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Orr, F. M. - Theory of Gas Injection Processes Episode 2

Tham khảo tài liệu 'orr, f. m. - theory of gas injection processes episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2.2. ONE-DIMENSIONAL FLOW 11 Table 2.1 Inventory of Equations and Unknowns Unknowns Equations Variable Number Equation Eq. Number Compositions Xij npnc Material balances 2.1.18 nc Saturations Sj np Phase equilibrium 2.1.21 nc np _ 1 Pressures Pj np Capillary pressure 2.1.20 np _ 1 Velocities Vj np Darcy s law 2.1.19 np Saturation sum 2.1.22 1 Mole fraction sum 2.1.23 np Total np nc 3 Total np nc 3 If capillary pressure differences are neglected Eq. 2.2.1 can be simplified by eliminating the pressure gradient from the expressions for the flow velocities. Phase flow velocities can then be written easily in terms of fractional flow functions fj defined by n Vj fjV fj Yj Vk j 1 np. 2.2.2 In Eq. 2.2.2 V is the total flow velocity defined as the sum of the phase flow velocities Vj. In one-dimensional flow in the absence of capillary pressure Darcy s law Eq. 2.1.19 becomes Vj _ -kkrj MAP Pmjg sin e Pj y dx j 1 np. 2.2.3 In Eq. 2.2.3 0 is the dip angle measured as the angle between the flow direction and a horizontal line. An expression for the fractional flow function fj is obtained by eliminating the pressure gradient from Eqs. 2.2.2 and Eq. 2.2.3. The expression for the pressure gradient can be obtained from any one of the np expressions for the phase flow velocities Eq. 2.2.3 . For the jth phase for example dp __ pmjg sin 0. 2.2.4 dx kkrj Substitution of Eq. 2.2.4 into Eq. 2.2.3 written for the nth phase gives Vn lp--PVj g sine pmj _ pmn . 2.2.5 krj pn pn Substitution of Eq. 2.2.5 into Eq. 2.2.2 gives the expression for the fractional flow of phase j 62 krj Pj kgsin Ỡ - krn 2 fj vnp k I1 2 pmj pmn . 2.2.6 Mn krn Pn V n i Pn y Substitution of Eq. 2.2.2 into Eq. 2.2.1 gives the one-dimensional version of the convectiondispersion equation for multicomponent multiphase flow 12 CHAPTER 2. CONSERVATION EQUATIONS d d Cnp d ỉứỸxpjSj TN 12 xijpjfjv - tKij T 0 dt j 1 dx j 1 dx i 1 Uc. 2.2.7 2.3 Pure Convection If the effects of dispersion can be neglected then Eq. 2.2.7 reduces