Gas Well Deliverability

Overview

Gas well performance differs fundamentally from oil wells due to:

• High flow velocities — Gas compressibility creates turbulence near wellbore
• Pressure-dependent properties — μg, ρg, Z vary significantly with pressure
• Non-Darcy flow — Inertial/turbulent effects at high rates
• Pseudo-pressure — Required for rigorous gas flow analysis

Gas well deliverability analysis determines:

• AOF (Absolute Open Flow) — Maximum theoretical rate
• Productivity — Relationship between rate and pressure drawdown
• Backpressure curve — Rate vs. (p²) or pseudo-pressure relationship
• Completion efficiency — Skin effects and damage quantification

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Darcy vs. Non-Darcy Flow

Darcy Flow (Laminar)

At low velocities, gas flow follows Darcy's law (linear relationship):

$$q_{sc} = \frac{k h}{\mu_g B_g} \frac{2\pi}{\ln(r_e/r_w) - 0.75 + s} (p_R - p_{wf})$$

Or in pseudo-steady state using real gas pseudo-pressure:

$$q_{sc} = \frac{k h}{1424 T} \frac{m(p_R) - m(p_{wf})}{\ln(r_e/r_w) - 0.75 + s}$$

Where real gas pseudo-pressure is:

$$m(p) = 2\int_0^p \frac{p}{\mu_g Z} dp$$

Applicability: Low rate wells, high permeability, large wellbore radius.

Non-Darcy Flow (Turbulent)

At high velocities, inertial (turbulent) effects become significant. The Forchheimer equation adds a rate-squared term:

$$-\frac{dp}{dr} = \frac{\mu_g q}{kh} + \rho_g \beta q^2$$

Where:

• $\beta$ = turbulence factor (or inertial resistance coefficient), ft⁻¹
• Second term represents non-Darcy (turbulence) pressure drop

This leads to the backpressure equation:

$$p_R^2 - p_{wf}^2 = a q_{sc} + b q_{sc}^2$$

Or:

$$m(p_R) - m(p_{wf}) = a' q_{sc} + b' q_{sc}^2$$

Physical interpretation:

• $a$ term: Darcy (laminar) flow
• $b$ term: Non-Darcy (turbulent) flow
• At high rates, turbulence dominates (b term >> a term)

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Non-Darcy Coefficient (D)

The non-Darcy flow coefficient quantifies turbulence:

$$D = \frac{b}{a} = \frac{F k h \beta}{1424 T}$$

Where:

• $F$ = turbulence factor (dimensionless, typically 1-100)
• Higher D → More turbulence, steeper backpressure curve

Typical D values:

• Low perm tight gas (0.1 md): D = 0.001 to 0.01
• Moderate perm (1 md): D = 0.0001 to 0.001
• High perm (100 md): D = 0.00001 to 0.0001

Estimating D (Jones-Blount-Glaze Correlation)

$$D = 3.161 \times 10^{-12} \frac{\beta \gamma_g T}{k h r_w^2 \mu_g}$$

Where $\beta$ (turbulence factor) can be estimated from:

$$\beta = \frac{1.88 \times 10^{10}}{k^{1.2}}$$

Data needed: k, h, rw, T, γg, μg (all at average reservoir conditions).

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Backpressure Testing

Four-Point Test Procedure

1. Shut in well → measure pR (stabilized)
2. Flow at rate q₁ → measure pwf,1 (stabilized)
3. Flow at rate q₂ → measure pwf,2 (stabilized)
4. Flow at rate q₃ → measure pwf,3 (stabilized)
5. Flow at rate q₄ → measure pwf,4 (stabilized)

Calculating a, b (or n)

Plot $(p_R^2 - p_{wf}^2)$ vs. $q_{sc}$ on log-log paper.

Slope = n (deliverability exponent):

• n = 1.0 → Pure Darcy flow (no turbulence)
• n = 0.5 → Fully turbulent flow
• n = 0.6 to 0.8 → Typical gas wells

Backpressure equation:

$$q_{sc} = C (p_R^2 - p_{wf}^2)^n$$

Or modern form:

$$p_R^2 - p_{wf}^2 = a q_{sc} + b q_{sc}^2$$

Absolute Open Flow (AOF):

$$q_{AOF} = C (p_R^2)^n = C p_R^{2n}$$

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Pseudo-Steady State Gas Flow

For bounded reservoirs (closed drainage volume), use pseudo-steady state equations.

PSS Darcy Flow

$$q_{sc} = \frac{k_g h [m(p_R) - m(p_{wf})]}{1424 T [\ln(r_e/r_w) - 0.75 + s]}$$

Where:

• $r_e$ = external drainage radius, ft
• $r_w$ = wellbore radius, ft
• $s$ = skin factor (dimensionless)

PSS Non-Darcy Flow

$$m(p_R) - m(p_{wf}) = \frac{1424 T q_{sc}}{k_g h} [\ln(r_e/r_w) - 0.75 + s] + D q_{sc}$$

Combining:

$$m(p_R) - m(p_{wf}) = a q_{sc} + D q_{sc}^2$$

Where:

$$a = \frac{1424 T}{k_g h} [\ln(r_e/r_w) - 0.75 + s]$$

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Time to Pseudo-Steady State

Before PSS is reached, gas wells exhibit transient flow. Time to reach PSS:

For Gas (Low Compressibility Liquid)

$$t_{PSS} = \frac{380 \phi \mu_g c_t r_e^2}{k}$$

Where:

• $t_{PSS}$ = time to PSS, hours
• $\phi$ = porosity, fraction
• $\mu_g$ = gas viscosity, cP
• $c_t$ = total compressibility, psi⁻¹
• $r_e$ = drainage radius, ft
• $k$ = permeability, md

Typical values:

• High perm (100 md), small drainage (500 ft): tPSS ≈ 5 hours
• Low perm (0.1 md), large drainage (2000 ft): tPSS ≈ 2000 hours

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Skin Factor and Wellbore Effects

Total Skin Factor

$$s_{total} = s_{damage} + s_{perforation} + s_{partial\,penetration} + s_{deviation} + s_{turbulence}$$

Components:

• $s_{damage}$: Formation damage near wellbore
• $s_{perforation}$: Perforation geometry and density
• $s_{partial penetration}$: Limited perforated interval
• $s_{deviation}$: Deviated well effects
• $s_{turbulence}$: High-velocity non-Darcy effects

Effective Wellbore Radius

Positive skin reduces effective wellbore radius:

$$r_w' = r_w e^{-s}$$

Where:

• $r_w'$ = effective wellbore radius, ft
• $r_w$ = actual wellbore radius, ft
• $s$ = skin factor

Example:

• rw = 0.5 ft, s = +5 → rw' = 0.0034 ft (147× reduction!)
• rw = 0.5 ft, s = -3 → rw' = 10.0 ft (20× increase from stimulation)

Equivalent Skin for Fractures

Hydraulically fractured wells can be represented by equivalent negative skin:

$$s_{frac} = \ln\left(\frac{r_w}{x_f/2}\right)$$

Where:

• $x_f$ = fracture half-length, ft

Example: xf = 200 ft, rw = 0.5 ft → sfrac = -6 (excellent stimulation).

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Drainage Radius Calculations

For circular drainage area:

$$r_e = \sqrt{\frac{A}{\pi}}$$

Where $A$ = drainage area, ft²

Common well spacings:

• 160 acres: re ≈ 1490 ft
• 80 acres: re ≈ 1053 ft
• 40 acres: re ≈ 745 ft

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Related Documentation

• WellFlow Overview — Well performance concepts
• Vogel IPR — Oil well IPR (different from gas)
• Vertical Flow Correlations — Tubing flow (Hagedorn-Brown, Gray)
• Gas Properties — Z-factor, viscosity, pseudo-pressure
• Horizontal Wells — Horizontal gas well productivity

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References

1. Lee, J. and Wattenbarger, R.A. (1996). Gas Reservoir Engineering. SPE Textbook Series Vol. 5. Richardson, TX: Society of Petroleum Engineers.

2. Economides, M.J., Hill, A.D., Ehlig-Economides, C., and Zhu, D. (2013). Petroleum Production Systems, 2nd Edition. Upper Saddle River, NJ: Prentice Hall. Chapter 4: Gas Well Deliverability.

3. Ahmed, T. (2019). Reservoir Engineering Handbook, 5th Edition. Cambridge, MA: Gulf Professional Publishing. Chapter 13: Gas Well Testing.

4. Guo, B., Lyons, W.C., and Ghalambor, A. (2007). Petroleum Production Engineering: A Computer-Assisted Approach. Burlington, MA: Gulf Professional Publishing.

5. Jones, L.G., Blount, E.M., and Glaze, O.H. (1976). "Use of Short Term Multiple Rate Flow Tests to Predict Performance of Wells Having Turbulence." SPE-6133-MS, presented at SPE Annual Fall Technical Conference, New Orleans, LA, October 3-6.