1. Lecture 2 Single Phase Flow Concepts

2. Wellbore Performance
Wellbore performance analysis involves establishing a relationship between
tubular size
wellhead and bottom-hole pressure
fluid properties
fluid production rate

3. Single-Phase Liquid Flow
Single-phase liquid flow exists in an oil well only when the wellhead pressure is above the bubble-point pressure of the oil, which is usually not a reality.
This is just a start point.
Multiphase flow usually dominates.

4. Conservation of Momentum
The rate of momentum out, minus the rate of momentum in, plus the rate of momentum accumulation in a given pipe segment must equal the sum of all forces on the fluids.

5. Conservation of Momentum
Conservation of mass

6. Flow to the Surface - Liquids
Taken as a macroscopic balance the changes at two selected points can be examined

7. Conservation of Momentum

8. Flow to the Surface - Liquids
Flow of slightly compressible liquids is described by the steady state mechanical energy balance
Steady state means that no properties are changing with time.

9. Kinetic Energy Term (KE) (acc)
The change in kinetic energy is the energy change brought about by a change in velocity between points 1 & 2.
For turbulent flow this is expressed as
Velocity is related to volumetric throughput or flux by

10. Potential Energy Term (PE) (el)
The change in potential energy is the change brought about by elevation change between point 1 & 2.
The flow may be inclined at an angle, or be horizontal.

11. Friction
Frictional energy loss is a function of velocity, viscosity, density, pipe size, & condition.
Correlations developed by Moody are widely used to obtain a Fanning friction factor based on Reynolds Number
Fanning friction factor
Moody friction factor

12. Moody Friction Factor

13. Moody Friction Factor

14. Pipe Roughness

15. Fanning Friction Factor
Many correlations exist. A good one is presented by Chen
Fanning friction factor = (1/4) Moody friction factor

16. Friction in Fittings
Friction loss in fittings is negligible in flow though long lines. However it may be important in short distance applications. If so it is added to the friction term as

17. Friction in Fittings

18. Mechanical Energy Balance for Liquids
Note the condition of no flow. Calc an example.

19. Example 1
Calculate the pressure change in a water injection well. The following data are known.

20. Example 2
Suppose that 1,000 bbl/day of 40 API, 1.2 cp oil is being produced through 2 7⁄8-in., 8.6 lbm/ft tubing in a well that is 15 degrees from vertical. If the tubing wall relative roughness is 0.001, calculate the pressure drop over 1,000 ft of tubing.

21. Liquid Pressure Traverse
A pressure traverse is a plot of the flowing pressure versus position in the pipe.
For the case of a slightly compressible liquid the plot is a straight line from P1 to P2 across L1 to L2.
You can verify this result by breaking the flow length into increments and computing and plotting the end point of each increment.

22. Production Well Example

23. Single-Phase Gas Flow
The first law of thermodynamics (conservation of energy) governs gas flow in tubing. The effect of kinetic energy change is negligible because the variation in tubing diameter is insignificant in most gas wells. With no shaft work device installed along the tubing string, the first law of thermodynamics yields the following mechanical balance equation:

24. Single-Phase Gas Flow
Since dz is Sin(θ).dL
The gas density from the gas law can be expressed as
The velocity can be written in terms of volumetric flow rate at the standard condition as

25. Main Equation for Gas Flow in Wellbores

26. Solution
This equation contains Three variables that are functions of position
Z (compressibility factor)
Temperature
Pressure
A simple approach is to use average values of temperature and Z over a segment of the pipe.
There are two options for temperature in a segment between points 1 and 2

27. Method to Calculate P in a Segment
T_average can be obtained from T1 and T2
Z_average is calculated from T_average and P1
In the segment, we can calculate P2
Z_average can be updated with new P in the segment which is (P1+P2)/2
We repeat the calculate until convergence.
P1, T1
P2, T2

28. Solution
For the special case of horizontal well

29. Oil Field Unit P1 P2

30. Example 3