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NTNU. Author: professor Jon Kleppe. Assistant producers: Farrokh Shoaei Khayyam Farzullayev. Control element. Mass balance:. u r. Conservation of mass. D X. or. Conservation of momentum. Semi-empirical Darcy's equation:. Forchheimer equation:.

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  1. NTNU Author: professor Jon Kleppe Assistant producers: Farrokh Shoaei Khayyam Farzullayev

  2. Control element Mass balance: u r Conservation of mass DX or

  3. Conservation of momentum • Semi-empirical Darcy's equation: • Forchheimer equation: where n is proposed by Muscat to be 2. • Brinkman equation:

  4. Constitutive equation for porous materials: • rock compressibility: • Constitutive equations for fluids : • fluid compressibility :

  5. Black Oil Model • The main parameters : Formation Volume Factor for each fluid, B: Solution Gas-Oil Ratio, Rso: density of oil at reservoir conditions: ρoS : density of oil at standard conditions ρgS : density of gas at standard conditions

  6. Bw Bg Bo Rso P P P P mw mg mo P P P Typical pressure dependencies Pb Pb

  7. FlowEquation • For a : • Single phase flow • One-dimensional • Horizontal system • Assuming • Darcy's equation to be applicable. • The cross sectional area is constant. • The flow equation becomes:

  8. Initial and Boundary Conditions 1. Initial conditions (IC) : • The initial state of the primary variables of the system: • For non-horizontal systems: Where Pref isreference pressure & ρ is fluid densities

  9. 2. Basic types of BC’s: • Pressure conditions (Dirichlet conditions) • Rate conditions (Neumann conditions) • Dirichlet conditions: Applied to the simple linear system described above: • Neumann conditions: Specify the flow rates at the end faces of the system:

  10. Before production:

  11. BC: k = 0 BC: 1) Pbh = constant 2) Q = constant BC: q = 0 BC: 1) Pbh = constant 2) Qinj = constant After production:

  12. Multiphase Flow • Continuity equation for each fluid phase : • Darcy equation for each phase : • Oil density equation: roL: the part of oil remaining liquid at the surface roG : the part that is gas at the surface

  13. the oil equation only includes the part of the oil remaining liquid at the surface which is: ρOL • The continuity equation for gas has to be modified to include solution gas as well as free gas which is: ρG & ρOG • The oil continuity equation: • The gas continuity equation:

  14. After substituting the Darcy's equations Black Oil fluid properties and Well rate terms The flow equations become: Where:

  15. Non-horizontal Flow • One-dimensional, inclined flow: u D a x • Darcy equation: which can be written: • dip angle: α • hydrostatic gradient : g=rg

  16. Multidimensional Flow • One-phase • Three-dimensional flow • Cartesian coordinates • Corresponding Darcy equations :

  17. Rectangular coordinates: y r x z z j r q Coordinate Systems • Cylindrical coordinates: • Spherical coordinates:

  18. Questions 1. Write the mass balance equation (one-dimentional, one-phase). 2. Write the most common relationship between velocity and pressure, and write an alternative relationship used for high fluid velocities. 3. Write the expression for the relationship between porosity and pressure. 4. List 3 commonly used expressions for relating fluid density to pressure. 5. Describe briefly Black Oil model. 6. Sketch typical dependencies of the standard Black Oil parameters. 7. Write Darcy equation for one-dimentional, inclined flow. 8. Write continuity equation for one-phase, three-dimensional flow in cartesian coordinates.

  19. References • Kleppe J.: Reservoir Simulation, Lecture note 2

  20. About this module • Title:REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS(PDF) • Author: • Name: Prof. Jon Kleppe • Address: NTNU S.P. Andersensvei 15A 7491 Trondheim • Website • Email • Size: 450 Kb

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