Response of a vertical hydraulic fracture intersecting a poroelastic formation bounded by semi-infinite impermeable elastic layers

The transient response of a vertical fracture, uniformly pressurized by a fluid, is analyzed. The fracture extends over the full height of an horizontal poroelastic layer bounded by two semi-infinite impermeable elastic regions. The study focuses on the calculation of the fracture volume Af(t), given an arbitrary pressurization history pf(t). The first step in solving this general problem is to reduce it to a superposition of two modal responses, corresponding to the sudden application of a constant normal stress and a constant pore pressure in the fracture. In a second step, the transient variation of the fracture volume (response function) for each of the loading modes is determined numerically using a finite-element formulation. These modal response functions are first calculated for the particular case where the drained elastic coefficients of the permeable layer are the same as those of the impermeable bounding barriers. The results are then extended to the more general case characterized by a contrast of elastic properties between the permeable and impermeable layers. The factors influencing the fracture response are examined, with special attention being given to the role of the poroelastic constants, and the stiffness ratio between the layers. This work is applicable to the PKN model for hydraulically-driven fracture propagation. It is concluded that poroelastic effects related to this model can be both greater than previous estimates would suggest and occur more rapidly if the permeable layer of rock is bounded by stiffer layers.