This paper describes the mathematical formulation of a Perkins-Kern-Nordgren (PKN) fracture model, that accounts for the existence of poroelastic effects in the reservoir. The poroelastic effects, induced by leak-off of the fracturing fluid, are treated in a manner consistent with the basic assumptions of the PKN model, by means of a transient influence function. The fracture model is formulated in a moving coordinates system and solved using an explicit finite difference technique. The numerical algorithm has the following features: fixed mesh, adaptive control of the time step, and unconstrained fracture length during shut-in. Numerical simulation with this model indicates that poroelastic processes could be responsible for a significant increase of the treatment pressure, but that they have virtually no influence on the fracture length and fracture width.
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Detournay et al. (1990) studied this question.