Axisymmetric modal propagation in cased boreholes depends on the bonding between the casing and the backing medium, either cement or formation. In the zero frequency limit, if some tangential stress is transmitted from the casing to the backing medium (by a perfect or partial bond), there is no propagating casing wave. There is only a single tube wave and its speed is easily derived from static analysis. When the shear stress on the back surface of the casing is zero, either due to perfect lubrication (accounted for by the boundary condition) or the presence of a fluid annular layer behind the casing. the casing is said to be unbonded and the. problem must be approached dynamically. To analyze the physical behavior of both the tube and casing wave, consider the canonical situation in which both arise, the case of a perfectly lubricated fluid-filled casing fit snugly into a borehole in a homogeneous isotropic formation. The dispersion relation for arbit rary frequency is derived from the boundary conditions on the casing; it is the condition that the determinant of constants in the general solution vanish. Modal phase and group velocities along with associated mode shapes may be calculated over any frequency range. In the zero frequency limit, the dispersion relation becomes a quadratic equation, giving explicit formulae for the speeds of the casing wave and of the predominantly fluid borne tube wave. These exact zero frequency solutions, which clarify the contribution of the various elements in the system to the behavior of the waves, are neither plane strain nor plane stress solutions. Under the usual conditions of thin fast casing, the tube wave speed is very close to that found from a static approach assuming plane stress in the casing; the casing wave phase speed is slightly less than the extensional plate wave speed in the casing medium. The dispersion relation for partially bonded casing at small frequency depends on the zero frequency relation for unbonded casing.


Article metrics loading...

Loading full text...

Full text loading...

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error