ÜBER DEN ZUSAMMENHANG ZWISCHEN SCHALLGESCHWINDIGKEIT UND POROSITÄT BEI ERDSCHICHTEN*

Abstract

AbstractThere are various methods of measuring the porosity of rocks. The direct methods are based on the definition of the porosity ratio. The indirect procedures, which have often been adopted purely for economic reasons require interpretation methods that relate certain measured quantities to the porosity ratio.In the present article, the relationship between acoustic velocity and porosity will be examined. The interpretation method most commonly in use today is based on an approximative equation developed by Wyllie, Gregory and Gardner. This equation obtains the travel time of acoustic waves in a porous medium saturated with liquid by linear interpolation with respect to the porosity n between the travel time T0 in the compact medium (n= o) and the travel time T1 in the pure liquid (n= 1).As observed values of the travel time T differ from those that follow from this linear equation, it seems obvious to attempt a higher degree of approximation, e.g. by means of a polynomial of the third degree.Besides T0 and T1 other quantities are needed in order to obtain the coefficients of such a polynomial. These quantities must bear some relation to the physical constants of the rock. Only in the neighbourhood of n= o and n= 1 is it possible to make general statements concerning the behaviour of rocks and, in particular of the skeleton. Therefore, in the points n= o and n= i not only the values of the approximative function T=f(n) itself are given, but also those of its derivative with respect to n. The values of these derivatives ∂T/∂/Tn=0 and ∂T/∂/n‐1 depend on the properties of the material filling the pores and on data concerning the structure of the skeleton and the stresses in it. The considerations leading to a mathematical formulation of this dependence are dealt with in this article.The elastic constants of the skeleton were developed in Taylor series in n, that, for skeletons with low porosity were truncated after the linear term. The coefficients of n are functions of the constants of the material of the skeleton, the shape of the pores and the stresses in the skeleton. These functions were approximately computed for two limiting cases. For a more general application, it is useful to assume that these coefficients are the same for both Young's modulus and Poisson's ratio and then obtain the functional relation between the coefficient and the residual pressure from measurements performed in the laboratory with various kinds of rock. When this relation is established from such laboratory experiments, the cubic interpolation method can be applied.Whereas the linear interpolation method, as it has been in use up to now, presents a relation between porosity and acoustic velocity only, the cubic interpolation method makes it possible to determine the influence of pressure and the state of cementation of the rocks. Comparison with data obtained from direct measurements shows good agreement.