While there are no geophysical methods that can directly measure hydraulic conductivity, there are several methods that can provide indirect estimates. However, hydraulic conductivity is a highly variable, isotropic and often heterogeneous parameter and is thus very difficult to estimate in field conditions. Many studies have reported success in estimating spatial distributions of hydraulic conductivity from geophysical measurements, however there is no "accepted" approach or method for doing this. For this reason, estimating hydraulic conductivity from field geophysical measurements is considered the holy grail of hydrogeophysics.
Numerous methods have been used to estimate hydraulic conductivity, but the most relevant for near surface applications are electrical resistivity tomography (ERT), self potential (SP), induced polarization (IP), and nuclear magnetic resonance (NMR).
The electrical methods (ERT, SP, IP) provide a particularly intuitive example of how geophysics can be used to look at hydraulic properties of the subsurface. Electrical current flow primarily occurs through the pore space in soils and sediments (with the notable exception when electronically conductive minerals are present), which is analogous to how fluids flow through the subsurface. Blocked off or isolated pores that do not conduct fluid flow are also electrically disconnected, so these pores will not contribute to the measured signal from the electrical method, and narrow pores will cut off electrical current flow in a similar manner to how they cut off fluid flow.
NMR is a relative newcomer to the hydrogeophysics community and is unique as it is directly sensitive to hydrogen protons in pore fluid (this can be in water or in hydrocarbons, such as NAPLs). In addition to quantifying water content, NMR signals are sensitive to the size of the pores containing the pore fluids, which is the basis for estimating hydraulic conductivity from NMR.
Based on the physics behind each method, rock physics relationships can be developed that can turn a measured geophysical signal into hydraulic conductivity. This can be as simple as an empirical model that, for example, links measured electrical resistivity to hydraulic conductivity by some power law (K=A*rho^B, where K is estimated hydraulic conductivity, rho is measured resistivity, and A and B are fitting constants). This is the approach used by the oil industry to derive the Schlumberger-Doll Research equation, which relates NMR parameters to hydraulic conductivity and has been used for decades in reservoir characterization. An alternative approach is to use a mechanistic model are extremely complex and may not provide a significant improvement on a well-calibrated empirical model.
The primary advantage of empirical models is their simplicity: anyone can apply an empirical model if they have the correct input parameters. However, the simplicity of empirical models comes at the expense of universality; by simplifying a model to a handful of parameters, it becomes difficult to apply it to a broad range of geological media. This is because any geophysical measurement is sensitive to numerous properties of a geological medium (more on this later), many of which may not impact the hydraulic conductivity. An empirical model that works in a predominantly sandy-silty aquifer may fail completely in a clay-rich sediment or a sandstone. For this reason, empirical models typically need to be calibrated to a specific field site.
Calibrating empirical models involves comparing measured geophysical parameters (e.g. electrical resistivity, seismic velocity, NMR relaxation time) with direct measurements of hydraulic conductivity. These tests can be done at the field scale, but it is more common to calibrate a method by extracting samples of the relevant geological medium and running laboratory tests. While laboratory tests are far more accurate, such samples are necessarily very small and may not represent larger-scale geological properties or heterogeneity. This is the problem of scaling: how to relate geophysical measurements in a lab to measurements in a borehole to surface measurements. This is an unsolved problem in geophysics and while laboratory calibration is often an important step for interpreting geophysical field data, calibrating field models using laboratory measurements may not work at some sites.
Furthermore, each geophysical method faces its own set of drawbacks and limitations that MUST be understood before the method is applied. For example, ERT measurements are not ideal for estimating hydraulic conductivity in many systems because the measurements are also highly sensitive to things like subsurface temperature, saturation, pore fluid conductivity/chemistry and mineralogy. IP, a similar method to ERT, is less prone to errors due to pore fluid chemistry variation in the subsurface, but measurements are far more difficult and time consuming. Still, IP measurements are highly sensitive to variations in hydraulic conductivity in lab studies and there have been several field studies that show good hydraulic conductivty estimates from IP measurements. NMR is a very promising method, both as a borehole tool and as a surface instrument. However, NMR signals are also controlled by mineralogy, in addition to pore size. More importatnly, NMR measurements are completely insensitive to anisotropy.
Ultimately, we are not at a point where geophysicists can go into any field, deploy their instruments, and come away with good estimates of hydraulic conductivity. While geophysical methods may not be able to provide hydraulic conductivity estimates on their own, they may offer great benefits as complements to traditional hydrogeological measurements, offering better spatial resolution and the ability to identify small-scale heterogeneity in the subsurface. One potential avenue is the use of statistical inversion algorithms where hydrogeological and geophysical data are jointly inverted to provide more robust estimates of subsurface properties, such as hydraulic conductivity.