# Matrix, bulk and dry bulk density

Especially for borehole logging, it is important to note there are different definitions of density. The density (or more precisely, the volumetric mass density *ρ*), of a substance is its mass *m* per unit volume *V*:

The difference between matrix density, bulk density and dry bulk density is caused by different definitions of either the mass or the volume. The key in these differences is the porosity *φ*, which is the volumetric void space in the rock, or the space not occupied by solid materials.

**Matrix density** is defined as the density of the rock formation. The volume used to calculate the matrix density excludes pore space, the mass does not include the fluids in the pores.

**Bulk density** is the density of the mixture of rock and fluid. Both mass and volume include fluid. Normally, this density is used when activity concentrations are calculated using gamma spectrometry.

**Dry bulk density** is defined as the density with the fluid removed. The mass does not include the fluid, but the volume does include the pore space.

Formula's to relate different definitions of density are shown below:

## Example

For example, suppose we have a rock formation with matrix density 2.4 kg/l (i.e. when there is no void space, 1 liter of rock material would weigh 2.4 kg). Suppose the pore space is 25% (*φ* = 0.25), which is filled with water. In other words, 75% of any *volume* is occupied by rock, and 25% is occupied by water (with *ρ _{fluid} *= 1). In this case, the matrix density is 2.4 kg/l, the dry bulk density is 1.8 kg/l and the bulk density is 2.05 kg/l.

## Gamma spectroscopy

The activity concentration collected with a gamma spectrometer would refer to the activity concentrations for bulk density. If we assume all radioactive material is located in the rock formation and not in the fluid, we can convert the activity concentration to another density definition using the density values:

Suppose a borehole measurement in the rock formation from the example above returns an activity concentration of 100 Bq/kg U^{238} for the bulk. The activity concentration for the matrix would be 100 * 2.4 / 2.05 ≈ 117 Bq/kg.

## Medusa calibrations

Medusa calibrates borehole spectrometers against a bulk density of typically 2.32 kg/l, with 0 porosity (i.e. the dry bulk density and matrix density are the same as the bulk density). The borehole itself is empty (which is comparable to an air-filled borehole). For an empty borehole, the diameter of the borehole has no influence on the measurement. A measurement with a tool in an empty, 10 cm diameter borehole would yield the same results as a measurement with the same tool in an empty, 20 cm diameter borehole. Similarly, the density does not affect the measurement results as long as there is no water in the pore space of the material (i.e. when bulk density equals dry bulk density). In material with lower bulk density, there is less radiation from the same volume, but radiation from a larger volume would be able to reach the tool, yielding the same results.

In other words: **The Medusa calibrations are valid for any situation where the borehole is empty and where there is no water in the matrix material.** For all other situations, borehole corrections should be applied. The activity concentrations one would get for a borehole measurement, are always with respect to the bulk density. As shown before, conversion formula's can be applied to convert results to other definitions of density.

## Borehole corrections

In a water or mud filled borehole, the measurement is heavily affected by the fluid inside the borehole. This fluid will absorb radiation from the matrix, while the material itself might also contain radioactive materials. The diameter of both the borehole and tool determine the amount of borehole fluid, and therefore have an effect as well. Casing surrounding the borehole absorbs radiation from the material and needs to be taken into account. The borehole correction module in either Gamman or the GammaBase DLL can be used to compensate for all these parameters.