Imaging permeability structure within the highly permeable carbonate earth; inverse theory and experiment

In most oil reservoir rocks, the squirt flow characteristic frequency (f (sub sq) ) range of 10-100 kHz is much less than the Biot characteristic frequency (f (sub 0) ) range of 1 MHz-1 GHz. In contrast, carbonate aquifers have very high permeability (10-3000 d) and the two characteristic frequencies are reversed: f (sub 0) from 10 Hz to 10 kHz and f (sub sq) from 1 MHz to 100 MHz. The principal objective of this paper is to develop a pilot inverse method for robust imaging of the high-permeability structure within carbonate aquifers. An analytical approximate model called the super-k model is developed. The super-k model coincides numerically with the combined Biot and squirt-flow mechanism model when permeability is higher than 100 md and the frequency is lower than 100 kHz (the super-k regime). In the super-k regime, the pore fluid is always relaxed so that the attenuation due to the Biot mechanism is roughly four times larger than that of the Biot model. Also, empirical equations are developed that relate the stiffness and rigidity of the skeletal frame and porosity of limestone to compression and shear wave velocities measured from ultrasonic data. The super-k model is combined with the empirical elastic equations to derive a robust high permeability inverse model. Use of the super-k inverse model is illustrated in an acoustic crosswell test section of a limestone aquifer between depths of 300 and 480 m over a horizontal width of 11 m. The acoustically imaged permeability, constructed from 4-kHz velocity and attenuation tomograms, shows excellent agreement with the permeability data obtained hydraulically from pump tests and packer tests performed at four different depth intervals within the test section. The permeability image reveals that the high permeability channels run fairly randomly within the limestone aquifers.