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

…time dependent upscaling with the steady state one.
In the last Chapter 5 we present two engineering applications of established framework and results on computation of PSS PI. In 5.1 we generalize the famous Peaceman approach to g-*Forchheimer* equation…

…well bore, and evaluation of the PI of the well by
solving an auxiliary problem with reduced degrees of freedom.
In §5.2 we provide the analytical formula for the “skin factor”, used by engineers
to account for the nonlinearity of the *flow*. Our formula…

…depends on the flux Q, the
reservoir geometry, and the parameters of *Forchheimer* polynomial. Finally in 5.3
the values of PI for different geometries and orders of nonlinearities are computed.
The research of this dissertation is supported by NSF Grant No…

…and · W r,q (U ) , respectively. Here U ⊂ Rd .
In studies of flows in porous media, the three Forchheimer’s laws (two-term, power,
and three-term) are widely used. Darcy and *Forchheimer* laws can be written in the
vector forms as…

…the fluid.
where α =
• The *Forchheimer* two-term law
αu + β
(Bu, u)u = −Π∇p,
(2.3)
ρF Φ
where β = 1/2 ,F is the *Forchheimer* coefficient, Φ is the porosity, ρ is the
k
density of the fluid, and B = B(x) is a positive…

…definite tensor with bounded
entries depending, in general, on the spatial variable.
• The *Forchheimer* power law
au + cn
(Bu, u)n−1 u = −Π∇p,
6
(2.4)
Texas Tech University, Lidia Bloshanskaya, August 2013
where n is a real number…

…and to *Forchheimer* two-term law, respectively.
• The *Forchheimer* three-term law
Au + B
(Bu, u)u + C(Bu, u)u = −Π∇p.
(2.5)
Here A, B, and C are empirical constants.
We now introduce a general form of *Forchheimer*…

…equations.
Definition 2.0.1 (g-*Forchheimer* Equations).
g(x, |u|B ) u = −Π∇p,
(2.6)
here g(x, s) > 0 for all s ≥ 0 and |u|B = (Bu, u).
We will refer to (2.6) as g-*Forchheimer* (momentum)…