Full Record

Author | Bloshanskaya, Lidia |

Title | Mathematical model of well productivity index for Forchheimer flows in fractured reservoirs |

URL | http://hdl.handle.net/2346/58420 |

Publication Date | 2013 |

Degree | PhD |

Discipline/Department | Mathematics |

Degree Level | doctoral |

University/Publisher | Texas Tech University |

Abstract | Porous media (rocks, soils, aquifers, oil and gas reservoirs) plays an essential role in our modern environment. The pores of such material are usually filled with fluid, liquid or gas, and the flow of the fluids through the media is a subject of common interest of many different fields of study. In the middle of 19th century, Henry Darcy experimented on water filtration through sand and he eventually formulated the famous Darcy's law which relates the pressure gradient to the velocity of the fluid linearly. This empirical law laid the foundations for the quantitative theory of fluid dynamics. However, linear law has limited range of validity. In 20th century, Forchheimer proposed his equations to account for the nonlinearity of the flow. In this thesis we generalize the Forchheimer equations and examine the properties of the corresponding parabolic partial differential equations. The developed framework is used to study the well productivity index (PI) as a functional defined on the solutions of differential equations modeling non-linear flows. Petroleum engineers use the PI to characterize the well performance to manage the well reserves. We study the long term dynamics of the PI and its dependence on the nonlinearity and geometric parameters. The obtained results can be effectively used in reservoir engineering and can be applied to other problems modeled by the nonlinear diffusive equations. |

Subjects/Keywords | Porous media; Fractures; Nonlinear flow; Non-darcy; Forchheimer equation |

Contributors | Aulisa, Eugenio (committee member); Hoang, Luan (committee member); Ibragimov, Akif (Committee Chair) |

Language | en |

Rights | Unrestricted. |

Country of Publication | us |

Record ID | handle:2346/58420 |

Repository | ttu |

Date Retrieved | 2018-12-03 |

Date Indexed | 2018-12-06 |

Grantor | Texas Tech University |

Sample Search Hits | Sample Images

…*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
the impact of nonlinearity on its value. §4.2 presents one of the main results of this
work, namely we obtain the convergence of transient PI to corresponding PSS one.
This is established for…

…DMS-0908177,
and main results are published in journal papers and were presented on various national meetings, conferences, and seminars (see §6).
4
*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
Chapter 2
Mathematical modeling of…

…variable in Rd , d = 2 or 3; t time; p(x, t)
5
*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
pressure distribution; y âˆˆ Rd variable vectors related to âˆ‡p; s, Î¾ scalar variables;
â€¢ Î dimensionless (normalized) permeability…

…definite tensor with bounded
entries depending, in general, on the spatial variable.
â€¢ The Forchheimer power law
au + cn
(Bu, u)nâˆ’1 u = âˆ’Î âˆ‡p,
6
(2.4)
*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
where n is a real number…

…x5D; Sec. 2.3, and also [16]
Sec. 3.4). Substituting Eq. (2.7) in the continuity equation
dÏ
= âˆ’âˆ‡ Â· (Ïu),
dt
yields
dÏ dp
dÏ
= âˆ’Ïâˆ‡ Â· u âˆ’ u Â· âˆ‡p,
dp dt
dp
7
(2.9)
*Texas* *Tech* *University*, Lidia Bloshanskaya…

…as a function |âˆ‡p|
|u| = Gâˆ’1 (|âˆ‡p|)
(2.15)
Substituting equation (2.15) into (2.6) one obtains the following alternative form
of the g-Forchheimer momentum equation (2.6):
8
*Texas* *Tech* *University*…

…L pâˆ— ),
dâˆ’1
L
or the same
uâˆ— = âˆ’
ÎºLdâˆ’2 K(|âˆ‡âˆ— (Îº/L pâˆ— ) |) âˆ— âˆ—
âˆ‡ p = âˆ’K âˆ— (|âˆ‡âˆ— pâˆ— |)âˆ‡âˆ— pâˆ— .
Q
9
(2.20)
*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
Similarly Eq. (2.19) can be…

…x28;|y|)yi ) = K(|y|) Î´ij âˆ’
âˆ‚yj
|y| |y|g (s) + g 2 (s)
10
,
(2.27)
*Texas* *Tech* *University*, Lidia Bloshanskaya, August 2013
for i, j = 1, . . . , d, where s = Gâˆ’1 (|y|). This proves…