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Title Mathematical model of well productivity index for Forchheimer flows in fractured reservoirs
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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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 ∈ Rd variable vectors related to ∇p; s, ξ scalar variables; • Π dimensionless (normalized) permeability…

…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…

…x5D; Sec. 2.3, and also [16] Sec. 3.4). Substituting Eq. (2.7) in the continuity equation dρ = −∇ · (ρu), dt yields dρ dp dρ = −ρ∇ · u − u · ∇p, dp dt dp 7 (2.9) Texas Tech University, Lidia Bloshanskaya…

…as a function |∇p| |u| = G−1 (|∇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 p∗ ), d−1 L or the same u∗ = − κLd−2 K(|∇∗ (κ/L p∗ ) |) ∗ ∗ ∇ p = −K ∗ (|∇∗ p∗ |)∇∗ p∗ . Q 9 (2.20) Texas Tech University, Lidia Bloshanskaya, August 2013 Similarly Eq. (2.19) can be…

…x28;|y|)yi ) = K(|y|) δij − ∂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 = G−1 (|y|). This proves…