thornton, New Hampshire. (From C.C. Barton, U.S. Geological Survey, written commun., 1993.)

Azimuthal p-wave seismic-refraction surveys have been conducted at the square-array test site (Lieblich and others, 1991). Data were collected every 22.5° about the same centerpoint used in the square-array survey. The contrast between the measured seismic velocity and the refraction-line orientation (fig. 9) was interpreted as indicating a fracture and (or) foliation oriented at 022.5° (Lieblich and others, 1991).

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Figure 9. Seismic compressional wave velocity plotted against azimuth at Priest Field, U.S. Geological Survey fractured bedrock research site, Mirror Lake, New Hampshire. (From Lieblich and others, 1991, fig. 6.)

An azimuthal dc-resistivity survey using the Schlumberger array was conducted at the CO well field 100 m from the square-array test site (Haeni and others, 1993). Data were collected every 45° with a Bison 2390 resistivity system, and the AB/2 spacing was expanded from 3 m to 30 m. Qualitative analysis of these data support a fracture set and (or) foliation oriented at 037° (fig. 10); however, the anomalously large anisotropy indicates some cultural interference (for example, well casing or buried metal) within the data.

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Figure 10. Schlumberger array apparent resistivity plotted again azimuth for the 40- meter AB/2 spacing, collected at the Camp Osceola well field, U.S. Geological Survey fractured bedrock research site, Mirror Lake, New Hampshire. (From Haeni and others, 1993, fig. 4.)

Two azimuthal surveys using the Schlumberger array also were conducted at the square-array test site about the same centerpoint as the square array. One set of soundings was conducted every 22.5° using a Bison 2390 resistivity system. The AB/2 spacings were expanded from 3 m to 40 m. A second set of soundings was conducted every 15° with the ABEM Multimac system. The AB/2 spacings were expanded from 3 m to 40 m. The fracture orientation interpreted from each of these data sets is 352° (fig. 11).

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Figure 11. Schlumberger array apparent resistivity plotted again azimuth for the 40- meter AB/2 spacing, collected at Priest Field, U.S. Geological Survey fractured bedrock research site, Mirror Lake, New Hampshire.

Inspection of the data, field site, and survey design may explain the discrepancy between the Schlumberger and square-array data, and illustrate possible weaknesses of azimuthal Schlumberger array surveys for fracture detection at this site. Because the large current electrode separation of the Schlumberger array required a large surface area, a buried phone line was included within the array. This phone line was not located within the square array survey area. The largest MN/2 potential-electrode separations used for the Schlumberger array was 4 m, which may not have satisfied the theoretical requirement of large electrode spacing as a function of fracture spacing. In addition, heterogeneities present near potential electrodes may have affected the Schlumberger data. In view of the above, the square-array results are interpreted to indicate fracture conditions at the test site more accurately than the Schlumberger array results.

*Secondary Porosity*

The secondary porosity estimate interpreted from the 50-m data ranges from 0.008 to 0.08. This range encompasses values higher than average secondary porosities for crystalline rock, which are generally less than 0.02
(Freeze and Cherry, 1979). An estimate of the fracture porosity at well CO-1, made by measuring the number of fractures and the average fracture aperture, is 0.001 (P.T. Harte, U.S. Geological Survey, oral commun.,
1992). A reason for this difference is that secondary porosity estimates obtained from wells are sensitive to subhorizontal fractures and the dc-resistivity method is more sensitive to vertical fractures. Another
explanation is that foliation could be a significant contributor to anisotropy. The secondary porosity calculations from the square array assumed that fracturing is the only cause of anisotropy.

Azimuthal dc-resistivity methods are sensitive to a sloping bedrock interface (Habberjam, 1975), and a reliable method of correcting for the slope is not available. Results from seismic-refraction surveys at Priest Field indicate that the bedrock surface slopes at less than 5 (D.A. Lieblich, U.S. Geological Survey, oral commun., 1992). This should have a minimal effect on the anisotropy measurement and the interpreted secondary porosity.

The square-array dc-resistivity method was used to determine the orientation of bedrock fractures at the USGS bedrock research site in the Mirror Lake watershed of the Hubbard Brook Experimental Forest in Grafton County, New Hampshire. A primary fracture strike orientation of 030 was interpreted from a graphical analysis of square-array data and an orientation of 027° was interpreted from the analysis of the crossed square-array data. These values closely match the orientation of fracture trends mapped in bedrock (030°) near the sites of surface-geophysical surveys. Fracture-strike orientation interpreted from square-array data is supported by other geophysical data, including azimuthal p-wave seismic refraction (022.5°) and azimuthal Schlumberger dc-resistivity (037° at Camp Osceola). The estimated secondary porosity attributed to fracturing ranged from 0.01 to 0.10, which was higher than the porosity estimated at well CO-1 (0.001). Further comparison of secondary porosity estimates determined from square-array surveys with porosities determined from other sources is needed to assess the square-array secondary porosity calculations.

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Final copy as submitted to Ground Water for publication as: Lane, J.W., Jr., Haeni, F.P., and Watson, W.M., 1995, Use of a square-array direct-current resistivity method to detect fractures in crystalline bedrock in New Hampshire: Ground Water, v. 33, no. 3, p. 476-485.