Tube-Wave Interaction with a Fluid-Filled Circular Fracture of a Finite Radius

Tube waves in boreholes are used for characterizing formation properties and hydraulic properties of fluid-filled fractures and permeable zones intersecting wellbores. At low frequencies there is a well-known approximate formalism describing reflection/transmision of tube waves on layer boundaries, infinite fluid-filled fractures and small-diameter washouts. However for fractures or washouts of finite size one can only use numerical methods such as finite difference that are time-consuming and do not provide physical insights.<br>Here, we extend existing formalism to analyze reflection/transmission of tube waves on a circular fluid-filled fracture/washout of an arbitrary finite size. We break the problem into three tasks. First, conversion of tube waves into guided waves is modeled using method of Kostek et al. (1998). Second, we derive new analytical solution for reflection of diverging guided slow wave from a fracture tip. Finally, we derive conversion coefficient describing transformation of imploding guided into tube waves in a borehole. Combining three solutions, we obtain simple analytical representation of a total wavefield in the borehole as a superposition of upgoing and downgoing tube waves generated at the fracture intersection and borne by incoming guided waves and their multiples. New solution is in good agreement with finite difference computations.