An exact effective stress law for elastic deformation of rock with fluids

The exact expressions for the effective stress 〈σij〉 and, in particular, pressure 〈P〉 that cause elastic strain of material with pore fluids are, assuming only that Hook's law is valid, 〈σij〉 = σij - αPδij and 〈P〉 = Pc - αPp, where α = 1 - (K/Ks), Pc and Pp are confining and pore pressures, and K and Ks are the bulk moduli of the rock and grain, respectively. The equation for 〈P〉 was first suggested by Geertsma (1957) and by Skempton (1960) on empirical grounds. The expression does not depend directly on porosity, but when pores vanish the effective pressure 〈P〉 equals the confining pressure Pc, because then K=Ks. Thus the strain of a porous solid with pore pressure can be completely determined from the elastic modulus of the solid without pore pressure, if the effective stress law in the equation for 〈σij〉 is used. The exact expression for the effective stress describes quite accurately the measured strains in sandstone and granite samples at confining and pore pressures to 2.5 kb. The results are not applicable to inelastic processes, such as fracture, or elastic processes other than strain.