{"id":43,"date":"2022-07-14T00:09:10","date_gmt":"2022-07-14T00:09:10","guid":{"rendered":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/chapter\/geometric-factors\/"},"modified":"2022-07-23T04:39:16","modified_gmt":"2022-07-23T04:39:16","slug":"geometric-factors","status":"publish","type":"chapter","link":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/chapter\/geometric-factors\/","title":{"raw":"2.1 Geometric Factors","rendered":"2.1 Geometric Factors"},"content":{"raw":"
\r\n

We introduced the concept of the geometric factor (Equation\u00a03<\/a>) as the parameter that converts a measured resistance to apparent resistivity. For surface arrays, the underlying math to calculate the geometric factor is fairly simple. Assuming a homogeneous and isotropic half space<\/em> (meaning the same electrical conductivity in the earth to infinite distance below a surface boundary) without any electrical sources, the geometric factor K<\/em>g<\/em><\/em><\/sub> for every quadripole can be calculated for surface arrays using Equation\u00a06a.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K_{g}=\\frac{2\\pi }{\\left [\\frac{1}{\\overline{AM}}-\\frac{1}{\\overline{AN}}-\\frac{1}{\\overline{BM}}+\\frac{1}{\\overline{BN}} \\right]}[\/latex]<\/td>\r\n(6a)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

[latex]\\overline{AM}[\/latex], [latex]\\overline{AN}[\/latex], [latex]\\overline{BM}[\/latex], and [latex]\\overline{BN}[\/latex] are the distances between electrodes A<\/em> and M<\/em>, A<\/em> and N<\/em>, B<\/em> and M<\/em>, and B<\/em> and N<\/em>, respectively. Current electrodes are defined as A<\/em> and B<\/em> and potential electrodes are M<\/em> and N<\/em>.<\/em> Current electrodes are defined as A<\/em> and B<\/em> and potential electrodes are M<\/em> and N<\/em> (Figure 5<\/strong><\/a>). These electrodes are often also called C<\/em>+, C<\/em>\u2212, P<\/em>+, and P<\/em>\u2212, respectively, in other literature, however the older A<\/em>, B<\/em>, M<\/em>, N<\/em> standard is used in this book. The geometric factor accounts for the arrangement of electrodes and allows one to calculate an apparent resistivity (Equation\u00a03<\/a>).<\/p>\r\n

For borehole geometries, the electrodes are located within the half-space rather than at the boundary at the Earth\u2019s surface. In this case, use of Equation\u00a06a is inappropriate, as it does not account for the no-flow boundary at Earth\u2019s surface. To account for the effect of the boundary on cross-well measurements, the method of images<\/em> from optics is invoked. This approach is analogous to the use of image wells in groundwater hydrology for analytical modeling of aquifer response to pumping. Here, imaginary image current electrodes<\/em> are introduced on the other side of the boundary, equidistant from the real current electrodes, to mathematically produce a no-flow condition at the Earth\u2019s surface and calculate a geometric factor for borehole arrays as shown in Equation\u00a06b.<\/p>\r\n\r\n\r\n\r\n\r\n
<\/td>\r\n[latex]\\displaystyle K_{g}=\\frac{4\\pi }{\\left [\\frac{1}{\\overline{AM}}+\\frac{1}{\\overline{A_{image}M}}-\\frac{1}{\\overline{AN}}-\\frac{1}{\\overline{A_{image}N}}-\\frac{1}{\\overline{BM}}-\\frac{1}{\\overline{B_{image}M}}+\\frac{1}{\\overline{BN}}+\\frac{1}{\\overline{B_{image}N}} \\right]}[\/latex]<\/td>\r\n(6b)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Here, \u201cimage<\/em>\u201d indicates the image current electrode. When the electrodes are all on the boundary, Equation\u00a06b simplifies to Equation\u00a06a. Also, limited burial is necessary before Equation\u00a06b simplifies to twice the result of Equation\u00a06a, as the distances from the potential electrodes to the true current electrodes and their images are approximately equal.<\/p>\r\n

Quadripoles with large geometric factors may produce small voltage differences, which are prone to measurement errors due to a lower signal-to-noise ratio. These are manifest (via propagation of errors) as higher relative errors in apparent resistivity data. A critical geometric-factor cutoff can be determined based on the average expected electrical conductivity of the subsurface and the instrument specifications. Based on Equation\u00a03<\/a>, for a given geometric factor and expected instrument error (in terms of voltage, inserted as the potential difference), we can calculate the expected error in apparent resistivity. Figure\u00a0<\/strong>6<\/strong> illustrates how error in measured potential difference translates into error in calculated apparent resistivity as a function of K<\/em>g<\/em><\/em><\/sub>. In this example, we consider an applied current of 50\u00a0mA and assume a 1-microvolt (\u03bcV) instrument accuracy (note the logarithmic scale). In practice, accuracy may be less. As evident in Figure\u00a0<\/strong>6<\/strong>, for large geometric factors or small assumed apparent resistivity, errors are larger relative to measurements.<\/p>\r\n

\"Graph<\/p>\r\n

Figure\u00a0<\/strong>6<\/strong>\u00a0<\/strong>-<\/strong>\u00a0Apparent resistivity error, as a percentage of the true apparent resistivity, as a function of geometric factor for three different values of resistivity (50, 500, and 5,000\u00a0\u03a9-m), assuming the voltage accuracy is 1\u00a0microvolt and the applied current is 50\u00a0milliamperes, where apparent resistivity error is given by K<\/em>g<\/em><\/em><\/sub>*<\/em>V<\/em>error<\/em><\/em><\/sub>\/I,<\/em> which is then compared to the assumed apparent resistivity in a relative sense. An error in the measured voltage translates into error in calculated apparent resistivity as a function of K<\/em>g<\/em><\/em><\/sub>. In practice, larger errors may occur in the field.<\/p>\r\n\r\n<\/div>","rendered":"

\n

We introduced the concept of the geometric factor (Equation\u00a03<\/a>) as the parameter that converts a measured resistance to apparent resistivity. For surface arrays, the underlying math to calculate the geometric factor is fairly simple. Assuming a homogeneous and isotropic half space<\/em> (meaning the same electrical conductivity in the earth to infinite distance below a surface boundary) without any electrical sources, the geometric factor K<\/em>g<\/em><\/em><\/sub> for every quadripole can be calculated for surface arrays using Equation\u00a06a.<\/p>\n\n\n\n
<\/td>\n[latex]\\displaystyle K_{g}=\\frac{2\\pi }{\\left [\\frac{1}{\\overline{AM}}-\\frac{1}{\\overline{AN}}-\\frac{1}{\\overline{BM}}+\\frac{1}{\\overline{BN}} \\right]}[\/latex]<\/td>\n(6a)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

[latex]\\overline{AM}[\/latex], [latex]\\overline{AN}[\/latex], [latex]\\overline{BM}[\/latex], and [latex]\\overline{BN}[\/latex] are the distances between electrodes A<\/em> and M<\/em>, A<\/em> and N<\/em>, B<\/em> and M<\/em>, and B<\/em> and N<\/em>, respectively. Current electrodes are defined as A<\/em> and B<\/em> and potential electrodes are M<\/em> and N<\/em>.<\/em> Current electrodes are defined as A<\/em> and B<\/em> and potential electrodes are M<\/em> and N<\/em> (Figure 5<\/strong><\/a>). These electrodes are often also called C<\/em>+, C<\/em>\u2212, P<\/em>+, and P<\/em>\u2212, respectively, in other literature, however the older A<\/em>, B<\/em>, M<\/em>, N<\/em> standard is used in this book. The geometric factor accounts for the arrangement of electrodes and allows one to calculate an apparent resistivity (Equation\u00a03<\/a>).<\/p>\n

For borehole geometries, the electrodes are located within the half-space rather than at the boundary at the Earth\u2019s surface. In this case, use of Equation\u00a06a is inappropriate, as it does not account for the no-flow boundary at Earth\u2019s surface. To account for the effect of the boundary on cross-well measurements, the method of images<\/em> from optics is invoked. This approach is analogous to the use of image wells in groundwater hydrology for analytical modeling of aquifer response to pumping. Here, imaginary image current electrodes<\/em> are introduced on the other side of the boundary, equidistant from the real current electrodes, to mathematically produce a no-flow condition at the Earth\u2019s surface and calculate a geometric factor for borehole arrays as shown in Equation\u00a06b.<\/p>\n\n\n\n
<\/td>\n[latex]\\displaystyle K_{g}=\\frac{4\\pi }{\\left [\\frac{1}{\\overline{AM}}+\\frac{1}{\\overline{A_{image}M}}-\\frac{1}{\\overline{AN}}-\\frac{1}{\\overline{A_{image}N}}-\\frac{1}{\\overline{BM}}-\\frac{1}{\\overline{B_{image}M}}+\\frac{1}{\\overline{BN}}+\\frac{1}{\\overline{B_{image}N}} \\right]}[\/latex]<\/td>\n(6b)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Here, \u201cimage<\/em>\u201d indicates the image current electrode. When the electrodes are all on the boundary, Equation\u00a06b simplifies to Equation\u00a06a. Also, limited burial is necessary before Equation\u00a06b simplifies to twice the result of Equation\u00a06a, as the distances from the potential electrodes to the true current electrodes and their images are approximately equal.<\/p>\n

Quadripoles with large geometric factors may produce small voltage differences, which are prone to measurement errors due to a lower signal-to-noise ratio. These are manifest (via propagation of errors) as higher relative errors in apparent resistivity data. A critical geometric-factor cutoff can be determined based on the average expected electrical conductivity of the subsurface and the instrument specifications. Based on Equation\u00a03<\/a>, for a given geometric factor and expected instrument error (in terms of voltage, inserted as the potential difference), we can calculate the expected error in apparent resistivity. Figure\u00a0<\/strong>6<\/strong> illustrates how error in measured potential difference translates into error in calculated apparent resistivity as a function of K<\/em>g<\/em><\/em><\/sub>. In this example, we consider an applied current of 50\u00a0mA and assume a 1-microvolt (\u03bcV) instrument accuracy (note the logarithmic scale). In practice, accuracy may be less. As evident in Figure\u00a0<\/strong>6<\/strong>, for large geometric factors or small assumed apparent resistivity, errors are larger relative to measurements.<\/p>\n

\"Graph<\/p>\n

Figure\u00a0<\/strong>6<\/strong>\u00a0<\/strong>–<\/strong>\u00a0Apparent resistivity error, as a percentage of the true apparent resistivity, as a function of geometric factor for three different values of resistivity (50, 500, and 5,000\u00a0\u03a9-m), assuming the voltage accuracy is 1\u00a0microvolt and the applied current is 50\u00a0milliamperes, where apparent resistivity error is given by K<\/em>g<\/em><\/em><\/sub>*<\/em>V<\/em>error<\/em><\/em><\/sub>\/I,<\/em> which is then compared to the assumed apparent resistivity in a relative sense. An error in the measured voltage translates into error in calculated apparent resistivity as a function of K<\/em>g<\/em><\/em><\/sub>. In practice, larger errors may occur in the field.<\/p>\n<\/div>\n","protected":false},"author":1,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-43","chapter","type-chapter","status-publish","hentry"],"part":118,"_links":{"self":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/43","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":11,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/43\/revisions"}],"predecessor-version":[{"id":437,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/43\/revisions\/437"}],"part":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/parts\/118"}],"metadata":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapters\/43\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/media?parent=43"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/pressbooks\/v2\/chapter-type?post=43"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/contributor?post=43"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/books.gw-project.org\/electrical-imaging-for-hydrogeology\/wp-json\/wp\/v2\/license?post=43"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}