an important relationship in the vadose zone is between K and water saturation, so identifying the degree of water saturation in the vadose zone is necessary.

Building on what Pat Quinn already posted. Because the hydraulic conductivity and resistivity (which I am interpreting to mean electrical resistivity) are both related to the degree of water saturation in the vadose zone, for a specific site, one could establish a relationship between the three. The utility of this would be that one could then use measurements of electrical resistivity to estimate hydraulic conductivity and water saturation as it varied over time.

That said, the variation in other parameters from site to site would limit the portability of such a relationship to other sites. For example parameters related to the composition of the rock/soil material, the pore space geometries, and the chemistry of the water in, and moving through, the vadose zone. These and more would be site dependent. Essentially, the required site characterization process would be the equivalent of establishing the relationship between electrical resistivity, hydraulic conductivity, and water saturation at the site.

If all one has is electrical resistivity, any inference about hydraulic conductivity would have to be based on suppositions about the earth materials and water saturation. If one knows the nature of the earth materials and the water saturation, then there are empirically based rules of thumb that allow for rough estimation of the hydraulic conductivity. I suppose that also having the electrical resistivity could be used to sharpen such and estimations. But your question mentioned accuracy. Thus, I am supposing that you want, or need, something a little better than that.

This a topic area where there is a lot of literature. Just a few random examples:

Article Electrical Resistivity—Hydraulic Conductivity Relationships ...

Article Geoelectric Sounding for Estimating Aquifer Hydraulic Conductivity

Article Hydraulic conductivity characterization of a karst recharge ...

Article Estimating Water Content of Soils from Electrical Resistivity

R. Kalinski and W. Kelly, "Estimating Water Content of Soils from Electrical Resistivity," Geotechnical Testing Journal 16, no. 3 (1993): 323-329. https://doi.org/10.1520/GTJ10053J

Article The Electrical Resistivity of the Vadose Zone — Field Survey

Article Electrical Resistivity Tomography of Vadose Water Movement

Article A geostatistically based inver model for electrical resistiv...

Please, see the attached file

I think that article answer your question

Thank you all for the helpful answer!

Regarding the paper posted by A. I. Ammar, as I read through it, it isn't clear what measurements of hydraulic conductivity were made for the study. The abstract and conclusion both mention 15 wells that are shown on the maps, but no other details are given. Perhaps A.I. Ammar can elaborate on how the hydraulic conductivity was measured at the 15 wells, or at any other locations. Or perhaps direct us to the companion paper where this is reported.

The paper does raise the question of what depths were of interest in the original question. I was thinking in terms of 10s of feet to 100s of feet for the vadose zone. The study by A.I. Ammar looked at the top 1.5m. If the area of interest is that shallow, then the characterization of the materials is a much simpler, and less costly, process than I was envisioning. Working shallow opens up a world of opportunities that are much less feasible at greater depths.

important topic

Empirical relation is specific for a certain area , It is not correct to use empirical relation if it is out of your area

The recently developed theoretical models can effectively replace traditional empirical models. For example, capillary fractal model and cylindrical electrical model. They characterized the geoelectric characteristics based on several typical pore morphologies, and there is a clear relationship between them and hydraulic parameters. For example, electro osmotic coupling. I believe this type of problem can be solved very well.