Choke Models for Multiphase, Gas, and Liquid Flow

Overview

A choke (also called a bean) is a restriction at the wellhead that controls flow rate. Choke performance modeling is essential for predicting rate from choke size and pressure, sizing chokes for target rates, and determining whether flow is critical or subcritical.

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Critical vs Subcritical Flow

The flow regime depends on the downstream-to-upstream pressure ratio:

$$r = \frac{P_2}{P_1}$$

• Critical flow ($r < r_c$): Fluid velocity reaches sonic speed at the throat. Rate depends only on upstream pressure and choke size.
• Subcritical flow ($r > r_c$): Flow is subsonic. Rate depends on both upstream and downstream pressures.

For an ideal gas with specific heat ratio $k$:

$$r_c = \left(\frac{2}{k+1}\right)^{k/(k-1)}$$

For natural gas ($k \approx 1.28$), $r_c \approx 0.55$. Most production chokes operate in critical flow because wellhead pressures are typically much higher than flowline pressures, providing stable rate control.

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Gilbert-Type Correlations (Critical Flow)

General Form

All Gilbert-type models share the same functional form:

$$q_L = \frac{C \cdot P_{wh}^a \cdot D^b}{GLR^c}$$

Where $q_L$ = liquid rate (STB/d), $P_{wh}$ = wellhead pressure (psig), $D$ = choke diameter (1/64 in.), $GLR$ = gas-liquid ratio (scf/STB), and $C$, $a$, $b$, $c$ are empirical constants.

Rearranged for wellhead pressure:

$$P_{wh} = \left(\frac{q_L \cdot GLR^c}{C \cdot D^b}\right)^{1/a}$$

Rearranged for choke size:

$$D = \left(\frac{q_L \cdot GLR^c}{C \cdot P_{wh}^a}\right)^{1/b}$$

Correlation Coefficients

Choosing Among Gilbert-Type Models

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Subcritical Flow Models

When the pressure ratio exceeds the critical value, Gilbert-type correlations are not valid and mechanistic models are needed.

Sachdeva Correlation (1986)

A mechanistic model based on conservation of mass and energy through the choke throat. The mass flow rate is:

$$G = C_D \cdot A \cdot \sqrt{\frac{2 \rho_1 (P_1 - P_t)}{1 - (A_t/A_1)^2}}$$

Where $C_D$ = discharge coefficient (0.75--0.85), $A$ = throat area, $\rho_1$ = upstream mixture density, and $P_t$ = throat pressure. For subcritical flow, $P_t = P_2$; for critical flow, $P_t$ equals the sonic pressure.

Advantages: Physically based, handles both critical and subcritical flow, accounts for gas-liquid slip.

Ashford-Pierce Correlation (1975)

A semi-empirical model combining the orifice equation with corrections for two-phase effects:

$$q_L = C_D \cdot A \cdot \sqrt{\frac{2g_c \cdot \Delta P}{\rho_{mix}}} \cdot f(GLR, P_1, P_2)$$

Accounts for gas expansion and solution gas liberation across the choke. Designed for subcritical conditions ($P_2/P_1 > 0.55$).

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Single-Phase Gas Chokes

Sonic (Critical) Flow

For dry gas at sonic velocity:

$$q_{sc} = 879.4 \cdot C_D \cdot D^2 \cdot P_1 \cdot \sqrt{\frac{k}{\gamma_g \cdot T_1 \cdot Z_1} \left(\frac{2}{k+1}\right)^{(k+1)/(k-1)}}$$

Where $q_{sc}$ = gas rate (Mscf/d), $D$ = choke diameter (in.), $P_1$ = upstream pressure (psia), $k$ = specific heat ratio, $\gamma_g$ = gas specific gravity, $T_1$ = upstream temperature (R), $Z_1$ = compressibility factor.

Subsonic Flow

When $P_2/P_1 > r_c$, rate depends on both pressures:

$$q_{sc} = 879.4 \cdot C_D \cdot D^2 \cdot P_1 \cdot \sqrt{\frac{2k}{(k-1)\gamma_g T_1 Z_1}\left[\left(\frac{P_2}{P_1}\right)^{2/k} - \left(\frac{P_2}{P_1}\right)^{(k+1)/k}\right]}$$

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Single-Phase Liquid Chokes

For incompressible liquid flow (Bernoulli-based orifice equation):

$$q_L = 22041 \cdot C_D \cdot D^2 \cdot \sqrt{\frac{\Delta P}{\gamma_L}}$$

Where $q_L$ = liquid rate (STB/d), $D$ = choke diameter (in.), $\Delta P$ = pressure drop (psi), $\gamma_L$ = liquid specific gravity (water = 1.0).

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Model Comparison

 Pressure Ratio (P2/P1)
 0.0          0.55           1.0
  |            |              |
  |  CRITICAL  |  SUBCRITICAL |
  |  (Sonic)   |              |
  |            |              |
  | Gilbert    |  Sachdeva    |
  | Ros        |  Ashford-    |
  | Baxendell  |  Pierce      |
  | Achong     |              |
  | Pilehvari  |              |
  |            |              |
  +------------+--------------+

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Applicability and Limitations

Gilbert-type correlations are valid when flow is critical, GLR is in the range 300--50,000 scf/STB, and choke size is 8/64 to 64/64 in. They do not account for water cut, temperature, or fluid properties explicitly.

Single-phase gas models require known upstream pressure, temperature, and Z-factor. Liquid models assume incompressible flow at choke conditions.

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Related Topics

• Surface Facilities Overview --- Model selection guide
• Pipeline Flow --- Downstream pressure loss calculations
• Pipe Flow Overview --- Wellbore multiphase flow correlations
• PVT Gas Properties --- Z-factor and gas viscosity for choke calculations

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References

1. Gilbert, W.E. (1954). "Flowing and Gas-Lift Well Performance." API Drilling and Production Practice, pp. 126-157.

2. Ros, N.C.J. (1960). "An Analysis of Critical Simultaneous Gas/Liquid Flow Through a Restriction and Its Application to Flowmetering." Applied Scientific Research, 9(1), pp. 374-388.

3. Sachdeva, R., Schmidt, Z., Brill, J.P., and Blais, R.M. (1986). "Two-Phase Flow Through Chokes." SPE-15657-MS, 61st Annual Technical Conference and Exhibition, New Orleans, Louisiana.

4. Ashford, F.E. and Pierce, P.E. (1975). "Determining Multiphase Pressure Drops and Flow Capacities in Down-Hole Safety Valves." Journal of Petroleum Technology, 27(9), pp. 1145-1152. SPE-5161-PA.

5. Baxendell, P.B. (1957). "Bean Performance --- Lake Wells." Shell Internal Report. Referenced in Beggs, H.D. (1991), Production Optimization Using Nodal Analysis.