Mathematical model of rock shaliness.

The shales contain much higher concentration of radioactive minerals comparing to clean sands and carbonates (see Table 1 below).

This is why the most common way to quantify the shale content is the intensity of the natural gamma-ray (GR) emission.

The fist step is to normalize the actual GR-tool readings  to the reference values in clean rocks   and pure shales  which is called Shale Index: 

I_{GR}(l) = \frac{GR_{log}(l) - GR_m}{GR_{sh} - GR_m}

where  – along-hole depth.

The model parameters   and    are calibrated for each facies individually.

The shale Index  is varying between 0 (for non-shally rocks) and 1 (for pure shales) but the actual shaliness may behave non-linearly between these extremes (especially for shallow, young reservoirs). 

This can be calibrated based on the available core data.

The table below summairzes some popular shaliness models:

The graphic image of different shales volume models is brought on Fig. 1.

References