Locating center of gravity in geological contexts

Abstract

This short article at first deduces center of gravity (C.G.) for horizontal single-layered and multi-layered rock bodies. Spatial variation of density is also incorporated using geologically realistic density–depth linear and exponential algebraic relations. These are the two well-known depth-wise density variation patterns. The derivations are then extended for dipping rock bodies. Location of C.G. depends on the (i) dimension (length, width and thickness) and (ii) dip amount of the geological body, (iii) the densities of the solid matrix and the pore fluid of one or more than one layers, (iv) gradient of density change with depth and along other two perpendicular directions-for individual layers, and for sedimentary units (v) on the porosity at the surface and (vi) the compaction constant. This work would be useful in gravity- and isostatic studies, and in deciphering stabilities of crustal blocks.

Appendix

The minor derivation in Mukherjee (2017a) is redone

Consider a porous sedimentary layer, and that the pore space is filled up by a single phase (Eq. 2–1 in pp. 32 of Rieke and Chilinganian 1974):

$${\rho _{bw}}={\text{ }}{\rho _m}-{\text{ }}\left( {{\rho _m}-{\text{ }}{\rho _f}} \right)\emptyset ,$$

(29)

where ρ _bw is the wet bulk density of sediments or the rock, ρ _m is the matrix or mineral or grain density, ρ _f is the fluid density, and ${\emptyset}$ is the porosity expressed as a fraction. If Athy’s (1930) empirical law works for the porosity reduction with depth in an exponential manner

$${\emptyset _z}={\emptyset _0}{{\text{e}}^{ - {\text{b}}z}},$$

(30)

where ${\emptyset _z}$ is the porosity at depth $z,\,{\emptyset _z}$ ₀ is the porosity at surface: z = 0, e is the exponential series, and b: a constant such that b ⁻¹ = λ is compaction constant.

Using the above two eqns,

$${\rho _{{\text{bw}}}}={\text{ }}{\rho _{\text{m}}}-{\text{ }}({\rho _{\text{m}}}-{\text{ }}{\rho _{\text{f}}}){\emptyset _0}{e^{ - {\text{b}}z}}.$$

(31)