Nowadays, hydraulic fracturing is largely used to produce energies (oil, gas and thermal energy) from impermeable formations. It requires the prediction the fracture shape and size that is a huge challenging problem of geomechanics. The most famous solutions to such problem are those of the PKN model that was developed by Perkins and Kern (1961) and Nordgren (1972). These solutions are usually considered as industrial standard to validate the numerical simulations (Detournay et al., 1990; Hossain et al., 2008).

Below, an online tool is provided, on the basis of the PKN model (see also Economides and Nolte, 2000),
for modeling the length L and width W of a fracture progagating in a 2D plan, that is parallel to the borehole's axis and perpendicular to the direction of the minimum borehole tangent strain (see the module
stress around a vertical wellbore or the module
stress on a horizontal borehole) within a rock layer of a given height H under a fluid injection rate Q_{0} at the fracture's inlet (see the figure below).
The injected fluid is assumed to be a Newtonian fluid that is characterized by a viscosity μ. The rock surrounding the fracture is assumed to be isotropic with given Young modulus and Poisson ratio. Fracture's faces are assumed to be impermeable (i.e. no fluid loss through the fracture's faces).

**References**

Perkins, T. K., & Kern, L. R. (1961). Widths of hydraulic fractures. Journal of Petroleum Technology, 13(09), 937-949.

Nordgren, R. P. (1972). Propagation of a vertical hydraulic fracture. Society of Petroleum Engineers Journal, 12(04), 306-314.

Economides, M., J., & Nolte, K. G. (2000). Reservoir stimulation (Vol. 18). M. J. Economides (Ed.). New York: Wiley.

Detournay, E., Cheng, A. D., & McLennan, J. D. (1990). A poroelastic PKN hydraulic fracture model based on an explicit moving mesh algorithm. Journal of energy resources technology, 112(4), 224-230.

Hossain, M. M., & Rahman, M. K. (2008). Numerical simulation of complex fracture growth during tight reservoir stimulation by hydraulic fracturing. Journal of Petroleum Science and Engineering, 60(2), 86-104.

**Propagation of a vertical fracture by injecting a fluid at the fracture inlet on the borehole wall.**