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We present a probe factor for a simple measurement device, which can be used to determine
*in-situ* electrical resistivity in soils or other penetrable bodies. The probe is primarily sensitive to the material immediately surrounding it and therefore is ideal for determining localized conductivities. The geometry of the probe can be scaled to effectively adjust the region of interest. The calibration, or “probe factor” is a function of the geometry, as well as the electrode configuration. Results are presented assuming a Wenner array configuration, however they can easily be extended to other geometries, such as the Schlumberger or dipole-dipole array.

Measurements of a materials electrical resistivity can provide useful information for, indirectly determining soil moisture [

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 [

The mathematical formulation of our probe is based on a Wenner [

Experimental results are presented for comparison of the derived, simulated, and measured probe factor.

The probe, shown in

In this derivation it is assumed that the electrical conductivity

where the probe is centered at

with

is shown in

The relationship between the apparent resistivity

where

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

With the ground interface included the expression for the probe factor is modified slightly and is given by,

where

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;

Method | Probe factor |
---|---|

Analytical | 1.365 m^{−1} |

Analytical w/ grounding | 1.382 m^{−1} |

COMSOL Multiphysics | 1.379 m^{−1} |

Experimental Meas. | 1.42 ± 0.677 m^{−1} |

Analytical, numerical, and experimentally derived probe factors for the as-built probe are presented in

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.

Munk, J., Petersen, T., Cullin, M. and Schnabel, W. (2017) Analysis of a Simple Probe for In-Situ Resistivity Measurements. Journal of Water Resource and Protection, 9, 1-10. http://dx.doi.org/10.4236/jwarp.2017.91001

Beginning with the equation of continuity for the current density

The current density at a point is related to the electric field through

Since the probe is sensitive only to the local region surrounding it we assume that

Employing a cylindrical coordinate system, specified by

with

Subject to the boundary condition that the potential vanish at

where

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,

where

Equation (13) is solved making use of the inverse Fourier transform. This gives,

where

Solving for

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

The final task is to relate the measured potential

then by definition,

Hence, the ratio between the measured voltage and applied current

Define

with the probe factor given by,

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

with

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

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