Acta mathematica scientia,Series B ›› 2014, Vol. 34 ›› Issue (1): 107-124.doi: 10.1016/S0252-9602(13)60130-7

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WELL-POSEDNESS OF A COMPRESSIBLE GAS-LIQUID MODEL FOR DEEPWATER OIL WELL OPERATIONS

Helmer A. FRIIS*|Steinar EVJE   

  1. International Research Institute of Stavanger (IRIS), Prof. Olav Hanssensvei 15, NO-4068 Stavanger, Norway; University of Stavanger, NO-4068 Stavanger, Norway
  • Received:2012-09-27 Online:2014-01-20 Published:2014-01-20
  • Contact: Helmer A. FRIIS,HelmerAndre.Friis@iris.no E-mail:HelmerAndre.Friis@iris.no;steinar.evje@uis.no
  • Supported by:

    The research of Helmer A. Friis has been supported by the Research Council of Norway under grant number 197739/V30 (“DMPL”), whereas the research of Steinar Evje has been supported by A/S Norske Shell.

Abstract:

The main purpose of this paper is two-fold: (i) to generalize an existence result for a compressible gas-liquid model with a friction term recently published by Friis and Evje [SIAM J. Appl. Math., 71 (2011), pp. 2014–2047]; (ii) to derive a uniqueness result for the same model. A main ingredient in the existence part is the observation that we can consider weaker assumptions on the initial liquid and gas mass, and still obtain an existence result. Compared to the above mentioned work, we rely on a more refined application of the estimates provided by the basic energy estimate. Concerning the uniqueness result, we borrow ideas from Fang and Zhang [Nonlinear Anal. TMA, 58 (2004), pp. 719–731] and derive a stability result under appropriate constraints on parameters that determine rate of decay toward zero at the boundary for gas and liquid masses, and growth rate of masses associated with the friction term and viscous coefficient.

Key words: two-phase flow, well model, gas-kick, weak solutions, Lagrangian coordi-nates, free boundary problem, friction term, uniqueness

CLC Number: 

  • 76T10
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