It is important to test water content of rock-soil mixtures efficiently of dry density, pore fluid conductivity, and soil/rock ratio on the relationship. In general, you do not have enough information to do this. A material's permittivity and its conductivity are fundamentally different quantities. We test the two relationships to predict the saturation exponent from the cementation exponent. The first one is.
The frequency dependence of the effective quadrature conductivity shows three domains. Between fp and a second critical frequency fd, the quadrature conductivity is generally well described by a plateau when clay minerals are present in the material.
Clay-free porous materials with a narrow grain size distribution are described by a Cole-Cole model. The characteristic frequency fd controls the transition between double layer polarization and the effect of the high-frequency permittivity of the material.
The Maxwell-Wagner polarization is found to be relatively negligible. For a broad range of frequencies below 1 MHz, the effective permittivity exhibits a strong dependence with the cation exchange capacity and the specific surface area. At high frequency, above the critical frequency fd, the effective permittivity reaches a high-frequency asymptotic limit that is controlled by the two Archie's exponents m and n like the low-frequency electrical conductivity.
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The unified model is compared with various data sets from the literature and is able to explain fairly well a broad number of observations with a very small number of textural and electrochemical parameters. It could be therefore used to interpret induced polarization, induction-based electromagnetic methods, and ground penetrating radar data to characterize the vadose zone. Introduction There are a number of electrical and electromagnetic EM geophysical methods providing nonintrusively information relative to the texture, water content, and interfacial electrochemistry in the shallow subsurface in an extremely broad range of frequencies 1 mHz—1 GHz.
These nonintrusive methods have many applications in hydrogeophysics to assess ground water resources and to monitor ground water quality or for vadose zone hydrogeology and agriculture just to cite few examples [e.
electromagnetism - The relation between permittivity and conductivity - Physics Stack Exchange
That said, there is not a single model able to explain the complex conductivity of unsaturated materials as well as their effective permittivity and their frequency dependences in the broad range of frequencies of interest in geophysics 1 mHz—1 GHz. There is a vast literature regarding this topic in interfacial electrochemistry and colloidal chemistry [e.
At very high frequencies, there are also predictive models describing the high-frequency permittivity of porous rocks. These models are empirically based like the Topp model for soils [Topp and Reynolds, ], based on upscaling methods like the volume-averaging method [Pride, ], or based on the effective differential medium approach or mixing laws see discussion in Jones and Or [ ], Cosenza et al.
Note that with the volume-averaging approach, the different phases are considered symmetrically while with the differential effective medium approach, one phase has to be chosen as the host for instance, grains immersed in a background fluid or fluid inclusion in a matrix.
The situation is much more confused in the intermediate range of frequencies 10 kHz—10 MHz. Effect of Different Dry Densities Figure 5 demonstrates the relationship between water content and the square root of the dielectric constant for samples 12and 5. Figure 6 compares the water content calculated by the TDR method using the calibration equation form of 2 with the directly measured value. Figure 7 shows the result using the Purdue model.
Among the groups, sample 1 with the dry density of 1. The calibration equation was obtained by fitting the data points of sample 5 with a dry density of 1. Relationship between square root of dielectric constant and gravimetric water content at different dry densities.
Comparison of gravimetric water content by TDR method and directly measured values at different dry densities. Relationship of Purdue equation at different dry densities. As shown in Figure 5the value of dry density has an apparent effect on the relationship between gravimetric water content and dielectric constant. The value of water content calculated by the calibration equation obtained for standard compaction sample will underestimate the result for a rock-soil mixture with lower dry density and overestimate the result for a rock-soil mixture with higher dry density.
Taking the calibration curve as the standard, for a given unit sample of rock-soil mixture with a fixed volumetric water content, with increase of dry density, the solid contribution to total dielectric constant increases causing a larger dielectric constant. This leads to the calculated result becoming larger as the actual value of gravimetric water content decreases. The calibration curve, therefore, overestimates the value of gravimetric water content and vice versa.
As shown in Figure 6even after calibration, the gravimetric water content result for the loose sample 1 is not ideal. The largest errors can be 0. The calibration curve, however, has good accuracy when the sample has a dry density close to that of the calibration sample. The above discussion makes it clear that the value of dry density has an obvious influence on 2 calculated result and that the effect can be allowed for by using the Purdue model as the calibration curve.
Effect of Pore Fluid Conductivity Figure 8 demonstrates the relationship between water content and the square root of the dielectric constant for samples 34and 5. Figure 9 compares the water content results when calculated by the TDR method using the calibration equation form of 2 and the directly measured value. The calibration equation was obtained by filling sample 5 with deionized water as the pore fluid.
Relationship between square root of dielectric constant and gravimetric water content at different pore fluid conductivities. Comparison of gravimetric water content by TDR method and direct measured value at different pore fluid conductivities.
As shown in Figure 8the water content value calculated by the calibration equation obtained using the sample with deionized water as the pore fluid overestimates the result for samples with higher pore fluid conductivities. The increase of pore fluid conductivity causes larger sample energy losses under the TDR waveform resulting in a lower dielectric constant, causing the result calculated by the calibration equation to become larger.
Comparing the result of 3 sample with 4no major differences are apparent, indicating that the value of pore fluid conductivity is not sensitive to the energy losses and has a limitation.
Above that limit, the value of the dielectric constant will not vary. This phenomenon was also discovered for MSW material in studies [ 2025 ]. Therefore, for the values of pore fluid conductivity encountered in geotechnical engineering, the influence of pore fluid conductivity can be ignored.
Result of Soil-Rock Ratio The influence of the soil-rock ratio on the relationship between water content and dielectric constant can be attributed to the different dielectric contribution of the various solids and the different structures and compactness of the sample compacted with the same compaction energy levels.
Relationship using the Purdue equation. As can be seen from Figure 10for a mixture with the same type of rock and soil material components, the data points show good coherence for mixtures with different soil-rock ratios. In order to quantify and evaluate the influence of the soil-rock ratio on the calculated water content, a regression analysis was conducted for the data points of sample 5.
Errors for water content of different soil-rock ratio. As shown in Table 5with increase of soil-rock ratio, the values of and SEE for each group do not contain any trends. The errors caused by material differences can be ignored if the calibration model has taken into account the effect of dry density.
As discussed above, the research result shows that the dielectric constant value for a rock-soil mixture is mainly determined by the water content and dry density.
The influences of pore fluid conductivity and soil-rock ratio are relatively small.How to Use Calculated pH from Conductivity to Validate Water Cycle pH Measurements
Relationship between the square root of the dielectric constant and gravimetric water content. As shown in Figure 11the data points for the rock-soil mixture are mainly located below the curve for soils at low water contents. This is because, at this condition, the dry density is the major factor influencing the value of dielectric constant except for the water content. After being compacted, a rock-soil mixture usually has a higher dry density.
The contribution of the solid phase to the dielectric constant is more than that for soils with the same volumetric water content, hence, leading to a larger dielectric constant.
The rock-soil mixture data points, therefore, are located below soil equations curve.
With an increase of water content, the difference is clearly reduced and the water content of the sample becomes the element determining the value of the dielectric constant. The empirical calibration equation for soil materials proposed by Zhao et al.