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# Wellbore flow performance

The pressure drop experienced in lifting reservoir fluids to the surface is one of the main factors affecting well deliverability. As much as 80% of the total pressure loss in a flowing well may occur in lifting the reservoir fluid to the surface. Wellbore flow performance relates to estimating the pressure-rate relationship in the wellbore as the reservoir fluids move to the surface through the tubulars.

## Pressure loss through the wellbore

The flow path through the wellbore may include flow through perforations, a screen and liner, and packers before entering the tubing for flow to the surface. The tubing may contain completion equipment that acts as flow restrictions, such as

• Profile nipples
• Sliding sleeves
• Subsurface flow-control devices

In addition, the tubing string may be composed of multiple tubing diameters or allow for tubing/annulus flow to the surface. At the surface, the fluid must pass through wellhead valves, surface chokes, and through the flowline consisting of surface piping, valves, and fittings to the surface-processing equipment. The pressure drop experienced as the fluid moves from the reservoir sandface to the surface is a function of the mechanical configuration of the wellbore, the properties of the fluids, and the producing rate.

Relationships to estimate this pressure drop in the wellbore are based on the mechanical energy equation for flow between two points in a system as written in Eq. 1.

....................(1)

In this relationship, α is the kinetic energy correction factor for the velocity distribution, W is the work done by the flowing fluid, and El is the irreversible energy losses in the system including the viscous or friction losses. For most practical applications, there is no work done by or on the fluid and the kinetic energy correction factor is assumed to be one. Under these conditions, Eq. 1 can be rewritten in terms of the pressure change as

....................(2)

This relationship states that the total pressure drop is equal to the sum of the change in potential energy (elevation), the change in kinetic energy (acceleration), and the energy losses in the system. This relationship can be written in the differential form for any fluid at any pipe inclination as

....................(3)

Methods to estimate the pressure drop in tubulars for single-phase liquid, single-phase vapor, and multiphase flow are based on this fundamental relationship.

With Eq. 3, the pressure drop for a particular flow rate can be estimated and plotted as a function of rate. In the typical application, the wellhead pressure is fixed and the bottomhole flowing pressure, pwf, is calculated by determining the pressure drop. This approach will yield a wellbore flow performance curve when the pressure is plotted as a function of rate as shown in Fig. 1. In this example, the wellhead pressure is held constant, and the flowing bottomhole pressure is calculated as a function of rate. This curve is often called a tubing-performance curve, because it captures the required flowing bottomhole pressure needed for various rates.

The following paragraphs summarize the basic approaches for estimating the pressure loss in the tubulars. Complete details of making these calculations are outside the scope of this section.

## Single-phase liquid flow

Single-phase liquid flow is generally of minor interest to the petroleum engineer, except for the cases of water supply or injection wells. In these cases, Eq. 3 is applicable where the friction factor, f, is a function of the Reynolds number and pipe roughness. The friction factor is most commonly estimated from the Moody friction factor diagram. The friction factor is an empirically determined value that is subject to error because of its dependence on pipe roughness, which is affected by pipe erosion, corrosion, or deposition.

## Single-phase vapor flow

There are several methods to estimate the pressure drop for single-phase gas flow under static and flowing conditions. These methods include:

• The average temperature and compressibility method[1]
• The original and modified Cullendar and Smith methods[2][3]

They require a trial-and-error or iterative approach to calculate the pressure drop for a given rate because of the compressible nature of the gas. These techniques are calculation intensive but can be implemented easily in a computer program. Lee and Wattenbarger[4] provide a detailed discussion of several methods used for estimating pressure drops in gas wells.

A simplified method for calculating the pressure drop in gas wells assuming an average temperature and average compressibility over the flow length was presented by Katz et al.[5]

....................(4)

where

....................(5)

and

....................(6)

This relationship can be solved directly if the wellhead and bottomhole pressures are known; however, in most applications, one pressure will be assumed and the other calculated. Thus, this method will be an iterative method as the compressibility factor is determined at the average pressure. Eq. 4 can be used to calculate the pressure drop for either flowing or static conditions.

## Multiphase flow

Much has been written in the literature regarding the multiphase flow of fluids in pipe. This problem is much more complex than the single-phase flow problem because there is the simultaneous flow of both liquid (oil or condensate and water) and vapor (gas). The mechanical energy equation (Eq. 3) is the basis for methods to estimate the pressure drop under multiphase flow; however, the problem is in determining the appropriate velocity, friction factor, and density to be used for the multiphase mixture in the calculation. In addition, the problem is further complicated as the velocities, fluid properties, and the fraction of vapor to liquid change as the fluid flows to the surface due to pressure changes.

Many researchers have proposed methods to estimate pressure drops in multiphase flow. Each method is based on a combination of theoretical, experimental, and field observations, which has led some researchers to relate the pressure-drop calculations to flow patterns. Flow patterns or flow regimes relate to the distribution of each fluid phase inside the pipe. This implies that a pressure calculation is dependent on the predicted flow pattern. There are four flow patterns in the simplest classification of flow regimes:[6]

• Bubble flow
• Slug flow
• Transition flow
• Mist flow, with a continually increasing fraction of vapor to liquid from bubble to mist flow

Bubble flow is experienced when the liquid phase is continuous with the gas phase existing as small bubbles randomly distributed within the liquid. In slug flow, the gas phase exists as large bubbles separating liquid slugs in the flow stream. As the flow enters transition flow, the liquid slugs essentially disappear between the gas bubbles, and the gas phase becomes the continuous fluid phase. The liquid phase is entrained as small droplets in the gas phase in the mist-flow pattern.

Poettman and Carpenter[7] were some of the earliest researchers to address developing a multiphase-flow correlation for oil wells, while Gray[8] presented an early multiphase correlation for gas wells. A large number of studies have been conducted related to multiphase flow in pipes. Brill and Mukerjee[9] and Brown and Beggs[10] include a review of many of these correlations. Application of the multiphase-flow correlations requires an iterative, trial-and-error solution to account for changes in flow parameters as a function of pressure. This is calculation intensive and is best accomplished with computer programs. Pressure calculations are often presented as pressure-traverse curves, like the one shown in Fig. 2, for a particular tubing diameter, production rate, and fluid properties. Pressure-traverse curves are developed for a series of gas-liquid ratios and provide estimates of pressure as a function of depth. These curves can be used for quick hand calculations.

## Nomenclature

 d = pipe diameter, L, in. El = energy loss per unit mass, L2/t2, ft-lbf/lbm fM = Moody friction factor in Eq. 5, dimensionless g = gravitational acceleration, L/t2, ft/sec2 gc = conversion factor, dimensionless, 32.2 ft-lbm/lbf-sec2 L = length, L, ft p = pressure, m/Lt2, psia pwf = bottomhole pressure, m/Lt2, psia pwh = wellhead pressure, m/Lt2, psia qg = gas flow rate, L3/t, Mscf/D s = skin factor, dimensionless S = defined by Eq. 6, m/L2t T = temperature, T, °R v = velocity, L/t, ft/sec W = work per unit mass, L2/t2, ft-lbf/lbm z = gas compressibility factor, dimensionless Z = elevation, L, ft γg = gas specific gravity, dimensionless Δp = pressure loss, m/Lt2, psia α = kinetic energy correction factor, dimensionless ε = absolute pipe roughness, L, in. ρ = fluid density, m/L3, lbm/ft3 ϕ = porosity, fraction

## References

1. Smith, R.V. 1950. Determining Friction Factors for Measuring Productivity of Gas Wells. Trans., AIME 189: 73.
2. Cullender, M.H. and Smith, R.V. 1956. Practical Solution of Gas-Flow Equations for Wells and Pipelines with Large Temperature Gradients. Trans., AIME 207: 281.
3. Oden, R.D. and Jennings, J.W. 1988. Modification of the Cullender and Smith Equation for More Accurate Bottomhole Pressure Calculations in Gas Wells. Presented at the Permian Basin Oil and Gas Recovery Conference, Midland, Texas, 10-11 March 1988. SPE-17306-MS. http://dx.doi.org/10.2118/17306-MS
4. Lee, W.J. and Wattenbarger, R.A. 1996. Gas Reservoir Engineering, 5. Richardson, Texas: Textbook Series, SPE.
5. Katz, D.L. et al. 1959. Handbook of Natural Gas Engineering. Nw York City: McGraw-Hill Publishing Co.
6. Orkiszewski, J. 1967. Predicting Two-Phase Pressure Drops in Vertical Pipe. J Pet Technol 19 (6): 829–838. SPE-1546-PA. http://dx.doi.org/10.2118/1546-PA
7. Poettman, F.H. and Carpenter, P.G. 1952. The Multiphase Flow of Gas, Oil and Water Through Vertical Flow Strings with Application to the Design of Gas-Lift Installations. Drill. & Prod. Prac., 257-317. Dallas, Texas: API.
8. Gray, H.E. 1974. Vertical Flow Correlation in Gas Wells. User’s Manual for API 14B, Appendix B. Dallas, Texas: API.
9. Brill, J.P. and Mukherjee, H. 1999. Multiphase Flow in Wells, 17. Richardson, Texas: Monograph Series, SPE.
10. Brown, K.E. and Beggs, H.D. 1977. The Technology of Artificial Lift Methods, 1. Tulsa, Oklahoma: PennWell Publishing Co.