Worked Example: Diagnosing Gas Well Liquid Loading and Sizing Velocity String Retrofit

As natural gas reservoirs deplete, reservoir pressure declines and gas superficial velocities inside the production tubing decrease. When the upward drag force exerted by the moving gas column falls below the downward gravitational force acting on entrained liquid droplets (condensate or formation brine), liquids can no longer be transported to the surface.

Liquids begin accumulating at the bottom of the wellbore, creating hydrostatic backpressure that progressively chokes the inflow until the well "dies" (liquid loading shut-in).

This worked example diagnoses liquid loading in a mature gas well using the Turner, Coleman, and Li droplet models, and sizes a velocity string retrofit to restore continuous liquid removal.

1. Wellhead Operating & Geometry Data

A mature gas producer completed with standard $3\text{-}1/2\text{ inch}$ tubing is exhibiting surging and production decline: * Wellhead Flowing Pressure ($P_{wh}$): $60.0\text{ barg}$ ($61.01\text{ bar(a)} = 884.9\text{ psia}$) * Wellhead Flowing Temperature ($T_{wh}$): $45.0^\circ\text{C}$ ($113.0^\circ\text{F} = 318.15\text{ K}$) * Current Production Rate ($Q$): $45,000\text{ Sm}^3/\text{day}$ ($1.589\text{ MMSCFD}$) * Current Tubing Size: $3\text{-}1/2\text{ in OD}$ ($ID = 76.0\text{ mm} = 2.992\text{ in}$, Flow area $A_t = 0.04886\text{ ft}^2$) * Produced Liquid Type: Formation water ($\rho_L = 1,000\text{ kg/m}^3 = 62.4\text{ lb/ft}^3$, Interfacial tension $\sigma = 60.0\text{ dynes/cm}$) * Gas Properties: Specific Gravity $G = 0.65$ ($M_{mix} = 18.83\text{ g/mol}$), $Z = 0.88$

2. Step 1: In-Situ Fluid Properties & Actual Velocity

A. In-Situ Gas Density ($\rho_g$):

Using the real gas law:

$$\rho_g = \frac{P \cdot M_{mix}}{Z \cdot R \cdot T} = \frac{6,101,000\text{ Pa} \times 0.01883\text{ kg/mol}}{0.88 \times 8.31446 \times 318.15\text{ K}} = \mathbf{49.29\text{ kg/m}^3} \quad (3.077\text{ lb/ft}^3)$$

B. In-Situ Actual Volumetric Flow Rate:

At $45,000\text{ Sm}^3/\text{day}$ ($0.5208\text{ Sm}^3/\text{s}$):

$$Q_{actual} = Q_{std} \times \left( \frac{P_{std}}{P} \right) \left( \frac{T}{T_{std}} \right) \left( \frac{Z}{Z_{std}} \right) = 0.5208 \times \left( \frac{1.01325}{61.01} \right) \left( \frac{318.15}{288.15} \right) \left( \frac{0.88}{1.0} \right) = \mathbf{0.00854\text{ m}^3/\text{s}}$$

C. Actual Gas Velocity in $3\text{-}1/2\text{ inch}$ Tubing:

The tubing cross-sectional flow area ($D = 0.076\text{ m}$):

$$A_t = \frac{\pi \times (0.076)^2}{4} = 0.004536\text{ m}^2$$

$$v_{actual} = \frac{Q_{actual}}{A_t} = \frac{0.00854\text{ m}^3/\text{s}}{0.004536\text{ m}^2} = \mathbf{1.88\text{ m/s}} \quad (6.17\text{ ft/s})$$

3. Step 2: Critical Droplet Velocity Models

To lift a suspended liquid droplet, the upward gas drag force ($F_D = C_D rac{\pi d_p^2}{4} rac{ ho_g v_c^2}{2}$) must exceed droplet gravitational weight ($F_W = rac{\pi d_p^3}{6} ( ho_L - ho_g) g$). Equating drag to buoyant weight and substituting the maximum stable droplet diameter from the critical Weber number ($We_{crit} = rac{ ho_g v_c^2 d_p}{\sigma} = 30$) yields the governing critical velocity relation:

$$v_{crit} = \left[ rac{40 \cdot g \cdot \sigma ( ho_L - ho_g)}{C_D \cdot ho_g^2} ight]^{0.25}$$

In SI units ($\sigma = 0.060\, ext{N/m}$, $ ho_L = 1{,}000\, ext{kg/m}^3$, $ ho_g = 49.29\, ext{kg/m}^3$, $g = 9.80665\, ext{m/s}^2$):

$$v_{terminal} = \left[ rac{40 imes 9.80665 imes 0.060 imes (1{,}000 - 49.29)}{0.44 imes (49.29)^2} ight]^{0.25} = \left[ rac{22{,}378.8}{1{,}068.99} ight]^{0.25} = \mathbf{2.14\, ext{m/s}} \quad (7.02\, ext{ft/s})$$

A. Turner et al. (1969) Formulation:

Turner assumed a spherical droplet with $C_D = 0.44$ and applied an empirical $20\%$ upward adjustment to ensure reliable field clearance:

$$v_{crit,Turner} = 1.20 imes v_{terminal} = 1.20 imes 2.14\, ext{m/s} = \mathbf{2.57\, ext{m/s}} \quad (8.43\, ext{ft/s})$$

(In standard USCS oilfield units, Turner's relation is written as $v_{crit} = 5.62 \cdot rac{( ho_L - ho_g)^{0.25}}{ ho_g^{0.5}} = 5.62 imes rac{(62.4 - 3.077)^{0.25}}{\sqrt{3.077}} = \mathbf{8.89\, ext{ft/s}} = 2.71\, ext{m/s}$).

B. Coleman et al. (1991) Formulation:

Coleman demonstrated that Turner's 20% multiplier over-predicts required velocities at moderate flowing pressures, validating that the unadjusted physical terminal velocity is sufficient for droplet lifting:

$$v_{crit,Coleman} = \mathbf{2.14\, ext{m/s}} \quad (7.02\, ext{ft/s})$$

C. Li et al. (2002) Flat Droplet Formulation:

Li showed that droplets deform into oblate spheroids with higher drag ($C_D pprox 1.0$), reducing critical lifting velocity:

$$v_{crit,Li} = \left(rac{0.44}{1.0} ight)^{0.25} imes 2.14\, ext{m/s} = \mathbf{1.74\, ext{m/s}} \quad (5.71\, ext{ft/s})$$

4. Step 3: Critical Unloading Flow Rate vs Actual Rate

The critical gas volumetric flow rate at standard conditions ($Q_{crit}$) required to maintain superficial velocity above $v_{crit}$ is:

$$Q_{crit} = v_{crit} imes A_t imes \left( rac{P}{P_{std}} ight) \left( rac{T_{std}}{T} ight) \left( rac{1}{Z} ight)$$

Diagnosis:

Under the primary industry benchmarks (Turner and Coleman), actual gas velocity ($1.88\, ext{m/s}$) is $12\%$ to $27\%$ below the velocity required to carry liquid droplets out of the wellbore. Liquids are falling back and loading the well.

5. Step 4: Velocity String Sizing Retrofit

To elevate gas velocity above $v_{crit}$ without altering reservoir deliverability, we decrease tubing cross-sectional flow area by retrofitting a smaller-diameter string inside the well:

Velocity String Options Comparison (at Current Rate $1.59\, ext{MMSCFD}$):

Engineering Decision:

• A $1\text{-}1/2\text{ inch}$ coiled tubing velocity string installed hanging inside the $3\text{-}1/2\text{ inch}$ completion boosts velocity by nearly $600\%$, suppressing liquid accumulation and stabilizing continuous flow down to $0.8\text{ MMSCFD}$.