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Title Mathematical model of well productivity index for Forchheimer flows in fractured reservoirs
URL
Publication Date
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
Date Indexed 2018-12-06
Grantor Texas Tech University

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…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)…

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