Direct Inversion of Measured Hydraulic
and Electric Transport Properties
into a Geometrical Rock Model
The prediction of the interior structure of sedimentary rocks from measured bulk physical data is an inverse problem with a huge number of unknowns, which can be solved only approximately using the Least Mean Squares Error, the Maximum Entropy or the Tikhonov Regularization methods. However, as has been shown by the success of the Cole-Cole model of the Induced Potential in rocks, or the Thomeer model of Mercury Injection, there are cases when an equivalent rock model with a few degrees of freedom can very well describe the behavior of a complex geologic system.
In this research we have shown that:
(a) The theory of Nabil Akbar (1994) provides an equivalent rock model which is able to reproduce the measured hydraulic and electric transport properties,
(b) The model has only three parameters (average radius r, average distance between nearest pores d, average throat radius d)
(c) These three parameters can be directly determined from the measured porosity f, hydraulic permeability k and cementation exponent m. of the rock, using simple analytic expressions.
The derivation of these formulae is based on effective medium theory (Yonezawa & Cohen, 1983) and on Perez-Rozales' (1982) non-standard theory of electric conduction in rocks.
Examples will be presented for the direct inversion of carbonate rock measurements. The mathematically derived rock model will be shown to well agree with the structure seen in thin sections under the microscope.