Journal of Vibration Testing and System Dynamics
Mixed Variational Problem for a Generalized DarcyForchheimer Model Driven by Hydraulic Fracture
Journal of Vibration Testing and System Dynamics 7(1) (2023) 1521
 DOI:10.5890/JVTSD.2023.03.003
Victor A. Kovtunenko
Institute for Mathematics and Scientific Computing, University of Graz, NAWI Graz, Heinrichstr.36,
8010 Graz, Austria
Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences,
630090 Novosibirsk, Russia
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Abstract
The model of a stationary flow in porous media stemming
from hydraulic fracking and accounting for inertial phenomena is considered.
The incompressible fluid is modeled by a nonlinear DarcyForchheimer (DF) equation
under mixed boundary conditions, which are appropriate for a fluiddriven fracture.
The classical DF equation is generalized with respect to a growth exponent $m$
and inhomogeneous coefficients.
Using mixed variational formulation of the problem for unknown fluid velocity
and fluid pressure, the wellposedness theorem is proved for arbitrary $m>1$.
The developed Lagrange multiplier formalism is advantageous
for optimal shape design of fractures.
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