Measurements of a materials electrical resistivity can provide useful information for, indirectly determining soil moisture  , assisting in the design of cathodic protection systems to prevent corrosion in buried metal structures  , deter- mining electrical substation grounding characteristics  , measuring subsurface hydrological properties  , and the extent of sub-seafloor sediment   in oceanographic studies, to name but a few. Electrical Resistivity Tomography (ERT) also provides a useful geophysical tool in imaging resistivity variations in the subsurface  .
This paper describes a simple probe to be used for in-situ resistivity mea- surements. We derive a probe factor based on the geometry of our device, which is given by an exact analytic solution of the boundary value problem. Our moti- vation for developing this probe was the need for an in-situ device to measure soil moisture over time within a free-draining lysimeter. Since the probe mea- sures electrical resistivity directly, it is also necessary to relate this value to soil moisture through experimental measurements  .
Analysis of this probe is similar to those used in bore-hole boundary value problems, specifically those employing an integral equation approach (see for example, Zhang  ; Tsang  ; Gianzero  ). The current application is inte- rested in material properties near the probe, and hence a more accurate representation of the probe current source(s) is required. The associated bo- undary value problem for the probe is solved in Appendix A, using an integral equation approach and assuming a homogeneous media where the conducting rings are placed over an insulated rod. The derivation in Appendix B includes the effect of a planar ground surface.
The mathematical formulation of our probe is based on a Wenner  style four point electrode configuration, however the results can easily be extended to other array geometries, such as a Schlumberger or a dipole-dipole  . For the Wenner method current is sourced through the outside points and the voltage is measured at the inside points, as shown Figure 1. The Wenner method has been adapted to very thin ring conductors around an insulating rod by Won  . In the case where the probe is to be driven into a soil mixture rather than a wet en- vironment, it is necessary to expand the conducting rings to have finite thickness to ensure a good electrical connection of the probe and the surrounding material. The conducting bands having a thickness requires a new derivation for the probe factor to relate the apparent resistivity to actual resistivity of the material.
Figure 1.A sketch of the resistivity probe (a) with the associated equipotential lines shown as solid and the electric field/current density lines shown as dashes shown in (b).
Experimental results are presented for comparison of the derived, simulated, and measured probe factor.
2. Mathematical Formulation
The probe, shown in Figure 1, consists of insulating rod of radius a, with four conducting rings of width with the outer and inner rings separated by a mean distance and, respectively. Placing a known current across the two outer conducting rings and measuring the resulting potential across the two inner rings, the resistivity/conductivity of the surrounding material can be deter- mined. This is achieved by first solving the associated boundary value problem, as derived in Appendix A.
In this derivation it is assumed that the electrical conductivity is constant in the vicinity surrounding the probe so that in cylindrical coordinates the electric potential is given by,
where the probe is centered at, is the measured current and
with, , and. Here the ground surface effect is neglected, however it is include later. Based on (1), a sketch of the potential as well as the current density, given by
is shown in Figure 1. Notable in the figure is the localization of the fields near the probe electrodes.
The relationship between the apparent resistivity, the applied current and measured voltage is given by,
where, with the “probe factor” defined as the quantity,
The integral given in (5) cannot be solved in closed form, and must therefore evaluated numerically. However, since the probe factor is solely a function of geometry, that is, it need only be calculated once for a given probe.
Figure 2 shows variations in the calculated probe factor for a probe radius and conductor thickness are independently varied from 0.5 cm to 2.0 cm, with the electrode spacing (a sensitivity analysis also indicated that the probe factor was generally less sensitive to variations in).
With the ground interface included the expression for the probe factor is modified slightly and is given by,
where is the depth to the center of the probe from the ground surface, and the subscript indicates that the air-earth interface is included. As expected the surface interface modifies only slightly when the probe is very near the surface. This is indicated in Figure 3, which shows the calculated as a fun- ction of probe depth for, and. Not- ing the scale used for Figure 3, it is likely that the ground interface can be negle- cted in most applications.
Figure 2. Resistivity probe factor, , as a function of probe radius and conductor thickness, with, and.
Figure 3. Resistivity probe factor, , as a function probe depth for cm, cm, and.
An experiment was conducted by submerging a custom built probe in a barrel of a salt water solution. Salt was incrementally added to the solution increasing its conductivity. The conductivity of the water was then measured and verified with a VWR Symphony conductivity probe (model number 11388 - 382). Once a voltage was applied to the outer rings of the probe, the electrical current through the outer rings and the voltage on the inner rings were measured with an Agilent Digital Multimeter (model number 34410a). The probe was submerged to app- roximately the same depth and readings were recorded for current and voltage. Next, the probe was then extracted, dried, and the process repeated for five measurements at each solution conductivity.
The probe constructed for experimental verification of (5) was designed and built in units of inches. Converted to centimeters they are;,
, , and. Surface to probe center depth was 21.59 cm. Simulations of this probe were also conducted in COMSOL Mul- tiphysics with a set material conductivity taken from one of the experimental measured levels.
Figure 4. Graph of experimentally measured probe resistance versus verified fluid resistivity. Experimentally derived probe factor at each known resistivity for each set of measurements along with a linear curve fit and average probe factor are also presented.
Table 1. Analytical, numerical, and experimentally derived probe factors for the as-built probe
Analytical, numerical, and experimentally derived probe factors for the as-built probe are presented in Table 1. The plot of the experimental measurements in Figure 4 shows that there is greater uncertainty in the measurements at higher resistances. The derived probe factor at each conductivity level is shown along with its uncertainty. The experimental differs from the computed value with the grounding effects by less than the standard deviation of the measure- ments. Simulations performed in COMSOL are also in agreement with the ana- lytical and experimentally derived values for the probe factor.
We derive a mathematical expression for the probe-factor of a simple device for use in it-situ resistivity measurements. Our derivation is based on a Wenner array configuration, however our results are easily extended to include other geometries, such as the Schlumberger and the dipole-dipole arrays. Comparisons of our mathematically derived probe-factor with measured, and numerically derived results show excellent agreement.
A. Uniform Ground
Beginning with the equation of continuity for the current density, which for the static case is given by,
The current density at a point is related to the electric field through, where is the electrical conductivity and, with the static electric potential.
Since the probe is sensitive only to the local region surrounding it we assume that is approximately constant so that the potential satisfies Laplace’s equa- tion, namely
Employing a cylindrical coordinate system, specified by, with the length of the probe is along the -axis it is clear that our solution must be independent of. Then the Laplacian reduces to,
with. Assuming a solution of the form,.
Subject to the boundary condition that the potential vanish at the general solution for the potential is then given by,
where (referred to as the “kernel function”  ) is an unknown function of to be determined from the boundary conditions. Equation (10) is a Fou- rier integral wherein by the inverse Fourier transformation can be deter- mined via,
Next boundary conditions are established on the surface of the probe, where it is assumed that the probe is insulated, except for the conducting bands, which are also assumed to be ideal conductors. Thus the surface of the probe represent a no--flow boundary, except at the two conducting bands where the current den- sity has only a radially directed component. Since the current density is given by
We obtain for the radially directed component at the surface of the probe,
where we have used that Watson 
Since the conducting rings constitute equipotential boundaries the current densities must also be constant on along their surface. We can therefore express the current density at the surface of the probe as,
is the measured current, is the mean separation between the two outer conducting rings, the ring width, as shown in Figure 1(a), with
Equation (13) is solved making use of the inverse Fourier transform. This gives,
Solving for in equation yields,
so that the electrical potential can be expressed as,
given that the term
within the integrand is odd with respect to the variable of integration. Hence only the term associated with contributes to the integral when evaluated over our limits of integration. The expression for given in (18) is preferable since it does not contain any imaginary terms, indicating that the potential is purely real, as it must be.
The final task is to relate the measured potential to the known current, applied across the two outer conducting rings. First define the term,
then by definition,
Hence, the ratio between the measured voltage and applied current is given by,
Define, then the apparent resistivity is given by
with the probe factor given by,
B. Uniform Half-Space
To include the effect of the ground surface, image theory is used to enforce the no-flow boundary at the ground-air interface. Assuming the probe is buried a depth as measured from the center of the probe to the ground surface,and assuming the -axis to be centered at the probe (as previously), then
with the modified kernel function with the ground effect included. In obtaining (25) the method of images was used as was the shifting property of the Fourier transform. Then, the potential in the lower half-space is given by,
where again the even/odd symmetry in our integrand was used to simplify the resulting integral. Following the same general procedure as in the previous case gives the modified probe factor,
with the subscript indicating the ground effect.
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