REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS NTNU Author: professor Jon Kleppe Assistant producers: Farrokh Shoaei Khayyam Farzullayev

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  Control element  Mass balance: XX or uu Conservation of mass

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems where n is proposed by Muscat to be 2.  Forchheimer equation:  Semi-empirical Darcy's equation: Conservation of momentum  Brinkman equation:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  Constitutive equation for porous materials:  rock compressibility:  Constitutive equations for fluids :  fluid compressibility :

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Solution Gas-Oil Ratio, R so : ρ gS : density of gas at standard conditions ρ oS : density of oil at standard conditions Formation Volume Factor for each fluid, B : Black Oil Model  The main parameters : density of oil at reservoir conditions:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Typical pressure dependencies BwBw PP BgBg gg P P ww oo P BoBo PP R so PbPb PbPb

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  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: Flow Equation

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  For non-horizontal systems: 1.Initial conditions (IC) :  The initial state of the primary variables of the system: Where P ref is reference pressure & ρ is fluid densities Initial and Boundary Conditions

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems 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:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Before production:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems BC: 1) P bh = constant 2) Q = constant BC: 1) P bh = constant 2) Q inj = constant BC: k = 0 BC: q = 0 After production:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Multiphase Flow  Continuity equation for each fluid phase :  Darcy equation for each phase :  Oil density equation:  oL : the part of oil remaining liquid at the surface  oG : the part that is gas at the surface

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  The oil continuity equation:  The gas continuity equation:  the oil equation only includes the part of the oil remaining liquid at the surface which is: ρ OL  The continuity equation for gas has to be modified to include solution gas as well as free gas which is: ρ G & ρ OG

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  After substituting the Darcy's equations Black Oil fluid properties and Well rate terms The flow equations become: Where:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems –dip angle: α –hydrostatic gradient : g=rg Non-horizontal Flow  One-dimensional, inclined flow: x u  D  Darcy equation: which can be written:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Multidimensional Flow  One-phase  Three-dimensional flow  Cartesian coordinates  Corresponding Darcy equations :

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Coordinate Systems  Rectangular coordinates: r z y x z r    Cylindrical coordinates:  Spherical coordinates:

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems Questions 1.Write the mass balance equation (one-dimentional, one-phase). 8.Write continuity equation for one-phase, three-dimensional flow in cartesian coordinates. 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.

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems References  Kleppe J.: Reservoir Simulation, Lecture note 2 Kleppe J.

REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS Conservation of mass Conservation of momentum Black Oil Model Flow Equation Initial and Boundary Conditions Multiphase Flow Non-horizontal Flow Multidimensional Flow Coordinate Systems  Title: REVIEW OF BASIC STEPS IN DERIVATION OF FLOW EQUATIONS (PDF) (PDF)  Author:  Name: Prof. Jon Kleppe  Address: NTNU S.P. Andersensvei 15A 7491 Trondheim  Website Website   Size: 450 Kb About this module