Introduction to the theory
The electrical permittivity of the bulk soil, e b, has been found to be a function of both soil water content, q , and the permittivity of the pore water e p (e.g. Nyfors and Vainikainen, 1989). In a similar way, the electrical conductivity of the bulk soil, s b, is a function of both q and the pore water conductivity s p.
Malicki et al. (1994) found a high degree of linear correlation between values of s b and e b for a broad range of soil types.
The following discussion proposes a theoretical basis for the relationship between s b and e b, and explains how this is used within the SigmaProbe to derive readings of pore water conductivity.
Bulk Soil Conductivity versus Pore Water Conductivity
The relative electrical permittivity of a dielectric material, e , is a complex quantity expressing the material’s response to the polarising effect of an applied electric field (E-field). It is defined as:
[1]
[2]where s
I is the specific ionic conductivity of the material and w
is the radian frequency in rad s-1. The frequency in Hz of the
applied E-field is 
.
The permittivity for free space is e
0 = 8.854 10-12 F m-1.
Now consider the water that can be extracted from the pores of the soil matrix. The permittivity and conductivity of the pore water will be denoted by the subscript p. The imaginary part of the complex permittivity of the pore water is e ''p. In soil science it is more practical to use the conductivity of the pore water, s p, which can be defined as:
[3]where s ip represents the ionic conductivity of the extracted pore water. Dielectric losses are frequency dependent and have a maximum at the relaxation frequency. The relaxation frequency of water is 17 GHz at 20°C (Kaatze and Uhlendorf, 1978). The operating frequency of the SigmaProbe is 30 MHz, and at that frequency e ''dp is negligable, so Eq. [3] can be reduced to:
[4]
Usually s
p is referred to as the EC (Electrical Conductivity)
of the pore water. Ionic conduction is a function of temperature. In the
case of a NaCl-water mixture, the conductivity increases by ~2.25 % per
°C. The values quoted for s
p are often corrected for temperature dependence to a temperature
of 20°C (or sometimes to 25°C). This temperature correction depends
on the ionic composition of the solution, and is not applied automatically
by the SigmaProbe.
The complex permittivity of the pore water, e p, is equal to that of pure water. The real part of the complex permittivity of the pore water e 'p = 80.3 at 20°C, with a temperature coefficient of about -0.37 per °C (Kaatze and Uhlendorf, 1981). By analogy with Eq. [1] we can write the following approximation for e p:
[5]The permittivity and conductivity of
the bulk soil will be denoted by the subscript b. The complex permittivity
of the bulk soil, e
b, is proportional to both e
p and a function of q
, g(q ). For
dry soil there is no water to facilitate ionic conduction, so the conductivity
of the bulk soil s
b »
0. However dry soil material is still polarisable, so 
and 
appears
as an offset to e
b.
By assuming that g(q ) takes into account the proportionality constant, it is reasonable to postulate the following form for the complex permittivity of the bulk soil:
[6]Note that is
a complex value and includes dielectric and ionic loss. However since s
b=0, we may approximate by
its real part 
. With this
and Eq. [5] substituted in Eq. [6], e
b can be written as:
[7]
An electrical model for a dielectric material such as soil between two electrodes is a lossy capacitor. We calculate the admittance, Y, of this soil-filled capacitor. The admittance is the inverse of its impedance, Z, and is a complex quantity which is proportional to e b of the bulk soil, and can be defined by:
[8]The equivalent circuit for such a lossy capacitor is a loss-free capacitor, C, with a conductor, G, in parallel. C represents the energy storage capability of the soil and is related to e 'b. G represents the energy loss and is related to s b. Y may be written in terms of C and G as:
[9]From Eq. [8] and Eq. [9] and with Eq. [1] to Eq. [7] in mind, the real and imaginary parts of Y can be found:
[10]and
[11]In terms of the measurable bulk quantities s b and e 'b:
[12]and
[13]From Eq. [12] and [13] the ionic conductivity of the pore water can be written as:
[14]
The model of Eq. [14] describes the
relationship between s
p of the pore water (the water that can be extracted from the
soil) and the values e
'b and s
b as measured in the bulk soil using a dielectric sensor. The
offset 
can be calculated
from the e
'b and s
b values measured at two arbitrary free water content values.
 |
 |
,
for two different soils.
The relationship between the the bulk
soil parameters e
’b and s
b and the corresponding pore water parameters e
’p and s
p is different when the the water present is bound to the soil
matrix rather than free water. The model of Eq. [14] cannot be used for
the conductivity due to ions moving through the lattice of ionic crystals
in a dry or almost dry soil. Therefore, the model is only valid for the
free water in the matrix. Thus 
is
not the value for e
'b if q
= 0. For sand the free water content corresponds to q
> 0.01 but for clay it can be q
> 0.12 (Dirksen and Dasberg, 1993). As a rule of thumb the model applies
for most normal soils and other substrates used for growing, such as rockwool,
if q
> 0.10.
The SigmaProbe
In Eqs [12] and [13], only the term g(q ) is affected significantly by the shape of the electrodes, by the contact between the electrodes and soil, and by the soil composition. This term is eliminated in Eq [14] due to its ratiometric form, and so measurements of pore water conductivity based on this equation are relatively insensitive to contact problems. The SigmaProbe has been developed to exploit this technique by making simultaneous readings of e 'b and s b within a small sampling volume and at the same frequency.
The probe is built around an ASIC developed specifically for dielectric sensors at IMAG-DLO. This operates as a vector voltmeter to make precision measurements of e 'b and s b, as shown in the following diagram:
Fig 2. Schematic
representation of the ASIC circuitry for measuring dielectric properties
| The small size of the ASIC enables all the electronics to be encapsulated in the handle of the SigmaProbe, which is designed for easy insertion into the soil. Detailed specifications of the SigmaProbe can be found in the "Water Relations" catalogue from Delta-T Devices. |  |
The model of Eq. [14] was evaluated for five different soils, glass beads of 0.2 mm diameter and a slab of rockwool. The soils were samples from the Dirksen and Dasberg (1993) experiment. Their compositions are listed in Table 1. The salinity of the soil samples were not changed.
The salinity of the rockwool slab and the glass beads were adjusted to s p = 0.3 S m-1 and s p = 0.1 S m-1 respectively, using water-NaCl solutions. Sufficient water was left on top of the saturated samples to measure s p of the soil solution. The SigmaProbe was used for the measurement of e 'b and s b, and of s p in the water left on top of the samples. The measurement of s p was checked using a laboratory 4-electrode conductivity meter at 1 kHz.
Each material was dried in ten steps
by slowly extracting solution from an initially saturated and thoroughly
mixed sample. In this way 10 water contents between q
= 0.10 and saturation were created. The change in q
was measured using a balance. Since the salinity of the pore water was
not allowed to change with q
, drying by evaporation was avoided. The measured s
p values are listed in the eighth column of Table
1. The average values with their standard deviations for s
p, at the ten q
steps, calculated according to Eq. [14] are listed in the last two columns.
The seventh column lists the measured s
p of the pore water extract. Comparison of the s
p values measured in the soil solution and the s
p values calculated from e
'b and s
b, justifies the model of Eq. [14]. The values found for 
at
which s
b = 0, are listed in the sixth column.
Table 1. Soil composition and validation results.
= 4.1 can be used as an average. In this case, calibration of the sensor
for s
p is not required.
