Hydraulic fracturing leads to a stationary flow in porous media, hence a model that takes inertial phenomena into account is examined. In order to simulate the incompressible fluid in a fluid-driven fracture, a nonlinear Darcy-Forchheimer (DF) equation is used with mixed boundary conditions. With the inclusion of a growth exponent m and inhomogeneous coefficients, the traditional DF equation is generalised. The well-posedness theorem is established for arbitrary m > 1 by using a mixed variational preparation of the problem with uncertain fluid velocity and fluid pressure. Optimal fracture shape design is aided by the proposed Lagrange multiplier formalism.

Phase-field theory, Hydraulic fracture propagation, Non-Darcy flow, Nonlinear deformations