11:00 am Wednesday, March 19, 2008
Mathematical Physics Seminar: Mathematical frame-work for non-linear Forchheimer flows in porous
media and applications by Akif Ibraguimov (Texas Tech) in RLM 12.166
In this talk, I will present some recent results on non-linear flows in porous media. These results were obtained in collaboration with E.Aulisa, M. Toda , J.Walton , A.Cakmak, A.Solynin, and P.Valko. The work is focused on certain theoretical aspects of non-linear non-Darcy flows in porous media, and their applications to reservoir and hydraulic engineering. The goal of this study is to develop a rigorous framework in order to study the dynamical processes associated to non-linear Forchheimer flows for compressible fluids.=20 It was shown that the so-called Forchheimer equation can be modeled as a generalized Darcy equation, where the permeability tensor depends on the gradient of pressure. This allows an essential simplification of original problem, and reduces the PDE system involving the velocity vector, pressure and gradient of pressure to one single quasi-linear parabolic equation of second order. We have proved an existence and uniqueness theorem for the so called pseudo-steady state (PSS) solution of the specific initial boundary value problem, which models a particular regime of the well performance in the reservoir. This solution appeared to be time-invariant with respect to a certain functional - namely, diffusive capacity. Moreover, the PSS solution serves as a global attractor for a class of IBVP with arbitrary initial data. In addition, it was observed that the PSS solution can be treated as a surface with prescribed constant mean curvature. On the other hand, the diffusive capacity brings a connection between two very different characteristics: the productivity index of the well, and the so-called torsional rigidity. It has also been observed that the diffusive capacity can be evaluated using a variational technique. The hereby obtained formulae, properties of fast flows and their geometric interpretation can be used as analytical tools to evaluate important technological parameters in reservoir engineering. Submitted by
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