At low frequencies, the fast compressional and shear wave velocities in poroelasticity are given by
v_+^2 = H/ and v_s^2 = /. We will assume that the dispersion (dependence of velocities on frequency) is negligible over the range of frequencies considered. Then, the adiabatic moduli can be determined from measured sound velocities according to the standard formulas (from linear elasticity)
= v_s^2 and K_u = (v_+^2 - 43v_s^2). When acoustic measurements are made on the porous medium while drained of fluid, we can also find the drained modulus Kd from the same formulas. The theory says that the shear modulus should be unaffected by the presence or absence of the fluid. Thus, changes in shear velocity should be due to the increase in density when a fluid fills the pores, but experience has shown (Gregory, 1976; Mavko and Jizba, 1991) this is not always true for low porosity rocks with microcracks.
Dividing (BandKs) through by , we find
1K_u - K_d = a + b_f, where the fluid compressibility is and the intercept a and slope b are given, respectively, by
a = (/K_s - /K_)/^2 and
b = ^2. The inversion problem then reduces to fitting modulus data to a straight line according to (linearfit), and subsequently computing Ks and from the slope and intercept formulas (b) and (a).