Q913 re1 w3 lec 9

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Transcript of Q913 re1 w3 lec 9

  • 1. Reservoir Engineering 1 Course (1st Ed.)

2. 1. SS Regime: R Flow, IC & SC Fluids A. SS Regime: R Flow, C Fluids2. Multiple-Phase Flow 3. Pressure Disturbance in Reservoirs 4. USS Flow Regime A. USS: Mathematical Formulation 3. 1. Diffusivity Equation A. Solutions of Diffusivity Equation a. Ei-Function Solution b. pD Solution c. Analytical Solution 4. Gaining Diffusivity Equation To simplify the general partial differential equation, assume that the permeability and viscosity are constant over pressure, time, and distance ranges. This leads to:Expanding the above equation gives:2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids5 5. Gaining Diffusivity Equation (Cont.) Using the chain rule in the above relationship yields:Dividing the above expression by the fluid density gives2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids6 6. Gaining Diffusivity Equation (Cont.) Recalling that the compressibility of any fluid is related to its density by:Define total compressibility, ct, as ct=c+cf, (the time t is expressed in days.)2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids7 7. Diffusivity Equation Diffusivity equation is one of the most important equations in petroleum engineering. The equation is particularly used in analysis well testing data where the time t is commonly recorded in hours. The equation can be rewritten as: Where k = permeability, md r = radial position, ft p = pressure, psia ct = total compressibility, psi1 t = time, hrs = porosity, fraction = viscosity, cp 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids8 8. Total Compressibility in Diffusivity Equation When the reservoir contains more than one fluid, total compressibility should be computed as ct = coSo + cwSw + cgSg + cf Note that the introduction of ct into diffusivity equation does not make it applicable to multiphase flow; The use of ct, simply accounts for the compressibility of any immobile fluids that may be in the reservoir with the fluid that is flowing.2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids9 9. Diffusivity Constant The term [0.000264 k/ct] is called the diffusivity constant and is denoted by the symbol , or:The diffusivity equation can then be written in a more convenient form as:2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids10 10. Diffusivity Equation Assumptions and Limitations The diffusivity equation is essentially designed to determine the pressure as a function of time t and position r. Summary of the assumptions and limitations used in diffusivity equation: 1. Homogeneous and isotropic porous medium 2. Uniform thickness 3. Single phase flow 4. Laminar flow 5. Rock and fluid properties independent of pressure2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids11 11. Flash Back: Laplaces Equation for SS Regime Notice that for a steady-state flow condition, The pressure at any point in the reservoir is Constant and Does not change with time, i.e., p/t = 0,:2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids13 12. Diffusivity Equation Solutions Based on the boundary conditions imposed on diffusivity equation, there are two generalized solutions to the diffusivity equation: Constant-terminal-pressure solution Constant-terminal-rate solution2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids14 13. Constant-Terminal-Pressure Solution The constant-terminal-pressure solution is designed to provide the cumulative flow at any particular time for a reservoir in which the pressure at one boundary of the reservoir is held constant. The pressure is known to be constant at some particular radius and the solution is designed to provide the cumulative fluid movement across the specified radius (boundary). This technique is frequently used in water influx calculations in gas and oil reservoirs.2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids15 14. Application of Constant-Terminal-Rate Solution The constant-terminal-rate solution is an integral part of most transient test analysis techniques, such as with drawdown and pressure buildup analyses. Most of these tests involve producing the well at a constant flow rate and recording the flowing pressure as a function of time, i.e., p (rw, t).2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids16 15. Constant-Terminal-Rate Solution In the constant-rate solution to the radial diffusivity equation, the flow rate is considered to be constant at certain radius (usually wellbore radius) and The pressure profile around that radius is determined as a function of time and position. These are two commonly used forms of the constant-terminal-rate solution: The Ei-function solution The dimensionless pressure pD solution2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids17 16. The Ei-Function Solution Assumptions Matthews and Russell (1967) proposed a solution to the diffusivity equation that is based on the following assumptions: Infinite acting reservoir, i.e., the reservoir is infinite in size. (BC) The well is producing at a constant flow rate. (BC) The reservoir is at a uniform pressure, pi, when production begins. (IC) The well, with a wellbore radius of rw, is centered in a cylindrical reservoir of radius re. (The Ei solution is commonly referred to as the line-source solution.) No flow across the outer boundary, i.e., at re. 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids19 17. Ei-Function Solution Employing the above conditions, the authors presented their solution in the following form:Where p (r, t) = pressure at radius r from the well after t hours t = time, hrs k = permeability, md Qo = flow rate, STB/day 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids20 18. Exponential Integral The mathematical function, Ei, is called the exponential integral and is defined by:For x > 10.9, the Ei (x) can be considered zero for all practical reservoir engineering calculations.2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids21 19. Values of the Ei-Function2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids22 20. Exponential Integral Approximation The exponential integral Ei can be approximated (with less than 0.25% error) by the following equation when its argument x is less than 0.01: Ei (x) = ln (1.781x) Where the argument x in this case is given by:(t = time, hr, k = permeability, md)2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids23 21. Behavior of the Pwf When the parameter x in the Ei-function is less than 0.01, the log approximation can be used in the EiFunction Solution to give:For most of the transient flow calculations, engineers are primarily concerned with the behavior of the bottom-hole flowing pressure at the wellbore, i.e., r = rwWhere k = permeability, md, t = time, hr, ct = total compressibility, psi1 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids24 22. Pressure Profiles as a Function of Time Most of the pressure loss occurs close to the wellbore; accordingly, near-wellbore conditions will exert the greatest influence on flow behavior. 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids25 23. The Dimensionless Pressure Drop (pD) Solution The second form of solution to the diffusivity equation is called the dimensionless pressure drop. Well test analysis often makes use of the concept of the dimensionless variables in solving the unsteady-state flow equation. The importance of dimensionless variables is that they simplify the diffusivity equation and its solution by combining the reservoir parameters (such as permeability, porosity, etc.) and thereby reduce the total number of unknowns.2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids28 24. Dimensionless Pressure Drop Solution Introduction To introduce the concept of the dimensionless pressure drop solution, consider for example Darcys equation in a radial form as given previously by:Rearrange the above equation to give:2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids29 25. Dimensionless Pressure It is obvious that the right hand side of the above equation has no units (i.e., dimensionless) and, accordingly, the left-hand side must be dimensionless. Since the left-hand side is dimensionless, and (pe pwf) has the units of psi, it follows that the term [Qo Bo o/ (0.00708kh)] has units of pressure. 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids30 26. pD in Transient Flow Analysis In transient flow analysis, the dimensionless pressure pD is always a function of dimensionless time that is defined by the following expression:2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids31 27. Dimensionless Time In transient flow analysis, the dimensionless pressure pD is always a function of dimensionless time that is defined by the following expression:The above expression is only one form of the dimensionless time. Another definition in common usage is tDA, the dimensionless time based on total drainage area.Where A = total drainage area = re^2, re = drainage radius, ft, rw = wellbore radius, ft 2013 H. AlamiNiaReservoir Engineering 1 Course: USS regime for Radial Flow of SC Fluids32 28. Dimensionless Radial Distances The dimensionless pressure pD also varies with location in the reservoir as represented by th