The interaction of tube waves with borehole fractures, Part II: Analytical models

Abstract

We develop a series of analytical models that can be used to interpret Stoneley‐wave reflections from fractures intersecting a borehole, thus facilitating the determination of effective fracture apertures from logs. The first model considers the combined effects of borehole enlargements (e.g., washouts) and fractures on the reflection coefficient of Stoneley waves. The result is expressed in terms of the washout volume, which can be obtained from a caliper log, as well as the fracture width. The predictions of this model are in excellent agreement with finite‐difference calculations. Next we develop an analytical elastic model that generalizes the rigid formation model. It also agrees with finite‐difference calculations. Finally, we establish the equivalence between a closely spaced multiple fracture and a permeable medium. In all the examples, the generalization of the result for the single fracture in a rigid formation can be accomplished with standard logging measurements.