Responsibilities for materials reliability and safety : G. P. Smedley. Metallurgist, Materials Technologist p. 505 (September 1979)

Abstract

Variations of the mechanical and transport properties of cracked and/or porous rocks under isotropic stress depend on both the confining pressure (Pc) and the pore-fluid pressure (Pp). To a first approximation, these rock properties are functions of the differential pressure, Pd = Pc − Pp; at least for low differential pressures. However, at higher differential pressures, the properties depend in a more complicated way upon the two pressures. The concept of effective pressure, Pe, is used to denote this variation and it is defined as Pe(Pc,Pp) = Pc − n(Pc,Pp)Pp. If n = 1 (and therefore, is independent of Pc and Pp), the effective pressure is just the differential pressure.We have used an asperity-deformation model and a force-balance equation to derive expressions for the effective pressure. We equate the total external force (in one direction), Fc, to the total force on the asperities, Fa, and the force of the fluid, Fp, acting in that same direction. The fluid force, Fp, acts only on the parts of the crack (or pore-volume) faces which are not in contact. Then, the asperity pressure, Pa, is the average force per unit area acting on the crack (or grain) contacts Pa = Fa/A=Fc/A−Fp/A= Pc − (1 −Ac/A)Pp, where A is the total area over which Fc acts and Ac is the area of contact of the crack asperities or the grains. Thus, the asperity pressure, Pa, is greater than the differential pressure, Pd, because Pp acts on a smaller area, A−Ac, than the total area, A. For elastic asperities, the area of contact Ac and the strain (e.g., crack and pore openings) remain the same, to a high degree of approximation, at constant asperity pressure. Therefore, transport properties such as permeability, resistivity, thermal conductivity, etc. are constant, to the same degree of approximation, at constant asperity pressure. For these properties, the asperity pressure is, very accurately, the effective pressure, Pc.Using this model, we find that the dynamic (undrained) elastic modulus (Mcr) of saturated cracks (rocks) at low effective pressure is given by Mcr = (1 − PpA′f)Ma + (1 − Af)Mf = −wdPc/dw, where Ma is the (dry-matrix or crack-) asperity modulus, Mf is the fluid's modulus, Af is the fractional area of contact (Af = Ac/A), A′f =dAf/dPa and w is a measure of the crack or pore openings. This simple model accounts for the dependence of the rock modulus (and elastic velocity) on: (1) the elastic properties of the fluid, (2) the elastic properties of the dry rock, and (3) the pore-fluid and confining pressures. Explicit expressions depend upon the choice of the asperity- (or grain-) deformation models and their contact distribution functions. The effective pressures for transport properties are different than the ‘effective pressure’ for the mechanical properties. Calculated results based on the ‘bed-of-nails’ model having power-law (or fractal) asperity-height distribution functions can be fitted quite well to experimental data with a minimum of fitting parameters.