Acta mathematica scientia, Series B >
WELL-POSEDNESS OF A COMPRESSIBLE GAS-LIQUID MODEL FOR DEEPWATER OIL WELL OPERATIONS
Received date: 2012-09-27
Online published: 2014-01-20
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.
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.
Helmer A. FRIIS , Steinar EVJE . WELL-POSEDNESS OF A COMPRESSIBLE GAS-LIQUID MODEL FOR DEEPWATER OIL WELL OPERATIONS[J]. Acta mathematica scientia, Series B, 2014 , 34(1) : 107 -124 . DOI: 10.1016/S0252-9602(13)60130-7
[1] Avelar C S, Ribeiro P R, Sepehrnoori K. Deepwater gas kick simulation. J Pet Sci Eng, 2009, 67: 13–22
[2] Brennen C E. Fundamentals of Multiphase Flow. New York: Cambridge University Press, 2005
[3] Breyholtz O. Managed Pressure Drilling: Improved Pressure Control through Model Predictive Control
[D]. University of Stavanger, 2011
[4] Delhaye J M, Giot M, Riethmuller M L. Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering. New York: Von Karman Institute, McGraw-Hill, 1981
[5] Evje S. Global weak solutions for a compressible gas-liquid model with well-formation interaction. J Diff Eqs, 2011, 251(8): 2352–2386
[6] Evje S. Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells. SIAM J Math Anal, 2011, 43(4): 1887–1922
[7] Evje S, Fjelde K K. Hybrid flux-splitting schemes for a two-phase flow model. J Comput Phys, 2002, 175(2): 674–701
[8] Evje S, Fjelde K K. On a rough AUSM scheme for a one dimensional two-phase model. Computers & Fluids, 2003, 32(10): 1497–1530
[9] Evje S, Karlsen K H. Global weak solutions for a viscous liquid-gas model with singular pressure law. Comm Pure Appl Anal, 2009, 8(6): 1867–1894
[10] Evje S, Fl°atten T, Friis H A. Global weak solutions for a viscous liquid-gas model with transition to single-phase gas flow and vacuum. Nonlinear Analysis TMA, 2009, 70: 3864–3886
[11] Faille I, Heintze E. A rough finite volume scheme for modeling two-phase flow in a pipeline. Computers & Fluids, 1999, 28: 213–241
/
| 〈 |
|
〉 |