Cased Borehole Effect on Downhole Seismic Measurements

Approximate and exact formulations are presented for the interaction of an incident wave with a cased borehole. In the approximate method, the borehole coupling theory is used to compute pressure in the fluid at a low frequency. The results are simple and explicit. They are useful in the study of cased borehole coupling and as well as borehole radiation. In the exact method, elastic potentials in each annulus are represented as a superposition of fundamental solutions to the Helmholtz equations. Continuity of displacements and stresses across layer boundaries are used to determine unknown coefficients. The global matrix method is employed to simultaneously compute these coefficients in individual layers. This method is advantageous over the ThomsonHaskell propagator matrix method in handling evanescent waves. Our results show that, in a cased borehole, the borehole effects on downhole seismic measurements are more significant than those in an open borehole, especially when the formation is soft and the casing is steel. For hard formations and frequency below 1 kHz, cased borehole influence on downhole geophone measurement is minimal, while at high frequencies, large discrepancies occur, especially at grazing incidence. For soft formations, both the pressure in the fluid and the solid displacement on the borehole wall show strong dependence on frequency and incidence angle, even at very low frequencies. Strong resonance occurs in the fluid for an SV incidence at angle 8 = cos-f3!CT where CT is the tube wave velocity in a cased borehole. This resonance is prominent even at a very high frequency and large incidence angle because the tube wave velocity is raised well above the formation shear velocity by the steel pipe. This behavior is very different from that in an open borehole. At a particular angle of incidence of a plane P wave, the pressure in the fluid is near zero at low frequencies. This angle is dependent on the casing thickness and can be computed exactly. In general the casing behaves like a shield in such a way that the amplitude of both pressure in the fluid and solid motion on the borehole wall are reduced compared to those in an open borehole. 138 Peng et a